TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

PHẦN V: BIẾN ĐỔI LƯỢNG GIÁC A – HỆ THỨC LƯỢNG GIÁC CƠ BẢN  tan  

sin  . cos 

 cot  

 tan .cot   1 .

 sin 2   cos2   1 .  1  tan2  

cos  . sin 

1 . cos2 

 1  cot2  

1 . sin2 

BÀI TẬP ÁP DỤNG Bài 1.

Chứng minh các đẳng thức sau 1/

cos2 x  sin2 x  1  2 sin2 x .

2/

2 cos2 x  1  1  2 sin2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

3  4 sin2 x  4 cos2 x  1 .

4/

sin x cot x  cos x tan x  sin x  cos x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

sin 4 x  cos 4 x  1  2 sin 2 x cos2 x .

6/

cos4 x  sin4 x  cos2 x  sin2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

4 cos2 x  3  1  2 sin x 1  2 sin x  . 8/

1  cos xsin

2



x  cos x  cos2 x  sin2 x .

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BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

................................................................................................................................................... ................................................................................................................................................... 9/

sin4 x  cos4 x  1  2cos2 x  2sin2 x 1 . 10/ sin 3 x cos x  sin x cos 3 x  sin x cos x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ tan2 x  sin2 x  tan2 x sin2 x .

12/ cot2 x  cos2 x  cot2 x cos2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 2.

Chứng minh các đẳng thức sau 1/

tan x  cot x 

1 . sin x cos x

2/

1  cos x sin x  . sin x 1  cos x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

1 1   1. 1  tan x 1  cot x

4/

   1  1  1  1   tan2 x  0 .    cos x  cos x 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

1  sin2 x  1  2 tan2 x . 2 1  sin x

6/

tan x tan y 

tan x  tan y . cot x  cot y

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

1  cot4 x 

2 1  . 2 sin x sin 4 x

8/

tan x 

cos x 1  . 1  sin x cos x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

1

1  cos x1  cot x  1  cos x . 2

10/

1  cos x 1  cos x 4 cot x   . 1  cos x 1  cos x sin x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/

sin x 1  cos x 2   . 1  cos x sin x sin x

12/

sin x  cos x  1 cos x  . sin x  cos x  1 1  sin x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

sin2 x  2 cos x  1 cos x  13/ . 2 1  cos x 2  cos x  cos x

14/

sin x  cos x  1 2 cos x  . 1  cos x sin x  cos x  1

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 3.

Chứng minh các đẳng thức sau

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1/

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

sin 6 x  cos6 x  1  3 sin2 x cos2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/







sin 6 x  cos6 x  sin2 x  cos2 x 1  sin2 x cos2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/





2

sin 8 x  cos 8 x  1  2 sin2 x cos 2 x  2 sin 4 x cos 4 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/







sin 8 x  cos8 x  sin2 x  cos2 x 1  2 sin2 x cos2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 4.

Chứng minh các đẳng thức sau 1/

1  sin x  cos x  tan x  1  cos x1  tan x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

1  tan x1  cot x sin x cos x  1  2 sin x cos x .

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

1  tan x cos

2

x  1  cot x  sin 2 x  sin x  cos x  . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

sin 2 x tan x  cos2 x cot x  2 sin x cos x  tan x  cot x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

sin 2 x tan 2 x  4 sin2 x  tan 2 x  3 cos2 x  3 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 5.

Chứng minh các đẳng thức sau 1/

sin2   cos2  tan   1  . 1  2 sin  cos  tan   1

2/

tan2   sin2   tan 6  . 2 2 cot   cos 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

cos2  sin2    1  sin  cos  . 1  tan  1  cot 

4/

tan   sin  1  . 3 sin  cos  1  cos 

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

1  tan2   cot2   2 . 2 2 sin  cos 

6/

1 3 tan2  2  tan    1. cos2  cos2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

tan2   tan2  sin2   sin 2  .  tan2  tan2  sin2  sin2 

8/

2    1  cos   1  cos      2 cot  .  1  sin   sin2   

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

1

1  cos  1  cot   1  cos  . 2

10/ 1  tan   tan2   tan3  

sin   cos  . cos3 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/

sin  cos  1  cot2    . sin   cos  cos   sin 1  cot2 

12/

tan2  1  cot2  1  tan4  .  . 1  tan2  cot2  tan2   cot2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6 | THBTN – CA

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 1  sin  1  sin    2 13/     4 tan  .  1  sin  1  sin  

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC 2

 1  cos  1  cos    2 14/     4 cot  .  1  cos  1  cos  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 6.

Rút gọn các biểu thức sau 1/ P1  sin 4   sin2  cos2  .

2/ P2  sin 4   cos4   cos2  .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/ P3  sin2   sin2  cot2  .

4/ P4  cos2   cos2  cot2  .

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................





5/ P5  1  sin2  cot2   1  cot2  .

6/ P6  sin2  tan  cos2  cot  2sin cos  .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

P7 

2 cos2   1 . sin   cos 

8/

P8 

1  cos  2

sin 



1 . 1  cos 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

P9 

cot   cot  . tan   tan 

10/ P10  tan  

cos  . 1  sin 

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ P11 

sin x  tan x  sin x cot x . tan x

12/ P12 

cos x tan x sin2 x

 cot x cos x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ P13 

cos2 x  cot2 x 2

2

sin x  tan x

14/ P14 

.

1  sin2 x cos2 x 2

cos x

 cos2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

sin3 x  sin x cos2 x  cos x 15/ P15  . 1  2 sin x cos x

2

16/ P16

sin x  cos x 

1

tan x  sin x cos x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/ P17 

sin2 x  sin2 x tan2 x 2

2

2

cos x  cos x tan x

.

18/ P18 

cot2 x  cos2 x 2

cot x

 1  cos2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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19/ P19 

cos2 x  sin2 y 2

2

sin x sin y

 cot2 x cot2 y .

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

20/ P20 

1 sin x cos x cot x   . sin x 1  cot x 1  tan x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 21/ P21 

 1  cos x 1  sin x 1  1  cotx  1  .   tan x  cotx . 22/ P22  1  cotx    sin x cos x sin x  sin x 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 23/ P23  1  tanx cos2 x 1  cotx sin2 x .

24/ P24 

  , 0  x  . 2 sinx  cot2 x  cos2 x  1

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 7.

Biến đổi các biểu thức sau thành tích số 1/

A  2 cos2 x  1 .

................................................................................................................................................... 2/

B  3  4 sin2 x .

................................................................................................................................................... 3/

C  sin x cos x  cos2 x  1 .

................................................................................................................................................... 4/

D  sin2 x  sin x cos x  1 .

...................................................................................................................................................

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BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

E  1  sin x  cos x  tan x .

................................................................................................................................................... 6/

F  tan x  cot x  sin x  cos x .

................................................................................................................................................... 7/

G  cos x tan2 x  1  cos x  .

................................................................................................................................................... 8/





H  3  4 cos2 x  sin x 2 sin x  1 .

................................................................................................................................................... ................................................................................................................................................... 9/

I  sin2 x  3 cos2 x  6 cos x  2 sin x .

................................................................................................................................................... ................................................................................................................................................... 10/ J  cos3 x  sin3 x  sin x  cos x . ................................................................................................................................................... ................................................................................................................................................... 11/ K  cos3 x  cos2 x  2 sin x  2 . ................................................................................................................................................... ................................................................................................................................................... 12/ L  cos2 x  sin3 x  cos x . ................................................................................................................................................... 13/ M  1  cos x  cos2 x  sin x 1  cos x  . ................................................................................................................................................... 14/ N  2 cos3 x  2 cos2 x  sin x  1 . ................................................................................................................................................... 15/ O  cos3 x  sin3 x  2 sin2 x  1 . ................................................................................................................................................... 16/ P  2 sin x  12 cos x  2 sin x  1  3  4 cos2 x . ...................................................................................................................................................

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17/ Q  2 cos x  1sin x  cos x   1 . ................................................................................................................................................... 18/ R  4 sin3 x  3 cos3 x  3 sin x  sin2 x cos x . ................................................................................................................................................... ................................................................................................................................................... 19/ S  1  sin x  tan2 x  1  cos x . ................................................................................................................................................... 20/ T  2  5 sin x  3 1  sin x  tan2 x . ................................................................................................................................................... 21/ U  2 sin x cos x  2 sin2 x  3 sin x  cos x  1 . ...................................................................................................................................................





22/ V  tan x  3 cot x  4 sin x  3 cos x . ................................................................................................................................................... ................................................................................................................................................... 23/ X  3 sin x  2 cos x  3 tan x  2 . ................................................................................................................................................... 24/ Y  2  tan x  sin x   3 cot x  cos x   5 . ................................................................................................................................................... 25/ Z  3 cot x  cos x  5 tan x  sin x  2 . ................................................................................................................................................... ................................................................................................................................................... Bài 8.

Chứng minh các biểu thức sau không phụ thuộc vào biến x 1/

A  cos4 x  sin4 x  2 sin2 x .

................................................................................................................................................... ................................................................................................................................................... 2/

B  sin4 x  sin2 x cos2 x  cos2 x .

................................................................................................................................................... ...................................................................................................................................................

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BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

C  cos4 x  sin2 x cos2 x  sin2 x .

................................................................................................................................................... ................................................................................................................................................... 4/









D  cos4 x 2 cos2 x  3  sin 4 x 2 sin2 x  3 .

................................................................................................................................................... ................................................................................................................................................... 5/

E  sin6 x  cos6 x  2 sin4 x  cos4 x  sin2 x .

................................................................................................................................................... ................................................................................................................................................... 6/

F  sin x.

1 1  , 1  cos x 1  cos x

 0  x  

  . 4 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G  sin 4 x  4 cos2 x  cos 4 x  4 sin2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

H  cos2 x cot2 x  5 cos2 x  cot2 x  4 sin2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

I  1  cot x  sin 3 x  1  tan x cos3 x  sin x  cos x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................







10/ J  sin4 x  cos 4 x  1 tan2 x  cot2 x  2 .

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................



 



11/ K  3 sin8 x  cos 8 x  4 cos6 x  2 sin 6 x  6 sin 4 x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................









12/ L  sin4 x 1  sin2 x  cos 4 x 1  cos2 x  5 sin 2 x cos2 x  1 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................



  2



13/ M  2 sin4 x  cos4 x  sin2 x cos2 x  sin 8 x  cos8 x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 9.

Chứng minh các biểu thức sau không phụ thuộc vào biến x 1/

A

2 cot x  1  . tan x  1 cot x  1

................................................................................................................................................... ...................................................................................................................................................

2/

B



1  tan2 x 2

4 tan x



2



1 2

4 sin x cos2 x

.

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3/

C

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

1  tan2 x  1  tan2 x 1  cot2 x . tan x







................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

1  cot x D

2

2

4/

cot2 x



1 sin2 x cos2 x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E

1  sin6 x cos6 x



3 tan2 x cos2 x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

F

tan2 x  cos2 x sin2 x



cot2 x  sin2 x cos2 x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G

cot2 x  cos2 x 2

cot x



sin x cos x . cot x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

H

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sin4 x  cos4 x  1 sin6 x  cos6 x

.

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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B – ĐƯỜNG TRÒN LƯỢNG GIÁC – CUNG GÓC LƯỢNG GIÁC CUNG LIÊN KẾT 1. Đường tròn lượng giác và dấu góc lượng giác sinx

Cung

1

phần tư

π

I

II III IV

+

+

–

–

+

–

–

+

+

–

+

–

+

–

+

–

2

Giá trị LG (II)

(I)

π

0 1 2π

O

-1

cosx

(IV)

(III) 3π 2

(Nhất cả – Nhì sin – Tam tan – Tứ cos)

-1

2.Cung liên kết

Cung đối nhau

Cung bù nhau

cos(a)  cos a

sin(   a)  sin a

  sin   a   cos a 2 

sin(a)   sin a

cos(   a)   cos a

  cos   a   sin a 2 

tan(a)   tan a

tan(   a)   tan a

  tan   a   cot a 2 

cot(a)   cot a

cot(   a)   cot a

  cot   a  tan a  2 

Cung hơn kém

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

Cung phụ nhau

Cung hơn kém

sin(   a)   sin a

  sin   a   cos a 2 

cos(   a)   cos a

  cos   a    sin a 2  

tan(   a)  tan a

  tan   a    cot a 2  

cot(   a)  cot a

  cot   a    tan a 2 

 2

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BÀI TẬP ÁP DỤNG MỐI LIÊN HỆ GIỮA ĐƠN VỊ ĐỘ VÀ RADIAN Bài 10.

Đổi số đo các cung sau ra độ, phút, giây: 2 3 5  7 13 3 15  , , , , , ,  , 2, , . 3 4 9 5 3 3 4 7 15

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 11.

Đổi số đo các góc sau ra radian rad : 900, 360, 150 , 720 , 2700, 2400, 5400, 7500 , 210 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 12.

Đổi số đo độ của các cung tròn sau thành số đo radian (chính xác đến phần nghìn): 21030 ', 75054 ', 12036 ' , 15030 ',  105045 ' 30 '', 27 038 ' 49 '' .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 13.

Điền các giá trị thích hợp vào ô trống

00

150

300

450

600

750

900

1200

1500

1800

 16

 8

 6

5 12

 2

2 3

3 4

5 6



4 3

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Radian

7 4

 12

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC



5 6

Độ

 2

2 3

3 4

1350

500

8100

TÍNH ĐỘ DÀI CUNG TRÒN Bài 14.

Một đường tròn có bán kính 10 cm  . Tính độ dài các cung có số đo là 2  3 7 13 3 3 , , , , ,  2, , 450, rad, 75 0 . 3 18 5 3 6 4 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 15.

Kim giờ của đồng hồ dài 7 cm  , kim phút dài 10 cm  . Tính quãng đường kim phút, kim giờ đi được trong 30 phút.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 16.

Bánh xe có đường kính (tính cả lốp) là 55 cm . Nếu xe chạy với vận tốc 40 km /h thì trong một giây bánh xe quay được bao nhiều vòng ?

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 17.

Kim phút và kim giờ của đồng hồ lớn nhà Bưu Điện Hà Nội theo thứ tự dài 1, 75 m  và 1, 26 m  . Hỏi trong 15 phút, mũi kim phút và mũi kim giờ vạch trên

cung tròn có độ dài bằng bao nhiêu mét ? ................................................................................................................................................... 18 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... BIỂU DIỄN NGỌN CUNG LƯỢNG GIÁC Bài 18.

Biểu diễn các góc sau lên đường tròn lượng giác gốc A: 300 ,  450, 1200,  1200, 3300, 6300, 7500 ,  12500,

7 4 15 2010 ,  ,  750 0, 11250 , , . 4 3 2 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 19.

   Xác định điểm cuối của cung có số đo: k2, k, k , k , k , k   . 2 3 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 20.

Xác định điểm ngọn của các họ nghiệm sau đây trên đường tròn lượng giác với k  .

1/ x  k .

2/ x 

  k2 . 2

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3/ x  4k .

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

4/ x 

  k . 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/ x  

  k2 . 6

6/ x  

  k . 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/ x 

k . 3

8/ x 

 k2  . 3 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/ x 

 k3  . 4 4

10/ x 

 k  . 6 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 21.

   và sđ AN   k với Cho hai điểm M và N sao cho sđ AM k   . Tìm k   để: 6 798

1/ M trùng với N.

2/ M đối xứng với N qua tâm O.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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Bài 22.

Hai góc lượng giác có số đo radian

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

35 m và với m là số nguyên có thể có cùng 3 5

tia đầu và tia cuối hay không ? ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

GÍA TRỊ LƯỢNG GIÁC CỦA MỘT CUNG (GÓC) Bài 23.

Xác định dấu của các giá trị lượng giác hoặc biểu thức 1/

sin x, cos x, tan x, cot x với   x 

3 . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

  3        sin  x  , cos   x , tan  x   , cot x   với 0  x  .  4  2  2  2 2   

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/





A  sin 400.cos 2900 .

4/





B  sin 250 .cos 1700 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/





C  sin 2250. tan1300.cot 1750 .

6/





D  cos1950. tan 2690.cot 980 .

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7/





E  sin 500.cos 3000 .

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

8/

F  sin 2150. tan

21 . 7

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

G  cot

 2  3 . sin   .  3 5

10/ H  cos

4  4 9 . sin . tan .cot . 5 3 3 5

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 24.

Tính giá trị còn lại của góc x, biết 1/

sin x 

1 với 900  x  1800 . 2

2/

sin x  

4 với 2700  x  3600 . 5

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

sin x  

3 3 với   x  . 5 2

4/

cos x 

1  với 0  x  . 4 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

cos x 

3 với 0  x  900 . 5

6/

cos x  

5 với 1800  x  2700 . 13

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

cos x 

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2 5

với 

  x  0. 2

8/

cos x 

4 với 2700  x  3600 . 5

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

sin x 

5  với  x  . 13 2

10/ sin x  

1 với 1800  x  2700 . 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ tan x  3 với   x 

3 . 2

12/ tan x  2 với

  x  . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ tan x  

1  với  x   . 2 2

14/ cot x  3 với   x 

3 . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 15/ tan x 

3 3 với   x  . 4 2

16/ tan x   2 với

  x  . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/ cot x 

2  với 0  x  . 3 2

18/ cot x   3 với

  x  . 2

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19/ cot x  3 với

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

3  x  2 . 2

20/ cot150  2  3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 25.

Tính giá trị của các biểu thức lượng giác sau 1/ Cho tan x  2 . Tính: A1 

5 cot x  4 tan x 2 sin x  cos x , A2  . 5 cot x  4 tan x cos x  3 sin x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/ Cho cot x  2 . Tính: B1 

3 sin x  cos x sin x  3 cos x , B2  . sin x  cos x sin x  3 cos x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/ Cho cot x  2 . Tính: C1 

2 sin x  3 cos x 2 , C2  . 2 3 sin x  2 cos x cos x  sin x cos x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 24 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/ Cho tan x  2 . Tính: D1 

D3 

3 sin x  2 cos x 2 sin x  3 cos x , D2  , 4 sin x  5 cos x 5 sin 3 x  4 cos3 x

sin 3 x  cos3 x sin x  5 cos x 8 cos3 x  2 sin3 x  cos x , D  , D  , 4 5 sin x  sin2 x cos x sin 3 x  2 cos3 x 2 cos x  sin3 x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/ Cho sin x 

3  cot x  tan x , 0  x  . Tính E  . 5 2 cot x  tan x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/ Cho sin x 

8 tan2 x  3 cot x  1 1 , 900  x  1800 . Tính F  . tan x  cot x 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

cot x  3 tan x 2 7/ Cho cos x   . Tính G  . 2 cot x  tan x 3

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/ Cho sin x 

tan x  cos x tan x cos x 2 , 0  x  900 . Tính H1  , H2   cos x cot x . ........ cot x 3 sin2 x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

cot x  tan x sin x 4  9/ Cho cos x   ,  x   . Tính I1  , I2  cot x  . cot x  tan x 1  cos x 5 2 ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 26.

Cho sin x  cos x 

5 . Hãy tính giá trị của biểu thức sau 4

1/ A  sin x.cos x .

ĐS: A 

9 . 32

................................................................................................................................................... ................................................................................................................................................... 2/ B  sin x  cos x .

ĐS: B  

7 . 4

................................................................................................................................................... ...................................................................................................................................................

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3/ C  sin3 x  cos3 x .

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

ĐS: C  

41 7 . 128

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 27.

Cho tan x  cot x  3 . Hãy tính giá trị của biểu thức sau 1/ A  tan 2 x  cot2 x .

ĐS: A  11 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/ B  tan x  cot x .

ĐS: B   13 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/ C  tan 4 x  cot4 x .

ĐS: C  33 13 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 28.

Cho sin x  cos x  m . Hãy tính theo m giá trị các biểu thức 1/ A  sin x cos x .

2/ B  sin 3 x  cos 3 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/ C  sin 4 x  cos4 x .

4/ D  sin x  cos x .

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................................................................................................................................................... 5/ E  tan 2 x  cot2 x .

6/ F  sin 6 x  cos 6 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 29.

Tính sin x, cos x, tan x, cot x . Biết rằng 1/ sin x  cos x  2 .

2/ sin x  cos x  2 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/ sin x  cos x 

1 . 5

ĐS:

4 3 4 3 ; ; ; . 5 5 3 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/ tan x  cot x  4 .

ĐS:

1 2 2 3

;

2 3 ; 2  3; 2  3 . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 30.

Cho tan x  2 cot x  1 . Hãy tính 1/ A  tan2 x  cot2 x .

2/ B  tan 3 x  cot3 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/ C  tan 4 x  2 cot4 x . 28 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 31.

Tính giá trị của các biểu thức sau đây khi 1/ Cho 3 sin 4 x  cos4 x 

3 7 . Tính A  sin 4 x  3 cos 4 x . ĐS: A  . 4 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/ Cho 3 sin 4 x  cos4 x 

1 . Tính B  sin 4 x  3 cos 4 x . ĐS: B  1 . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/ Cho 4 sin 4 x  3 cos4 x  C

7 . Tính C  3 sin 4 x  4 cos 4 x . 4

ĐS:

7 57 C . 4 28

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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CUNG GÓC LIÊN KẾT Bài 32.

Biểu diễn các số đo của cung dưới dạng   k2 với k   . 3 5 11 14 21 , , , ,  . 2 4 3 5 6

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 33.

Biểu diễn các số đo của cung dưới dạng   k3600 với k   . 3700 , 5120, 7650,  10000, 12340 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 34.

Biễu diễn các số đo của cung dưới dạng   k2 với k   . 8 11 7 18 27 25 , , , ,  , . 3 5 2 7 10 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 35.

Dùng cung liên kết (không dùng máy tính), hãy tính các giá trị sau 1/

sin 1500 .

2/

cot1350 .

3/

cos 2250 .

4/

tan 2100 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

cot2250 .

6/

sin 2400 .

7/

cos 3150 .

8/

tan 3000 .

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9/





cot 13800 .

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

10/ cos11 .

11/ sin13 .

12/ tan10 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ cot

7 . 6

11 . 3

14/ cos

15/ cot

25 . 4

16/ sin

17 . 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/ cos

29 . 6

18/ tan

 16  . 19/ cos   3 

45 . 4

 31  . 20/ sin   2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  19   . 21/ cot   4 

 26   . 22/ cos   3 

 159  . 23/ tan  4  

 115  . 24/ sin  6  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 36.

Diễn tả giá trị lượng giác của các góc sau bằng giá trị lượng giác của góc x 1/





sin x  900 .

2/





cos 1800  x . 3/





sin 2700  x . 4/





sin x  1800 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/





cos x  5400 .

6/





cot 1800  x . 7/





sin x  5400 . 8/





tan 3600  x .

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/





cos 4500  x .













10/ sin2 2700  x . 11/ cos3 900  x . 12/ cot5 1800  x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 37.

Diễn tả giá trị lượng giác của các góc sau bằng giá trị lượng giác của góc x 1/

cot  x   .

2/

sin   x  .

3/

tan 2  x .

4/

cot 3  x  .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

sin x  7 .

6/

tan  x  5 .

7/

 5  sin   x  . 2 

8/

 3  cos   x . 2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

 3  cot  x   .  2 

 5  10/ cos  x   .  2 

 11   7  11/ tan   x  . 12/ sin  x   .  2 2   

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ sin2   x .

14/ cos9   x .

15/ cot11 x  3 . 16/ cos4  x  3 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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17/ cot2 x  5 .

18/ cos6 x   .

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     19/ cos8 x   . 20/ cos5   x . 2   2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   21/ sin2012  x   .  2 

   5  9  7 22/ tan2 x   . 23/ cos2013 x   . 24/ sin1991   x  .    2 2  2  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  11  . 25/ cos2015  x   2 

 11  9  26/ cot2  x   . 27/ tan11   x . 28/ cos x  2012 .  2   2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 38.

Rút gọn (đơn giản) các biểu thức 1/

  A  cos x    sin  x   .  2 

................................................................................................................................................... ................................................................................................................................................... 2/

        B  cos   x   sin   x   cos   x  sin   x  . 2  2  2  2 

................................................................................................................................................... ................................................................................................................................................... 3/

 7   3  C  2 cos x  3 cos   x  sin   x   tan   x  .  2  2  

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................................................................................................................................................... 4/

   3    D  2 sin   x   sin(5  x)  sin   x   cos   x  . 2  2  2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

 7   3  E  2 cos x  3 cos(  x)  5 sin   x   cot   x  . 2  2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

  3   F  sin 5  x  cos x    cot 3  x  tan   x  . 2   2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

    11  3  G  cos 15  x  sin x    tan   x  cot   x  . 2   2   2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

   3  H  sin   x   cos   x  cot 2  x   tan   x .   2  2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

 3   3  I  cos 5  x   sin   x   tan   x  cot 3  x  . 2  2 

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

















10/ J  cos 2700  x  2 sin x  4500  cos x  9000  2 sin 2700  x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................      3  11/ K  sin2 x    sin2 x    sin 2 x    sin2 x   . 4  2  4    

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 12/ L  sin2

   2 5 7  sin2  sin2  sin2  sin2  sin2 . 3 6 9 9 8 8

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   13/ M  cos2017 x  cos2017   x  . sin2016   x   sin2015   x  .  2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   3   14/ N  sin6   x  cos6 x    2 sin 4 x  2  sin 4 x    cos2 x   . 2  2   

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...................................................................................................................................................  19  tan   x  . cos 36  x .sin x  5  2  15/ O  .  9  sin   x  . cos x  99  2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   85  3    cos 207  x  sin 2 33  x  sin 2 x   . 16/ P  sin  x   2  2  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 39.

Rút gọn và tính giá trị của biểu thức (không dùng máy tính) 1/

A  cos 00  cos 200  cos 400  ......  cos180 0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

B  cos 200  cos 400  cos 60 0  ......  cos1800 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  cos100  cos 400  cos 70 0  ......  cos170 0 .

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................................................................................................................................................... 4/

D  tan 200  tan 400  tan 600  ......  tan 180 0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  cot150  cot 300  cot 450  ......  cot165 0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

F  sin 50  sin 100  sin 150  ......  sin 3600 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G  cot1950  cot2100  cot225 0  ......  cot 345 0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

H  cot150.cot 350.cot 550. cot 750 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

I  tan 100.tan 200.tan 300......tan 800 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 10/ J  tan 10.tan 20.tan 30...... tan 890 . ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ K  sin 2 28 0  sin 2 360  sin 2 540  cos2 152 0 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 12/ L  cos2 20  cos2 4 0  cos2 60  ......  cos2 880 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ M  sin 2 100  sin 2 200  sin2 30 0  ......  sin2 900 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 14/ N  cos2 100  cos2 200  cos2 300  ...  cos2 180 0 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 15/ O  sin 200  sin 400  sin 60 0  ...  sin 340 0  sin 3600 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 40.

Rút gọn và tính giá trị của biểu thức (không dùng máy tính) 1/





A  cos 3150 .sin 7650 .

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

B  sin 320.sin 148 0  sin 3020.sin 1220 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  sin 8100. cos 5400  tan 1350. cot 5850 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/









D  sin 8250.cos 150  cos 750.sin 5550 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  2 tan 5400  2 cos11700  4 sin 9900 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

F





sin 234 0  cos 2160 0

sin144  cos 126

0

.tan 360 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G





sin 324 0  cos 6660 0

sin 1206  cos 36

0

. cot 360 .

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

H





sin 328 0 .sin 9580 cot 5720





  tan 212 

cos 5080 . cos 1022 0 0

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

I

  tan 142  . sin 108

cos 288 0 .cot 720 0

0

 tan 180 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

















10/ J  2 sin 7900  x  cos 12600  x  tan 6300  x .tan 12600  x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................





2 sin 25500. cos 188 0 1 11/ K   . tan 368 0 2 cos 6380  cos 98 0

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 12/

cos 44 L

0



 tan 2260 .cos 4060 cos 316

0

 cos 720.cot18 0 .

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................................................................................................................................................... 13/ M 





tan 460.sin 440  cot 1360 .sin 404 0 cos 316

0

 tan 360.tan 54 0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 14/ N 





sin 328 0 .sin 9580 cos 5720





  tan 212 

cos 5080 . cos 1022 0 0

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4 17  cos 3 6 15/ O  .  23 10    sin  cos   3  6 sin

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 16/ P 

sin 4, 8 .sin 5,7 cot 5,2



cos 6,7.cos 5, 8 tan 6,2

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 41.

Chứng minh rằng nếu A, B, C là ba góc của một tam giác thì 1/

sin B  sin A  C .

2/

cos A  B   cos C .

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................................................................................................................................................... 3/

sin

AB C  cos . 2 2

4/

cos B  C   cos A  2C .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

cos A  B  C   cos 2C .

6/

sin A  2B  C   sin B .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

cot A  B  C   cot2B .

8/

cos

3A  B  C   sin 2A . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

sin

A  B  3C  cos C . 2

10/ tan

A  B  2C 3C  cot . 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ cos

A  3B  C   sin B . 2

12/ cot

A  2B  C 3B  tan . 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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C – CÔNG THỨC CỘNG  sin a  b  sin a.cos b  sin b.cos a

 sin a  b  sin a. cos b  sin b. cos a .

 cos a  b  cos a.cos b  sin a. sin b

 cos a  b  cos a.cos b  sin a.sin b .

 tan a  b 

tan a  tan b . 1  tan a. tan b

 tan a  b 

tan a  tan b . 1  tan a. tan b

Hệ quả:   1  tan   tan     .  4  1  tan 

  1  tan   tan      .  4  1  tan 

BÀI TẬP ÁP DỤNG Bài 42.

Không dùng máy tính. Hãy tính giá trị của các biểu thức 1/ A  sin120.cos 480  cos120.sin 480 .

................................................................................................................................................... ................................................................................................................................................... 2/ B  cos 380. cos 220  sin 38 0.sin 220 . ................................................................................................................................................... ................................................................................................................................................... 3/ C  sin 100. cos 55 0  cos10 0. sin 55 0 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/ D  sin 360. cos 60  sin 1260. cos 84 0 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/ E  cos 1120. cos 230  sin 1120. sin 23 0 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/ F  sin 2000.sin 3100  cos 3400.cos 500 . ................................................................................................................................................... ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/ G  cos110.cos 210  cos 690.cos 790  cos100 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/ H  cos 680.cos 780  cos 220.cos120  sin1000 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................



 



9/ I  cos 530 .sin 337 0  sin 307 0.sin 1130 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 43.

Tính giá trị lượng giác của các cung góc sau 1/ 150 .

2/

750 .

3/ 1050 .

4/ 2850 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

19 . 12

6/

5 . 12

7/

7 . 12

8/

13 . 12

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 44.

Tính giá trị của các biểu thức sau

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1/

A

1  tan150 1  tan 150

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

.

2/

B

tan 250  tan 200 1  tan 250 tan 200

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C

sin100 cos 200  sin 200 cos100 cos 17 0 cos130  sin 17 0 sin130

.

4/

D

tan 2250  cot 810 cot 690

.

cot2610  tan 2010

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E

sin 730 cos 30  sin 87 0 cos17 0 cos1320 cos 620  cos 420 cos 28 0

.

6/

F

cot2250  cot 79o0 ot 710 cot2590  cot 2510

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G  cos2 750  sin2 750 .

8/

H  sin2 200  sin2 1000  sin2 1400 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

I  cos2 100  cos1100  cos2 1300 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 10/ J  tan 200.tan 800  tan 800.tan1400  tan1400. tan 200 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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................................................................................................................................................... 11/ K  tan100.tan 700  tan 700.tan1300  tan1300.tan1900 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 12/ L  sin1600 cos1100  sin 2500 cos 3400  tan1100 tan 3400 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................





 





13/ M  cos 700  cos 500 cos 3100  cos 2900  cos 400  cos1600 cos 3200  cos 3800 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 45.

Tính giá trị của các biểu thức sau 1/

   1 và 0  x  . A  cos x  x   biết sin x  3  2  3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

  12 3 và   x  . B  sin   x biết cos x    3 13 2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/





C  cos x  300 biết tan x  2 và 0  x  900 .

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

  3  D  tan x   biết sin x  và  x   .  3 5 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

  12 3  x  2 . E  cos   x biết sin x   và 3  13 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

 4 3  . F  cot x   biết sin x   và   x   4  5 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

  5   G  tan  x   biết cot   x   2 . 4   2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

 2 7  H  sin 2x   biết cot x  .  4 3 

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9/

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

I  cos a  b . cos a  b biết cos a 

1 1 và cos b  . 3 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 46.

Biết sin  

4 8 , 00    900 và sin   , 900    1800 . Hãy tính giá trị của 5 17









biểu thức A  cos    và B  sin    . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 47.

Biết sin  

8 5 , tan b  và ,  là các góc nhọn. Hãy tính giá trị của các biểu 17 12

thức A  sin    , B  cos    và C  tan    . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 48.

Cho 0  ,  

  ,     và tan . tan   3  2 2 . 2 4

1/ Hãy tính tan    ; tan   tan  .

2/ Tính tan ; tan  . Suy ra  và  .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 49.

Cho    

 . Tính giá trị của các biểu thức 3 2

2

1/ A  cos   cos   sin   sin  .

2

2

2/ B  cos   sin   cos   sin   .

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................................................................................................................................................... Bài 50.

Rút gọn các biểu thức 1/ A  sin x  3 cos x .

2/ B  3 sin 7x  cos 7x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................





3/ C  a sin x  b cos x, a 2  b2  0 .

   4/ D  3 sin x    sin x   3  

  . 6 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   5/ E  cos7x.cos5x  3 sin2x  sin7x.sin5x . 6/ F  3 cos 2x  sin 2x  2 sin 2x   . 6  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   7/ G  2 sin 2x    4 sin x  1 . 6  

  8/ H  sin2x  2 2 cos x  2sin x    3 . 4  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 51.

Rút gọn các biểu thức 1/

A  sin x cos 5x  cos x sin 5x .

................................................................................................................................................... 2/

B  sin 4x cot2x  cos 4x .

................................................................................................................................................... 3/

C  cos 6x tan 3x  sin 6x .

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4/

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

D  sin  x  y cos x  y  sin x  y cos x  y .

................................................................................................................................................... 5/



 





 



E  cos 400  x cos x  200  sin 400  x sin x  20 0 .

................................................................................................................................................... 6/



 





 



F  sin 140  2x cos 160  2x  cos 14 0  2x sin 16 0  2x .

................................................................................................................................................... 7/



 





 



G  sin x  100 cos 2x  800  sin x  100 0 cos 2x  100 .

................................................................................................................................................... ................................................................................................................................................... 8/

        H  sin  x   cos   x  sin   x  cos  x   .   4 3  3    4  

................................................................................................................................................... ................................................................................................................................................... 9/

   I  cos x   cos x  3   

     3    cos  x   cos  x   . 4  6  4   

................................................................................................................................................... ...................................................................................................................................................   9   5   5   10/ J  sin x   cos   x   sin   x  cos   x  .  3   4   4   3 

................................................................................................................................................... ...................................................................................................................................................        11/ K  cos  x   cos  x    cos x   cos x  3  4  6     

  . 4 

................................................................................................................................................... ...................................................................................................................................................      13  13  3    cos  x   cos  x   . 12/ L  cos  x   cos  x   3  4  6  4    

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Bài 52.

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

Rút gọn các biểu thức sau 1/

A

tan 3x  tan x . 1  tan x tan 3x

2/

B

tan 2x  1 . 1  tan 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C



tan 2x  cot 900  x



0



.

4/

1  cot 90  2x tan x

D

tan2 2x  tan2 x 1  tan2 2x tan2 x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 53.

Rút gọn các biểu thức sau 1/

  A  sin a  b  sin   a  sin b .  2 

................................................................................................................................................... ................................................................................................................................................... 2/

    B  cos   a  cos   b  cos a  b . 2  2 

................................................................................................................................................... ................................................................................................................................................... 3/

    1 C  cos   a cos   a   sin2 a .  4  4   2

................................................................................................................................................... ................................................................................................................................................... 4/

D  sin2 a sin2 b  cos2 a cos2 b .

................................................................................................................................................... ................................................................................................................................................... 5/

E

2 sin a  b cos a  b  cos a  b

 tan b .

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................................................................................................................................................... ................................................................................................................................................... Bài 54.

Rút gọn các biểu thức sau 1/

A  cos2 x  3 sin 2x  sin2 x .

2/

B  4 sin3 x  3 sin x  3 cos 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/









C  sin x  450  cos x  450 .

4/

D  tan 3x  tan x  sin 2x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 55.

Chứng minh các đẳng thức sau 1/

sin 2x  2 sin x cos x .

................................................................................................................................................... 2/

cos 2x  cos2 x  sin 2 x .

................................................................................................................................................... 3/

tan 2x 

2 tan x 1  tan2 x

.

................................................................................................................................................... 4/

sin 3x  3 sin x  4 sin 3 x .

................................................................................................................................................... 5/

cos 3x  4 cos 3 x  3 cos x .

................................................................................................................................................... 6/

   cos x  sin x  2 cos x    2 sin x   4  

  . 4 

................................................................................................................................................... ................................................................................................................................................... 7/

    cos x  sin x  2 cos x     2 sin x   . 4  4   

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................................................................................................................................................... ................................................................................................................................................... 8/

sin  x  y sin x  y  sin2 x  sin2 y  cos2 y  cos2 x .

................................................................................................................................................... ................................................................................................................................................... 9/

cos  x  y cos x  y  cos2 x  sin2 y  cos2 y  sin2 x .

................................................................................................................................................... ................................................................................................................................................... Bài 56.

Chứng minh các đẳng thức sau 1/

    sin   x  sin   x  2 sin x .  4   4 

................................................................................................................................................... ................................................................................................................................................... 2/

    4 sin  x   sin x    4 sin2 x  3 .  3   3 

................................................................................................................................................... ................................................................................................................................................... 3/

sin x sin y  z  sin y sin z  x   sin z sin x  y  0 .

................................................................................................................................................... ................................................................................................................................................... 4/

cos x sin y  z  cos y sin z  x   cos z sin x  y  0 .

................................................................................................................................................... ................................................................................................................................................... 5/

tan x  y  tan x  tan y  tan x  y tan x tan y .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

        tan 2x tan   x   tan 2x tan   x   tan   x  tan   x  1 . 6  3  3  6 

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................................................................................................................................................... ................................................................................................................................................... 7/

      2  2  tan x. tan  x    tan x  . tan x    tan x  . tan x   3 .     3 3 3 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

     cos x   . cos  x    cos x     3 4

   3  2 . cos x    1 3 .  6 4 4





................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

cos 70

o



 





 cos 50o cos230o  cos290o  cos 40o  cos160o cos 320o  cos 380o  0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 57.

Chứng minh các đẳng thức sau 1/

cos a  b cos a  b



cot a cot b  1 . cota cot b  1

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

sin a  b cos a cos b



sin b  c cos b cos c



sin c  a  cos c cos a

 0.

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3/

sin a  b sin a  b 2

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

 tan2 a  tan2 b .

2

cos a cos b

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

cos a  b cos a  b 2

2

cos a cos b

 1  tan 2 a tan2 b .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 58.

Chứng minh các hệ thức sau, với điều kiện cho trước 1/

Nếu cos a  b  0 thì sin a  2b  sin a .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

Nếu sin 2a  b  3 sin b thì tan a  b  2 tan a .

b  a  b  a  HD:  . 2a  b  a  b  a 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

Nếu tan a  2 tan b thì sin a  b  3 sin a  b .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

Nếu tan a. tan b  

1 thì cos a  b  2 cos a  b . HD: Khai triển giả thiết. 3

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

Nếu 5 sin b  sin 2a  b thì tan a  b 

3 tan a . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

Nếu sin b  sin a cos a  b thì 2 tan a  tan a  b . HD: b  a  b  a .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

Nếu cos 2a  b  1 thì tan a  b  tan a  2 tan

b . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

Nếu cos a  b  k cos a  b thì tan a tan b 

1 k . HD: 1 k

a  2b  a  b  b  .  a  a  b  b 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 59.

Chứng minh các đẳng thức sau 1/

tan a  tan b  tan c  tan a tan b tan c 

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sin a  b  c cos a cos b cos c

.

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

cos2 x  cos2  x  y  cos2 y  2 cos x cos y cos  x  y  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 60.

Chứng minh các biểu thức sau độc lập với biến x 1/

    A  sin2 x  cos   x  cos   x .  3  3  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

    B  cos2 x  cos2   x  cos2   x .  3   3 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

 2   2  C  sin2 x  sin2   x   sin2   x  . 3  3 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

  2  2  D  cos2 x  cos2 x    cos2 x   .  3  3  

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

F

a cos3 x  cos 3x a sin 3 x  sin 3x  , a  const . cos x sin x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 61.

    Chứng minh rằng: tan x tan   x tan   x  tan 3x . Từ đó tính giá trị của biểu 3  3 

thức P  tan 100 tan 500 tan 1100 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 62.

Cho tam giác ABC với A, B,C lần lượt là ba góc của tam giác. Chứng minh 1/

sin C  sin A. cos B  sin B. cos A .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

sin A  sin B cos C  sin C cos B .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

cos A  sin B sin C  cos B cos C .

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

sin C  tan A  tan B, cos A.cos B

A, B  90  . 0

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

tan A  tan B  tan C  tan A. tan B. tan C,

A, B, C  90  . 0

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

cot A. cot B  cot B. cot C  cot C. cot A  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

sin

A B C B C  cos cos  sin sin . 2 2 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

cos

A B C B C  sin cos  cos sin . 2 2 2 2 2

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

A B B C C A .tan  tan .tan  tan . tan  1 . 2 2 2 2 2 2

tan

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 10/ cot

A B C A B C  cot  cot  cot .cot .cot . 2 2 2 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ cot B 

cos C cos B  cotC  , sin B.cos A sin C.cos A

A  90  . 0

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 12/ cos

A B C A B C A B C A B C .cos .cos  sin sin cos  sin cos sin  cos sin sin . 2 2 2 2 2 2 2 2 2 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ sin2

A B C A B C  sin2  sin2  1  2 sin sin sin . 2 2 2 2 2 2

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................................................................................................................................................... ................................................................................................................................................... 14/ sin A cos B cos C  sin B cos C cos A  sin C cos A cos B  sin A sin B sin C . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 63.

Cho tam giác ABC với A, B,C lần lượt là ba góc của tam giác. Chứng minh 1/

tan A  tan B  tan C  3 3,  A, B, C nhọn.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

2/

tan2 A  tan2 B  tan2 C  9,  A, B, C nhọn.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

tan 6 A  tan 6 B  tan6 C  81,  A, B, C nhọn.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

tan2

A B C  tan2  tan2  1 . 2 2 2

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................................................................................................................................................... 5/

tan

A B C  tan  tan  3 . 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... HD:

1/, 2/, 3/, sử dụng tan A  tan B  tan C  tan A. tan B. tan C và bất đẳng thức Cauchy.

4 / sử dụng bất đẳng thức bổ đề a 2  b2  c2  ab  bc  ca và

tan

A B B C C A .tan  tan .tan  tan . tan  1 . 2 2 2 2 2 2

2  A B C   5 / khai triển tan  tan  tan  và sử dụng câu 3 / .  2 2 2

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D – CÔNG THỨC NHÂN – HẠ BẬC Công thức nhân đôi  sin 2  2 sin .cos  .  cos 2  cos2   sin 2   2 cos2   1  1  2 sin2  .  tan 2 

 cot2 

2 tan  1  tan2 

.

cot2   1 . 2 cot 

Công thức hạ bậc

Công thức nhân ba (mở rộng)

 sin2  

1  cos 2 . 2

 sin 3  3 sin   4 sin 3  .

 tan2  

1  cos 2 . 1  cos 2

 tan 3 

3 tan   tan 3  1  3 tan2 

.

BÀI TẬP ÁP DỤNG RÚT GỌN VÀ TÍNH GIÁ TRỊ CỦA BIỂU THỨC Bài 64.

Tính giá trị của các biểu thức sau (không dùng máy tính bỏ túi) 1/

A  cos2

   sin2 . 8 8

2/

B  3 cos100  4 cos 3 100 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/





C  sin 200 1  4 cos2 200 .

4/

D  4 sin3 400  3 sin1300 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  4 sin3 500  3 cos1400 .

6/

F

tan 150 1  tan2 150

.

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................................................................................................................................................... tan

7/

G

7 8

 1  tan 8

.

8/

2

H

1 sin 100



3 cos100

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

I

1 sin 10

0

 4 sin 700 .

10/ J  tan2

 5  tan2 . 12 12

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ K  tan2 360 tan2 720 .

12/ L  cos2 700  sin2 400 sin1000 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ M  cos2 200  2 sin2 550  2 sin 650 . 14/ N  sin 6 0.sin 420. sin 660. sin 78 0 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 65.

Tính giá trị của các biểu thức sau (không dùng máy tính bỏ túi) 1/

A  cos 360 cos 720 .

2/

B  cos

 2 cos . 5 5

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

   C  sin .cos cos . 8 8 4

4/

D  sin

   .cos .cos . 16 16 8

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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5/

E  sin100 sin 500 sin 700 .

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

6/

F  cos

 2 4 cos cos . 7 7 7

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G  cos

 4 5 cos cos . 7 7 7

8/

H  sin 60 cos120 cos 240 cos 48 0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

I  sin 60 sin 420 sin 660 sin 780 .

10/ J  sin50.sin150.sin250.... sin750.sin850 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ K  cos100.cos 200.cos 300...cos 700.cos 800 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 12/ L  8 tan180 cos180 cos 360 cos 720 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ M  cos 200 cos 400 cos 600 cos 800 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 14/ N  cos

 2 3 4 cos cos cos . 9 9 9 9

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15/ O  96 3 sin

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     cos cos cos cos . 48 48 24 12 6

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 16/ P  cos

2 4 8 16 32 .cos .cos .cos .cos . 31 31 31 31 31

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/ Q  cos

 2 4 8 16 cos cos cos cos . 33 33 33 33 33

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 18/ R  cos

 2 3 4 5 6 7 .cos .cos .cos .cos .cos .cos . 15 15 15 15 15 15 15

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 66.

Tính giá trị của các biểu thức sau (không dùng máy tính bỏ túi) 1/

A

cos 800  cos 200 cos 350 cos150  sin 350 sin 150

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

B  3 sin 150 cos150 

sin 600 sin4 150  cos4 150

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  tan 90  tan 270  tan 630  tan 810 .

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................................................................................................................................................... ................................................................................................................................................... 4/

D  4 tan

     2 tan  tan  cot . 8 16 32 32

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E





cos2 6960  tan 2600 tan 5300  cos2 156 0

.

tan2 2520  cot2 3420

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 67.

Rút gọn các biểu thức 1/

2

A  sin x  cos x  .

2/

B  1  4 sin2 x cos2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  sin x cos x cos 2x .

4/

D  cos 4 2x  sin 4 2x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

    E  cos2  x    sin2  x   . 2  2   

6/

F  sin x cos x cos 2x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

   G  4 sin x sin x   sin 2x  2   

  . 2 

8/

 x  x H  sin2     sin2    .  8 2   8 2 

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................................................................................................................................................... ................................................................................................................................................... 9/





I  sin 8x  2 cos2 450  4x .

10/ J  sin2x  cos2x 2cosxsinx  cosx 1.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 68.

Tính giá trị của các biểu thức sau (không dùng máy tính bỏ túi) 1/

A

2 cos2 x  1 . sin x  cos x

2/

B

1  2 sin2 2x . cos 2x  sin 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

2

C

1  tan x1  cot x

.

4/





D  1  tan2 x cot x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E

sin 2x cos2x  . sin x cos x

6/

F

cot x  tan x . cos 2x

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

7/

 3  1  sin   x   2  . G    1  sin   x   2 

8/

H

sin2 2x  4 sin2 x sin2 2x  4 sin2 x  4

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

I

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sin 4x cos 2x . . 1  cos 4x 1  cos 2x

10/ J 

sin 3x  cos 3x . sin x  cos x

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................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   1  cos   x  x  2  11/ K  tan    . .    4 2   sin   x   2 

12/ L  cot2x 

1  tan x . 2 sin 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

1  cos 2x  sin 2x 13/ M  . cot x . 1  cos 2x  sin 2x

14/ N 





2 sin 2x  2 cos2 x  1

cos x  sin x  cos 3x  sin 3x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  1 1 1     cot 8x . 15/ O  1  sin x  1  sin x, 0  x   . 16/ P   4  sin 2x sin 4x sin 8x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   17/ O  1  sin 2x  1  sin 2x,   x   4

  . 4 

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Bài 69.

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

Tính giá trị của biểu thức lượng giác, khi biết 1/

sin 2x, cos 2x khi sin x 

3  ,  x  . 5 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

sin 2x, cos 2x khi sin x  cos x  2 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

cos2x, sin 2x, tan 2x khi cos x  

5 3 ,x . 13 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

cos 2x, sin 2x, tan 2x khi tan x  2 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

4  3 sin x, cos x khi sin 2x   , x . 5 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

cos 2x, sin 2x, tan 2x khi tan x 

7 . 8

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

sin 2x, cos 2x, tan 2x khi tan x  2  3, 0  x 

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 70.

Tính giá trị của biểu thức lượng giác, khi biết 1/

A  sin 5 x cos x  cos5 x sin x khi biết x 

 . 16

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

B  sin4 x  sin3 x cos x  cos3 x sin x  cos4 x khi biết x 

 . 48

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  cos 4x khi biết tan x  cot x  3 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/









D  cos 2700  4x khi biết cot 450  x  2 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  sin4 x  cos4 x khi biết sin x  cos x  m với m   .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

F

tan x  cot x 3 khi biết sin x cos x  , tan x  cot x 4

 0  x  

  . 4 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 71.

Tính giá trị lượng giác của (không dùng máy tính bỏ túi)

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1/

cos150 .

2/

cos180 .

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

3/ sin 360 .

4/

cos 360 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 72.

Tính tan

1/

x khi biết 2

cos x 

4  , 0x . 5 2

2/

tan x 

24 3 , x . 7 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/ sin x  cos x 

7  , 0x . 2 6

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 73.

1   1   Cho tan x  , 0  x   và tan y  , 0  y   . 2  2  3  2 

1/

Tính x  y .

2/

sin  x  2y và cos 2x  y .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 74.

Tìm x khi biết

1/

 0  x    . 2   tan x  2  1

2/

 0  x    2 .   6 2 cos x  4 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 72 | THBTN – CA

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Bài 75.

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Tính theo cos 2x các biểu thức sau 1/

A  1  cos2 x .

2/

B  sin2 x cos2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C

1  sin2 x

.

cos2 x

4/

D

1  cot2 x 1  cot2 x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  sin6 x  cos6 x .

6/

F  sin6 x cos2 x  cos6 x sin2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 76.

Tính theo t  tan 1/

x các biểu thức sau 2

A  sin x .

2/

B  cos x .

................................................................................................................................................... ................................................................................................................................................... 3/

C

sin x . 3  2 cos x

4/

D  tan x  cot x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E

1  tan x . 1  cot x

6/

F

tan x  sin x . tan x  sin x

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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CHỨNG MINH ĐẲNG THỨC Bài 77.

Chứng minh các đẳng thức sau 1/

cos 4 x  sin 4 x  cos 2x .

2/





sin 4x  4 sin x cos x 1  2 sin 2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

cos2 2x  sin 2 x  cos x cos 3x .

4/

cos 4x  8 cos4 x  8 cos2 x  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

8 sin4 x  3  4 cos 2x  cos 4x .

6/

sin 4  cos4 x 

3 1  cos 4x . 4 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

sin4  cos4 x  6 cos2 x sin2 x  cos 4x . 8/

sin6 x  cos6 x 

5 3  cos 4x . 8 8

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

sin6 x  cos6 x 

15 1 x x 1 cos2x  cos 6x . 10/ sin6  cos6  cos x sin2 x  4 . 16 16 2 2 4





................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ cos8 x  sin 8 x 

7 1 cos 2x  cos 6x . 8 8





13/ cos3 xcos3xsin3 xsin3x 3 sin4 x cos4 x 2.

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 78.

Chứng minh rằng nếu a  b 

 thì 1  tan a1  tan b  2 . 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 79.

Chứng minh các đẳng thức sau 1/

1  cos 2x  tan x . sin 2x

2/

tan x  cot x 

2 . sin 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

cot x  tan x  2 cot2x .

4/

 x  cos x  cot    . 1  sin x  4 2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

1  cos x x .tan2  1 . 1  cos x 2

6/

1  cot2x  cot x sin 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

   x  2 . tan     cot      4 2   4 2  cos x

8/

tan 2x  tan x cos 2x  tan x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

    tan   x  tan   x  2 tan 2x .  4   4 

10/ cos3 x sin x  sin3 x cos x 

1 sin 4x . 2

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................................................................................................................................................... ................................................................................................................................................... 11/ cos 3x sin3 x  sin 3x cos3 x 

    3 sin 4x . 12/ tan x tan   x . tan   x  tan 3x .  3  3 4  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   2  13/ tanx  tanx    tanx    3tan3x . 14/ cotx  tanx 2tan2x  4tan4x  8cot8x .  3 3  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 80.

Chứng minh các đẳng thức sau 1/

 x 1  sin x  cot    . cos x  4 2 

2/

tan 2x 

1 1  2 sin2 x  . cos 2x 1  sin 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

2 sin 2x  sin 4x 2 cos x  cos 3x

 tan 2x cos x .

4/

6  2 cos 4x  tan2 x  cot2 x . 1  cos 4x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

1  2 sin2 x 1  tan x  . 1  sin 2x 1  tan x

6/

 1  x   tan  tan x  1  cos x 2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

  1  sin 2x  tan   x  .  4 cos 2x 

8/

2 sin x  sin 2x x  tan2 . 2 sin x  sin 2x 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... sin x  sin

2

9/

1  2 sin x  1.   2    2 tan   x  cos   x 4  4 

10/

x 2

1  cos x  cos

x 2

 tan

x . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/

1  cos x  cos2x  cot x . sin 2x  sin x

12/

1  cos 4x 1  sin 4x . cot x  tan x 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/

sin 2x cos x

1  cos 2x1  cos x

 tan

x . 2

14/

sin4 x  cos4 x  cos2 x 1 1  cos x 

 cos2

x . 2

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6  2 cos 4x 15/ cot x  tan x  . 1  cos 4x 2

2

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC 2  x x  4  16/ cot  tan   . 2 2  1  2 tan x cot2x 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/

cot2 2x  1  cos 8x cot 4x  sin 8x . 2 cot2x

18/ tan 2x 

1 1  2 sin2 x  . cos 2x 1  sin 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 19/ tan 4x 

1 sin 2x  cos 2x  . cos 4x sin 2x  cos 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 20/

cos x  sin x cos x  sin x   2 tan 2x . cos x  sin x cos x  sin x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... x x  sin 2 2  1  tan x . 21/ x x cos x cos  sin 2 2 cos

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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1  cos x  cos 2x  cos 3x 2 cos2 x  cos x  1

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

 2 cos x .

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  1  x x 23/ 2   cot 2x   cot  tan .  sin 2x 2 2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 24/

sin4 x  sin 2x  cos4 x  cos 2x . tan 2x  1

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 25/

sin2 2x  4 sin 2 x 2

2

sin 2x  4 sin x  4

 tan 4 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 26/

sin2 3x sin2 x



cos2 3x cos2 x

 8 cos 2x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 27/

cos2 x  cos 3x sin3 x  sin 3x   3. cos x sin x

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  1   1  tan x  1   sin 2x . 28/ 1  tan x     cos x  cos x  cos2 x

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 81.

Chứng minh các biểu thức sau đây không phụ thuộc vào biến số 1/

A

sin 2x  2 sin x x  tan2 . sin 2x  2 sin x 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

B

cos3 x  cos 3x sin 3 x  sin 3x  . cos x sin x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

1 C  4 sin 4 x  2 cos2x  cos 4x . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

D  3 cos 2x  5 sin4 x  4 sin2 x cos2 x  cos4 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E

tan2 x  1 cot x  cos 4x cot2x  sin 4x . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

   F  sin 4 x  sin 4 x    sin 4 x   4  

   3    sin 4 x   . 2  4  

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................................................................................................................................................... BIẾN ĐỔI THÀNH TÍCH Bài 82.

Biến đổi thành tích các biểu thức sau 1/

A  2 sin x  12 cos x  sin x   sin 2x  cos x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

B  2 cos x  12 sin x  cos x   sin 2x  sin x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  2 sin x  12 cos 2x  2 sin x  3  4 sin2 x  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

D  2 cos x  12 cos x  2 sin x  1  4 sin2 x  3 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  2 sin x  13 cos 4x  2 sin x  4  4 cos2 x  3 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

F  1  sin x  cos x  sin 2x  cos 2x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G  1  cos x  cos2 x  sin x 1  cos x .

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................................................................................................................................................... 8/









H  1  sin2 x cos x  sin 2x  1  cos2 x sin x  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 83.

Biến đổi thành tích các biểu thức sau 1/

A  sin 3 x cos x  cos 3 x sin x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

B  sin3 x  cos3 x  sin x  cos x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  sin3 x  cos3 x  cos 2x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

D  cos2 x  sin3 x  cos x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  cos3 x  cos2 x  2 sin x  2 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

F  cos3 x  sin3 x  2 sin2 x  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 84.

Biến đổi thành tích các biểu thức sau

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

A  sin 2x  cos 2x  cos x  sin x .

................................................................................................................................................... 2/

B  sin 2x  cos 2x  3 sin x  cos x  2 .

................................................................................................................................................... 3/

C  cos 2x  sin 2x  3 sin x  cos x  2 .

................................................................................................................................................... 4/

D  sin 2x  cos 2x  2 cos x  sin x  3 .

................................................................................................................................................... 5/

E  9 sin x  6 cos x  3 sin 2x  cos 2x .

................................................................................................................................................... ................................................................................................................................................... 6/

F  sin 2x  2 cos 2x  4 cos x  sin x  1 .

................................................................................................................................................... ................................................................................................................................................... 7/

G  2 cos x  1sin x  cos x   1 .

................................................................................................................................................... ................................................................................................................................................... 8/

H  2 sin 2x  cos 2x  7 sin x  2 cos x  4 .

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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E – CÔNG THỨC BIẾN ĐỔI LƯỢNG GIÁC  Công thức biến đổi tổng thành tích ● cos a  cos b  2 cos

ab ab .cos . 2 2

● cos a  cos b   2 sin

ab ab .sin . 2 2

● tan a  tan b 

● tan a  tan b 

● sin a  sin b  2 sin

ab a b .cos . 2 2

● cot a  cot b 

● sin a  sin b  2 cos

ab a b .sin . 2 2

● cot a  cot b 

sin a  b cos a. cos b sin a  b cos a.cos b sin a  b sin a.sin b sin b  a  sin a.sin b

.

.

.

.

 Hệ quả  ● sin   cos   2.sin   

      2. cos    . 4  4  

   ● sin   cos   2 sin      2 cos     4

  . 4 

 Công thức biến đổi tích thành tổng ● cos a.cos b 

1 cos a  b  cos a  b .  2 

● sin a.sin b 

1 cos a  b  cos a  b .    2

● sin a.cos b 

1 sin a  b  sin a  b .  2  BÀI TẬP ÁP DỤNG CÔNG THỨC BIẾN ĐỔI TỔNG THÀNH TÍCH

Bài 85.

Tính giá trị của biểu thức 1/

A

cos 2x  cos 4x khi x  200 . sin 4x  sin 2x

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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2/

B

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

cos x. cos13x  khi x  . cos 3x  cos 5x 17

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C

cos x. cos10x  khi x  . cos 2x  cos 4x 13

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

D

tan 2x  sin 2x 2 khi tan x  . tan 2x  sin 2x 15

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E

sin x  sin 2x  sin 3x 1   khi sin x  , x . cos x  cos 2x  cos 3x 3 4 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 86.

Rút gọn biểu thức 1/

A

1  2 cos x . 1  2 cos x

2/

B

3  2 cos 3x 3  2 cos 3x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C

1  2 cos 2x 3  2 sin 2x

.

4/

D

2  2 sin 2x 2  2 sin 2x

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E

sin 5x  sin 3x . 2 cos 4x

6/

F

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G

sin x  y sin x  sin y

.

8/

H

sin x  sin y . cos x  cos y

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

cos x  sin x I . cos x  sin x

10/ J 

sin2 4x  sin2 2x cos2 x  cos2 2x

.

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

sin2 4x 11/ K  . 2 cos x  cos 3x  cos 5x

12/ L 

sin 2x . tan x  cot 2x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ M 

tan 3x  tan 5x . cot 3x  cot 5x

14/ N 

tan 2x  cot2x . 1  tan 2x. tan 4x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 15/ O 

1  sin 2x  cos 2x . 1  sin 2x  cos 2x

16/ P 

1  sin 4x  cos 4x . 1  cos 4x  sin 4x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/ Q 

sin 2x  2 sin 3x  sin 4x . cos 3x  2 cos 4x  cos 5x

18/ R 

sin x  sin 4x  sin 7x . cos x  cos 4x  cos 7x

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19/ S 

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

cos 2x  sin 4x  cos 6x . cos 2x  sin 4x  cos 6x

1  cos x  cos 2x  cos 3x

20/ T 

2 cos2 x  cos x  1

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 21/ U 





2 sin 2x  2 cos2 x  1

cos x  sin x  cos 3x  sin 3x

.

22/ V 

sinx  y  sinx sinx  y  sinx



cosx  y  cosx cos x  y  cosx

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 23/ X 

cos 7x  cos 8x  cos 9x  cos 10x sin 4x  sin 5x  sin 6x . 24/ Y  . sin 7x  sin 8x  sin 9x  sin 10x cos 4x  cos 5x  cos 6x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 87.

Biến đổi thành tích các biểu thức sau đây 1/

A  cos 3x  cos x .

2/

B  sin 3x  sin 2x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  cos 4x  cos x .

4/

D  sin 5x  sin x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  1  sin 2x .

6/

F  1  sin x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G  1  2 cos x .

8/

H  2 sin 2x  1 .

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9/

I  3  2 cos 2x .

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

10/ J  sin a  b  sin a  b .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ K  cos2 x  cos2 y .

12/ L  1  sin x  cos 2x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ M  1  sin x  cos x .

14/ N  cos x  sin 2x  cos 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 15/ O  sin 3x  sin x  sin 2x .

16/ Q  cos x  cos 2x  sin 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/ R  sin x  sin 2x  sin 3x .

18/ S  cos x  cos 2x  cos 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 19/ T  2 sin 2x  cos 5x  cos 9x .

20/ U  sin 3x  2 sin 2x  sin x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 21/ V  cos x  cos 3x  2 cos 5x .

22/ X  cos 460  cos 220  2 cos 780 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 23/ Y  cos

 2 3  cos  cos . 7 7 7

24/ Z  cos

2 4 6  cos  cos . 7 7 7

................................................................................................................................................... 88 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... 25/ W  sin 700  sin 200  sin 500 .

26/   cos 5x  cos 7x  cos   6x .

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   27/   cos   5x   sin x  cos 3x .  2 

28/   sin x  sin y  sin

xy . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 29/   sin x  sin y  sin x  y .

30/   cos x  cos y  sin x  y .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 31/   cos x  cos y  cos x  y  1 .









32/   cos 600  x  cos 600  x  cos 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 33/   1  cos 2x  cos 4x  cos 6x .

34/   sin 2x  sin 4x  sin 6x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 35/   sin 5x  sin 6x  sin 7x  sin 8x .

36/   cos 5x  cos 8x  cos 9x  cos12x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 88.

Biến đổi thành tích các biểu thức sau đây 1/

1  cos x  cos 2x  cos 3x .

2/

sin x  sin 3x  sin 7x  sin 5x .

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................................................................................................................................................... 3/

sin x  sin 2x  sin 5x  sin 8x .

4/

cos 7x  sin 3x  sin 2x  cos 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

cos 9x  cos 7x  cos 3x  cos x .

6/

cos10x  cos 8x  cos 6x  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

sin 350  cos 400  sin 550  cos 200 .

8/

sin 570  sin 590  sin 930  sin 610 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

sin 470  sin 610  sin110  sin 250 .

10/ cos 5x  3 cos 7x  3 cos 9x  cos 11x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ 1  sin x  cos 5x  sin 7x  2 cos2

3x . 2

12/ sin 3x  sin x  sin2x  2 1  cos x cos x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ sin x  sin 2x  sin 3x  1  cos x  cos 2x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 14/ 1  sin x  cos 3x  cos x  sin 2x  cos 2x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 15/ sin x  sin 2x  sin 3x  cos x  cos 2x  cos 3x . ................................................................................................................................................... 90 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... Bài 89.

Biến đổi thành tích các biểu thức sau đây 1/

cos2 x  cos2 2x  cos2 3x  1 .

2/

sin2 x  sin2 2x  sin2 3x 

3 . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

sin 2 3x  sin 2 2x  sin2 x .

4/

sin 2 x  cos2 2x  cos2 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

sin2 x  2 sin2 2x  sin2 3x .

6/

sin2 4x  cos2 6x  sin 10, 5  10x  .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

sin2 3x  cos2 4x  sin2 5x  cos2 6x .

8/

3 cos2 x  cos2 2x  cos2 3x  cos2 4x  . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

 5 9x cos2 x  cos2 2x  cos2 3x  cos2 4x  2 . 10/ cos3x  sin7x 2sin2     2cos2 .  4 2  2

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   cos3xcos  5x      2  11/ sin  2x cot3x  sin 2x  2 cos5x . 12/ 2sin2x  cos  3x  . cos5x 2  2 

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Bài 90.

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

Chứng minh 1/

sin 750  cos 750 

6 . 2

2/

cos120  cos 480  sin 180 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

sin 650  sin 550  3 cos 50 .

4/

tan 2670  tan 930  0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

cos 850  cos 350  cos250  0 .

6/

tan 90  tan270  tan 630  tan 810  4 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

cos 240  cos 48 0  cos 840  cos120 

1 . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

tan 90  tan150  cot150  cot90  tan 270  cot270 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

1 1 cos x  cos 3x  cos 5x  8 sin2 x cos3 x . 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 10/ sin x 1  2 cos 2x  2 cos 4x  2 cos 6x  sin 7x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 92 | THBTN – CA

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11/ 1  4 cos x  6 sin 2x  4 sin x  16 sin 2x sin4

x . 2

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................



 



12/ 8 sin2 x sin x  600 sin x  600  cos 4x  cos 2x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/

2

sin x  cos x



 



 cos 4x  4 sin 2x sin x  150 cos x  150 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 91.

a b c Cho a  b  c . Chứng minh: sin a  sin b  sin c  4 cos cos sin . 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 92.

Tính các giá trị của biểu thức 1/

A  sin6

   cos6 . 24 24

................................................................................................................................................... 2/

B  tan2

   cot2 . 12 12

................................................................................................................................................... 3/

C  cos

 3 5 7 9  cos  cos  cos  cos . 11 11 11 11 11

................................................................................................................................................... ................................................................................................................................................... 4/

D  cos

2 4 6 8 10  cos  cos  cos  cos . 11 11 11 11 11

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................................................................................................................................................... 5/

E  cos 00  cos

 2 3 4 5 6  cos  cos  cos  cos  cos . 7 7 7 7 7 7

................................................................................................................................................... ................................................................................................................................................... 6/



F  sin 400  cos 100

  cos 40 2

0

 sin 100



2

 cos1400 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 93.

Trong ΔABC có ba góc làn lượt là A, B,C . Chứng minh rằng 1/

sin A  sin B  sin C  4 cos

A B C cos cos . 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

sin A  sin B  sin C  4 sin

A B C sin cos . 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

cos A  cos B  cos C  1  4 sin

A B C sin sin . 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

sin 2A  sin 2B  sin 2C  4 sin A sin B sin C .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

1  cos 2A  cos 2B  cos 2C  4 cos A cos B cos C .

................................................................................................................................................... ................................................................................................................................................... 94 | THBTN – CA

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................................................................................................................................................... 6/

sin2 A  sin2 B  sin2 C  2 1  cos A cos B cos C .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

cos2 A  cos2 B  cos2 C  1  2 cos A cos B cos C .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... CÔNG THỨC BIẾN ĐỔI TÍCH THÀNH TỔNG Bài 94.

Biến đổi thành tổng các biểu thức sau 1/

A  sin

 2 sin . 5 5

2/

B  sin 5x cos 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  sin

3  cos . 4 6

4/

D  sin

 7 cos . 12 12

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  sin x  y cos x  y .

6/



 



F  sin x  300 cos x  300 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G  2 sin x sin 2x sin 3x .

8/

H  8 cos x sin 2x sin 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

 I  sin x  

     sin x   cos 2x . 6   6 

10/ J  4 cos a  b cos b  c cos c  a  .

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 95.

Tính giá trị của biểu thức 1/

A  cos 750 cos150 .

................................................................................................................................................... 2/

B  sin

 5 sin . 12 12

................................................................................................................................................... 3/

C  sin

11 5 cos . 12 12

................................................................................................................................................... 4/

D  sin 200. sin 400. sin 800 .

................................................................................................................................................... 5/

x 5x E  sin .sin , khi x  600 . 4 4

................................................................................................................................................... 6/

F  sin 200 sin 400 sin 600 sin 800 .

................................................................................................................................................... ................................................................................................................................................... 7/

G  24.sin

 5 7 11 .sin .sin .sin . 24 24 24 24

................................................................................................................................................... ................................................................................................................................................... 8/

H  sin 50.sin150.sin 250.......sin 650 sin 750.sin 850 .

................................................................................................................................................... ................................................................................................................................................... 9/

I  sin2 100  cos 700 cos 500 .

................................................................................................................................................... ...................................................................................................................................................

1 10/ J  cos100 cos 500  cos 50 cos250  cos100 . 2 96 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... 11/ K  sin2 500  sin2 100  sin 500 sin100 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 12/ L  cos2 730  cos2 470  cos 730.cos 470 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ M  sin2 500  sin2 700  cos 500 cos 700 . ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

1 14/ N  cos100 cos 500  cos 50 cos250  sin100 . 2 ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................





15/ O  2 cos 220  cos 440 

sin 550 sin 110

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 16/ P 

sin 600 4

0

4

sin 15  sin 75

0

 3 sin 150 sin 750 .

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  1 cos 400  17/ Q  2 2 sin 100 2 sin 350   . 2 cos 50 sin 50  

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................................................................................................................................................... ................................................................................................................................................... 18/ R 

1 sin 10

0

 4 sin 700 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 19/ S 

1 sin 10

0



3 cos100

.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 20/ T  cos

 2 3  cos  cos . 7 7 7

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 21/ U  cos

2 4 5 2 4 6 8  cos  cos  cos  cos  cos   cos . 7 7 7 5 5 5 5

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 22/ V  cos

2 4 6 8  cos  cos  cos . 9 9 9 9

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 96.

Rút gọn biểu thức A  2 sin x cos x  cos 3x  cos 5x  . Từ đó suy ra giá trị của biểu thức B  cos

 3 5  cos  cos . 7 7 7

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 98 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... Bài 97.

Rút gọn các biểu thức sau 1/

A  cos11x cos 3x  cos17x cos 9x .

................................................................................................................................................... ................................................................................................................................................... 2/

B  sin18x cos13x  sin 9x cos 4x .

................................................................................................................................................... ................................................................................................................................................... 3/

C  sin x sin 3x  sin 4x sin 8x .

................................................................................................................................................... ................................................................................................................................................... 4/

D  sin 2x sin 6x  cos x cos 3x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

E  cos 3x cos 6x  cos 4x cos 7x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/



 



F  sin x sin 600  x sin 600  x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/



 



G  8 cos x cos 600  x cos 600  x  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

1 H  cos x cos 2x sin 3x  sin12x . 4

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................................................................................................................................................... ................................................................................................................................................... 9/

I  4 sin 2x sin 5x sin 7x  sin 4x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 10/ J  sin 2x sin 6x cos 4x 

1 cos12x . 4

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

1 11/ K  sin x sin 2x sin 3x  sin 4x . 4 ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................     12/ L  4 cos x sin   x sin   x  cos 2x .  6   6 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ M  sin 4x sin 10x  sin 11x sin 3x  sin 7x sin x . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 14/ N  cos

x 3x cos  sin x sin 3x  sin 2x sin 3x . 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 15/ O  sin

7x 3x x 5x cos  sin cos  sin 2x cos 7x . 2 2 2 2

................................................................................................................................................... 100 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... 16/ P  cos 2x  cos 4x  cos 6x  4 cos x cos 2x cos 3x  2 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/ Q  sin x 1  2 cos 2x  2 cos 4x  2 cos 6x  . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 18/ R 

sin 5x  2 cos 2x  cos 4x . sin x

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................        19/ S  sin2 x    sin2 x    sin cos   2x  . 4  4  12    12 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 98.

Chứng minh các đẳng thức sau 1/

cos120  cos180  4 cos150 cos 210 cos 240  

3 1 . 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

tan 90  tan 630  tan 810  tan 270  4 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

tan 300  tan 400  tan 500  tan 600 

8 3 cos 200 . 3

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/



 



cos 2x  600 cos 2x  600 

1 1 cos 4x  . 2 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

2

sin x  cos x



 



 cos 4x  4 sin 2x sin x  150 cos x  150 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

    3 sin2 x  sin2   x  sin x sin   x   . 3  3  4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

sin a sin b  c  sin b sin c  a   sin c sin a  b  0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

cos2 x  2 cos a cos x cos a  x   cos2 a  x   sin2 a .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 9/

sin 8 x  cos8 x 

7 1 cos 2x  cos 6x . 8 8

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 10/ sin 8 x  cos8 x  102 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 11/ tan

3 2  4 tan  11 . 11 11

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 12/ tan 200 tan 400 tan 800  3 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 13/ 8 sin3 180  8 sin2 180  1 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 14/ sin 4

 3 5 6 3  sin4  sin 4  sin 4  . 16 16 16 16 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 15/ tan100 tan 250  tan 250 tan 550  tan 550 tan100  1 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 16/ tan 150 tan 250 tan 350 tan 850  1 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 17/ tan 200  tan 400  tan 80 0  3 3 . ................................................................................................................................................... ................................................................................................................................................... ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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................................................................................................................................................... 18/ tan 100  tan 50o  tan 600  tan 700  2 3 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 19/ tan 20 0  tan 400  tan 800  tan 60 0  8 sin 400 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 20/ tan 6 200  33 tan 4 20 0  27 tan 2 20 0  3  0 . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 99.

Tính các góc của ΔABC biết rằng 1/

BC 

 1 và sin B.sin C  . 3 2

ĐS: B 

   , C , A . 2 6 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

BC 

2 1 3 và sin B.cos C  . 3 4

ĐS: A 

 5  , B , C . 3 12 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 100. Chứng minh điều kiện cần và đủ để ΔABC vuông 1/

cos 2A  cos 2B  cos 2C  1 .

2/

tan 2A  tan 2B  tan 2C  0 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 104 | THBTN – CA

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................................................................................................................................................... 3/

b c a   . cos B cos C sin B.sin C

4/

cot

B ac  . 2 b

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 101. Chứng minh điều kiện cần và đủ để ΔABC cân: 1/

a tan A  b tan B  a  b tan

AB . 2/ 2

2 tan B  tan C  tan2 B. tan C .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

sin A  sin B 1   tan A  tan B . cos A  cos B 2

4/

cot

C 2 sin A.sin B  . 2 sin C

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 102. Chứng minh bất đẳng thức, từ đó suy ra điều kiện cần và đủ để ΔABC đều: 1/

sin A  sin B  sin C 

3 3 . 2

HD: Cộng sin

 vào VT. 3

................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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2/

cos A  cos B  cosC 

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

3 . 2

HD: Cộng cos

 vào VT. 3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

tan A  tan B  tan C  3 3 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

cos A.cos B.cos C 

1 . 8

HD: Biến đổi cos A.cos B.cos C 

1 về dạng hằng đẳng thức. 8

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 103. Cho cos   cos a cos b với a, b 

 a a b  k . Chứng minh: tan tan  tan2 . 2 2 2 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 104. Cho sin 2a  b  5 sin b . Chứng minh:

2 tan a  b tan a

 a    k 2  3 với  .   a  b   l 2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 105. Cho tan a  b  3 tan a . Chứng minh: sin 2a  2b  sin 2a  2 sin 2b . ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 106. Rút gọn các biểu thức sau 106 | THBTN – CA

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1/

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

A  cos   cos 3  cos 5  ...  cos 2n  1  .

ĐS: A 

sin 2n . 2 sin 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

B  sin

n  1  .  2 3  sin  sin  ...  sin n n n n

ĐS: B  cot

 . 2n

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/

C  cos

2n  1  .  3 5  cos  cos  ... cos n n n n

ĐS: C   cos

 . n

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

D

1 1 1    ...  , a  . ĐS: D  1  5 . cos a.cos 2a cos 2a.cos 3a cos 4a.cos 5a 5

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

  1  1  1  1   1  1  ... 1   . E  1      cos x  cos 2x  cos 3x   cos 2n 1 x 

ĐS: E 

tan 2n1 x . x tan 2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 6/

F

1 1 1 1    ...  . sin 2x sin 4x sin 8x sin 2n x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

G  sin3

x x x x  3sin 3  32 sin3  ......  3n1 sin 3 n . 3 3 3 3

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 107. Tìm phần nguyên của biểu thức: M  tan4

 3  3 2   tan 4  2 tan2  tan2  . 4 4 7 7  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Bài 108. Chứng minh các bất đẳng thức sau 1/

sin 3x cos x  cos 3x sin x  cos 2x  2 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

3 sin 3x  cos 2x cos x  sin 2x sin x  2 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3/



 



sin x sin x  600 sin x  600 

1 . 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4/

cos x cos 3x  sin 2x sin 4x  1 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5/

cos 2x cos x  sin x sin 3x  sin 2 x cos 3x  1 .

................................................................................................................................................... ...................................................................................................................................................

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................................................................................................................................................... Bài 109. Tìm giá trị lớn nhất và giá trị nhỏ nhất của các biểu thức sau 1/

y  3 sin x  cos x  2 .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2/

y  cos x 

1 , cos x

    x   2

  . 2 

................................................................................................................................................... ................................................................................................................................................... 3/

y  sin x cos x cos 2x cos 4x .

................................................................................................................................................... ................................................................................................................................................... 4/

 y  sin4 x  cos4 x, 0  x  

  . 2 

................................................................................................................................................... ................................................................................................................................................... 5/

y  sin x  2  sin2 x .

................................................................................................................................................... ................................................................................................................................................... 6/

y  cos4 x  sin4 x  cos2 x sin2 x .

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7/

y  sin4 4x  cos2 x 

5 cos2x . 4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 8/

y  2 sin6 x  2 cos6 x  sin 4 x  cos4 x  cos 2x .

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................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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BÀI TẬP TỔNG ÔN – BIẾN ĐỔI LƯỢNG GIÁC § 1. BIEÁN ÑOÅI LÖÔÏNG GIAÙC 1. Tính giaù trò cuûa bieåu thöùc löôïng giaùc BT 1. Tính giá trị biểu thức lượng giác khi đề bài cho sin: a)

2 3

Tính A  (1  3cos 2)  (2  3cos 2), biết sin   

ĐS: A 

14  9

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... b)

2 3

Tính A  sin4   cos4  , biết sin 2  

7 9

ĐS: A  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... c)

Cho

tan   3      và sin    Tính A  2 5 1  tan 2 

ĐS: A  

12  25

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... d)

Cho

 4     và sin    Tính A  sin 2(  ). 2 5

ĐS: A  

24  25

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................................................................................................................................................... ................................................................................................................................................... e)

Cho    

3 4 1  cot  và sin     Tính A   2 5 1  cot 

ĐS: A  7.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... f)

Cho 0   

 3 cot   tan  và sin    Tính A   2 5 cot   tan 

ĐS: A 

25  7

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... g)

1 3

Cho 90o    180o và sin    Tính A 

2 tan   3cot   1  tan   cot 

ĐS: A 

26  2 2  9

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... h)

 2



1 3

Cho    ;   có sin    Tính A  sin 2  cos 2. 

ĐS: A  

74 2  9

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 112 | THBTN – CA

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Cho góc  thỏa

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

 4 cos 2     và sin    Tính A   2 5 1  cos 

ĐS: A  

7  40

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... j)

Cho 0   

1  sin 2  cos 2  2 và sin    Tính A   sin   cos  2 3

ĐS: A 

2 5  3

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... k)



Tính A  cos   

  1   , biết rằng: sin   và 0     3  2 3

ĐS: A 

6 3  6

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... l)



Tính A  cos   

  12   , biết rằng: sin   và    . 4  13 2

ĐS: A 

7 2  26

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 4 6 1 m) Cho  thỏa tan   cot   0 và sin    Tính A  sin 2. ĐS: A    25 5 ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... n)

Cho  thỏa 0   

   1 và sin    Tính A  tan     4 2 2 

ĐS: A  2  3.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... o)

  3  Tính A  tan    , biết rằng: sin   và    . 

3

5

2

ĐS: A 

48  25 3  11

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... p)

Cho

 7   1     và sin(   )    Tính A  tan      2 3  2 

ĐS: A  2 2.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... q)

Cho sin  

sin   sin 2  sin 3 1   và     Tính A   cos   cos 2  cos 3 3 4 2

ĐS: A  

4 2  7

................................................................................................................................................... ................................................................................................................................................... 114 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... BT 2. Tính giá trị biểu thức lượng giác khi đề bài cho cos: a)

Cho

sin   4      và cos     Tính A  cot   1  cos  2 5

5 3

ĐS: A  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... b)

Cho

cot  3 2     2 , cos    Tính A  2 3 1  cot 2 

ĐS: A  

2 5  9

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... c)

Cho cos   

  3 3 và      Tính A  sin     3 5 2 

ĐS: A 

4  3 3  10

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... d)

Cho cos   

  3 3 và      Tính A  sin     6 5 2 

ĐS: A 

34 3  10

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... e)

Cho cos   

  4 3 và      Tính A  tan     4 5 2 

1 7

ĐS: A   

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... f)

Cho cos 2  

  4   và     Tính A  cos 2    4 5 4 2 

ĐS: A  

2  10

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... g)

Cho

3 5    2 và cos    Tính A  tan 2. 2 13

ĐS: A 

120  119

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... h)

Cho    

3 4  3 và cos     Tính A  sin  cos  2 5 2 2

ĐS: A 

39  50

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 116 | THBTN – CA

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

................................................................................................................................................... i)

 2

4 5



Cho     0 và cos    Tính A  sin   

     cos     4  4  

ĐS: A  

49  50

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... j)

 2

3 5

Cho     0, cos    Tính A  5  3tan   6  4cot . ĐS: A  4.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... k)

 2

Cho     0 và cos 3  1. Tính A  2 3 sin   8 cos3 .

ĐS: A  4.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... l)

1 3

1 4

Cho cos 4   Tính A  cos 6   sin 6   

ĐS: A  1.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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        13 m) Cho  2  Tính A  tan 2   2 tan a  1  4 cot   1 . ĐS: A    3 3 cos   5 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... BT 3. Tính giá trị biểu thức lượng giác khi đề bài cho tan và cotan a)

Cho tan   2 thỏa

 2 sin   3cos     . Tính A   2 5cos   7 sin 

ĐS: A  

1  19

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... b)

Cho tan   2. Tính A 

sin   sin   3cos 3  3

ĐS: A 

10  11

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... c)

Cho tan   2. Tính A 

sin   2cos 3   cos   2 sin 3 

4 7

ĐS: A  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 118 | THBTN – CA

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TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG d)

Cho tan   3. Tính A 

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

3sin   2 cos   5 sin 3   4 cos 3 

ĐS: A 

70  139

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... e)

Cho tan   2. Tính A 

8 cos 3   2 sin 3   cos   2 cos   sin 3 

3 2

ĐS: A   

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... f)

Cho tan   2. Tính A 

sin 2  cos 4  1

ĐS: A  

9  10

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... g)

Cho

     3    2 , tan     1. Tính A  cos     sin . ĐS: A        2 4 6 2  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... h)

tan   2    5   Cho   Tính A  sin 2   sin     sin   2  3      2    2   2

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ĐS: A 

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... i)



3 4

Cho tan     Tính A  2  cos 2     sin 2  

3   2 

ĐS: A 

64  25

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... j)

 5







Cho cot     2. Tính A  tan     4  2  

ĐS: A  3.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................       2  2  k) Cho   Tính A  sin     cos .      6   cot      3 3  

ĐS: A  

3  2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... BT 4. Tính biểu thức lượng giác khi đề bài cho dạng hỗn tạp:

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

90o   0  a) Cho   Tính A  sin  1  tan2  cos  1  cot2 . ĐS: A  2.  sincos  2 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... b) Cho 0   

 3 và      Tính A  1  tan    1  tan    4 4

ĐS: A  2.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  1  sin 4  cos 4   3    c) Cho 1  sin 4  cos 4  Tính A  2 sin     cos .  6    0    2 

ĐS: A 

3  2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

121 | THBTN

TÀI LIỆU HỌC TẬP TOÁN 10- CHẤT LƯỢNG d) Cho

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

cos x  cos 4 x  cos 7 x 1   Tính A  cos8 ? sin x  sin 4 x  sin 7 x 2

3 5

ĐS: A   

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   5 e) Cho sin   cos   ,    ;   Tính A  tan 2.  4 2  4

ĐS: A  

9 7  35

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... f)

Cho sin   cos  

5  và 0     Tính A  sin   cos . 4 2

ĐS: A  

3  2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

3 g) Cho 3sin 4   cos 4    Tính A  sin 4   3cos 4 . 4

7 4

ĐS: A  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 7 h) Cho 4 sin 4   3cos 4    Tính A  3sin 4   4 cos 4 . 4

ĐS: A 

7 57  A  4 28

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... i)

Cho    

3 và sin   2 cos   1. Tính A  2 tan   cot . 2

1 6

ĐS: A  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... j)

Cho tan   cot   4. Tính A  sin   cos .

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ĐS: A 

6  2

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 90 o    180o k) Cho   Tính A  tan   cot . tan   cot   1 

ĐS: A   5.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 2. Thu goïn bieåu thöùc löôïng giaùc  3

  3   x  tan   x  cot(2   x). ĐS: A  cos x.  2  2 

BT 5. Thu gọn: A  cos(   x)  2 sin 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... BT 6. Thu gọn các biểu thức sau đây: 124 | THBTN – CA

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TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG

a) P 

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

  , 0  x     2  sin x  cot x  cos x  1

2

2

ĐS: P  

sin x  cos 2 x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  7

  3   x  tan   x   2   2 

b) P  2 cos x  3cos(   x)  sin 

ĐS: P  cot x.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... c)

    11  3  P  cos(15  x)  sin  x    tan   x cot   x   2   2   2 

ĐS: P  1.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... d) P  cos(2700  x)  2 sin( x  4500 )  cos( x  9000 )  2 sin(2700  x). ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................      3  e) P  sin 2 x    sin 2  x    sin 2  x    sin 2 ( x   ). 

4



2



4

ĐS: P  2  sin 2 x.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... f) P  cos7 x.cos 5 x  3 sin 2 x  sin7 x.sin 5x.



ĐS: P  2 cos 2 x  

   3 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 







g) P  3 cos 2 x  sin 2x  2 sin  2 x    6



ĐS: P  2 2 sin2x  

   12 

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

h) P 

  sin 2 x  2 2 cos x  2 sin  x    3  4 2 sin x  3



ĐS: P  2 cos x  1.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 







1

i) P  cos   x  cos   x  sin 2 x. 4  4  2

1 2

ĐS: P  cos 2 x 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... j)

        P  sin  x   cos   x  sin   x cos  x      4  4  3  3   

ĐS: P  0.

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................   9   5   5   k) P  sin  x    cos   x  sin   x  cos   x  

3

4



4



 3



ĐS: P 

2 6  4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... l)

P  3 cos 2 x  5 sin 4 x  4 sin 2 x cos 2 x  cos 4 x

ĐS: P  2.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 







m) P  1  sin x  1  sin x ,  0  x    4

x 2

ĐS: P  2 cos 

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... n) P 

tan 2 x  1 cot x  cos 4 x cot 2 x  sin 4 x 2

ĐS: P  0.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... o) P 

sin 2 4 x  2 cos x  cos 3x  cos 5x

ĐS: P  4 sin2 x cos x.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... q) P 

cos 2 x  sin 4 x  cos 6 x  cos 2 x  sin 4 x  cos 6 x

ĐS: P 

2 sin 2 x  1  2 sin 2 x  1

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... r)

P

1  cos x  cos 2x  cos 3 x  2 cos 2 x  cos x  1

ĐS: P  2 cos x.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... s) P 

2(sin 2 x  2 cos 2 x  1)  cos x  sin x  cos 3x  sin 3 x

ĐS: P 

1  sin x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... t)

    P  sin 2 x  sin 2   x   sin x sin   x   3 3    

3 4

ĐS: P  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 130 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 















w) P  tan 2 x tan   x  tan 2 x tan   x  tan   x tan   x  ĐS: P  1. 6  3  3  6  ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 3. Chöùng minh ñaúng thöùc löôïng giaùc BT 7. Chứng minh các đẳng thức lượng giác sau: a)



1  cos x sin x   sin x 1  cos x







b) sin   x  sin   x  2 sin x. 4  4 

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  1  sin  1  sin   2 c)     4 tan . 1  sin    1  sin  2

d)

 1  cos   (1  cos )2   1  2 cot . 2  sin   sin  

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... e) tan x 

cos x 1   1  sin x cos x

3 4

1 4

f) sin 4  cos 4 x   cos 4 x.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 5 8

3 8

g) sin 6 x  cos6 x   cos 4 x.

h) tan x  cot x 

2  sin 2 x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... i)

  1 1 1 1 1 1 x    cos x  cos , x  0;    2  2 2 2 2 2 2 8

j)

sin 2 2 x  4 sin 2 x  tan 4 x. sin 2 2 x  4 sin 2 x  4

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 132 | THBTN – CA

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... k)

1  cos x  cos 2 x  cos 3x  2 cos x. 2 cos 2 x  cos x  1

l)

cos x  sin x cos x  sin x   2 tan 2x. cos x  sin x cos x  sin x

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... m)

sin 2 x cos x x  tan  (1  cos 2 x)(1  cos x) 2

n)

1  2 sin 2 x  1.   2      2 tan   x cos   x  4  4  

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 







o) tan   x  tan   x  2 tan 2 x. 4  4 









p) tan x  tan   x  tan   x  tan 3 x. 3  3 

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................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... 

q) tan x  tan x  

   2    tan  x    3 tan 3x. 3  3  

1 2

r) cos 3 x sin x  sin 3 x cos x  sin 4 x.

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................  3



 2  3  x    3  2

s) cos 2 x  cos 2   x   cos 2  

t) sin 75o  cos75o 

6  2

................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................

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MỘT SỐ BÀI TẬP TRẮC NGHIỆM – RÈN LUYỆN Câu 1.

Theo định nghĩa trong sách giáo khoa, A. Mỗi đường tròn là một đường tròn lượng giác. B. Mỗi đường tròn có bán kính R  1 là một đường tròn lượng giác. C. Mỗi đường tròn có bán kính R  1 , tâm trùng với gốc tọa độ là một đường tròn lượng giác. D. Mỗi đường tròn định hướng có bán kính R  1 , tâm trùng với gốc tọa độ là một đường tròn lượng giác. Hướng dẫn giải Lý thuyết : sách giáo khoa: Đường tròn lượng giác là đường tròn định hướng có tâm O, bán kính R  1 .

Câu 2.

Cho biết câu nào sai trong số các câu sau đây? Theo định nghĩa trong sách giáo khoa trên đường tròn lượng giác  với M , N thuộc đường tròn đều là góc lượng giác. A. Mỗi góc MON  với M , N thuộc đường tròn đều là góc lượng giác và có phân biệt B. Mỗi góc MON

điểm M là điểm đầu, N là điểm cuối đều là góc lượng giác.  với M , N thuộc đường tròn đều là góc lượng giác và có phân biệt C. Mỗi góc MON

tia đầu OM , tia cuối ON là điểm cuối đều là góc lượng giác.  với A 1;0  và N thuộc đường tròn đều là góc lượng giác D. Mỗi góc MON

Hướng dẫn giải Góc lượng giác tạo bởi cung lượng giác nên đáp án A sai Câu 3.

Trên đường tròn cung có số đo 1 rad là? A. Cung có độ dài bằng 1

B. Cung tương ứng với góc ở tâm 600

C. Cung có độ dài bằng đường kính

D. Cung có độ dài bằng nửa đường kính

Hướng dẫn giải Theo khái niệm trong sgk chọn đáp án D Câu 4.

Theo sách giáo khoa ta có: A. 1 rad  10

B. 1 rad  600

C. 1 rad  1800

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 180  D. 1 rad      

0

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Hướng dẫn giải Xem lại sách giáo khoa Đại Số 10 trang 136 chọn đáp án D Câu 5.

Theo sách giáo khoa ta có: 0

0

A.  rad  1

B.  rad  60

C.  rad  180

0

 180  D.  rad      

0

Hướng dẫn giải 0

0

 180   180  0 Do 1 rad      .   180 chọn đáp án C.      

Câu 6.

Trên đường tròn bán kính r=5, độ dài của cung đo A. l 

 . 8

B. l 

r 8

C. l 

 là: 8

5 8

D. kết quả khác A

Hướng dẫn giải Độ dài cung AB có số đo cung AB bằng n độ:  l  r.n  5. 8

Câu 7.

O

l

n B

Trên đường tròn bán kính r=15, độ dài của cung có số đo 500 là: 180 A. l  750. B. l  15.  C. l 

15 . 180

D. l  15.

180 .50 

Hướng dẫn giải A

Nôi dung hướng dẫn giải .r.n 0 15.500 l  1800 180

O

l

n B

Câu 8.

Trên đường tròn lượng giác, A. cung lượng giác có điểm đầu A và điểm cuối B chỉ có một số đo. B. cung lượng giác có điểm đầu A và điểm cuối B chỉ có hai số đo sao cho tổng của chúng bằng 2. C. cung lượng giác có điểm đầu A và điểm cuối B chỉ có hai số đo hơn kém nhau 2. D. cung lượng giác có điểm đầu A và điểm cuối B có vô số đo sai khác nhau 2.

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Hướng dẫn giải Nôi dung hướng dẫn giải Đáp án D Câu 9.

Trên đường tròn lượng giác, A. góc lượng giác có điểm đầu A điểm cuối B chỉ có một số đo. B. góc lượng giác có điểm đầu A điểm cuối B chỉ có hai số đo sao cho tổng của chúng bằng 2 . C. góc lượng giác có điểm đầu A điểm cuối B chỉ có hai số đo hơn hoặc kém nhau 2 .

D. góc lượng giác có điểm đầu A điểm cuối B có vô số số đo sai khác nhau 2 . Câu 10. Trên đường tròn lượng giác với điểm gốc A , cung lượng giác có số đo 550 có điểm đầu A xác định A. chỉ có một điểm cuối M .

B. đúng hai điểm cuối M .

C. đúng 4 điểm cuối M .

D. vô số điểm cuối M . Hướng dẫn giải

Vì cung lượng giác có số đo xác định, điểm đầu A xác định nên chỉ có một điểm cuối M . Câu 11. Trên đường tròn lượng giác với điểm gốc là A , cung AN , có điểm đầu là A , điểm cuối là N A. chỉ có một số đo.

B. có đúng hai số đo.

C. có đúng 4 số đo.

D. có vô số số đo. Hướng dẫn giải

Trên đường tròn lượng giác với điểm gốc là A , cung AN , có điểm đầu là A , điểm cuối là N có vô số số đo, các số đo này sai khác nhau 2 . Câu 12. Lục giác ABCDEF nội tiếp đường tròn lượng giác có gốc là A , các đỉnh lấy theo thứ tự đó và các điểm B, C có tung độ dương. Khi đó góc lượng giác có tia đầu OA , tia cuối OC bằng: A. 1200 .

B. 2400 .

C. 1200 hoặc 2400 .

D.

1200  k 360 0 , k   .

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BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

Hướng dẫn giải

Theo bài ra ta có  AOC  120o nên góc lượng giác có tia đầu OA , tia cuối OC có số đo bằng 1200  k 360 0 , k   . Chọn D. Câu 13. Trên đường tròn lượng giác có điểm gốc là A . Điểm M thuộc đường tròn sao cho cung lượng giác AM có số đo 450 . Gọi N là điểm đối xứng với M qua trục Ox , số đo cung lượng giác AN bằng: A. 450 .

B. 3150 .

C. 450 hoặc 3150 .

D. 450  k 3600 , k   . Hướng dẫn giải

Vì số đo cung AM bằng 450 nên  AOM  450 , N là điểm đối xứng với M qua trục Ox nên  AON  450 . Do đó số đo cung AN bằng 45o nên số đo cung lượng giác AN có số đo là 45o  k 360o , k   . Chọn D. Câu 14. Trên đường tròn với điểm gốc là A . Điểm M thuộc đường tròn sao cho cung lượng giác AM có số đo 600 . Gọi N là điểm đối xứng với điểm M qua trục Oy , số đo cung AN là: A. 120o .

B. 2400

C. 1200 hoặc 2400 .

D. 1200  k 360 0 , k   . Hướng dẫn giải

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  600 nên  Ta có  AON  600 , MON AON  1200 . Khi đó số đo cung AN bằng 1200 .

Câu 15. Trên đường tròn lượng giác vớ điểm gốc là A . Điểm M thuộc đường tròn sao cho cung lượng giác AM có số đo 750 . Gọi N là điểm đối xứng với điểm M qua gốc tọa độ O , số đo cung lượng giác AN bằng: A. 2550 .

B. 1050 .

C. 1050 hoặc 2550 .

D. 1050  k 3600 , k   . Hướng dẫn giải

Ta có

 AOM  750 ,

  1800 MON

nên cung lượng giác

AN

có số đo bằng

1050  k 360 0 , k  

Câu 16. Trên đường tròn lượng giác với điểm gốc là A , điểm M thuộc đường tròn sao cho



cung lượng giác AM có số đo 135O . Gọi N là điểm đối xứng của M qua trục Oy ,

 số đo cung AN là A. 45O .

B. 315O .

C. 45O hoặc 315O .

D.

45O  k 360O ,

k 

Hướng dẫn giải Vẽ sơ bộ hình biểu diễn và xác định vị trí của N , từ đó chọn đáp án D. 45O  k 360O , k   . ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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3 và góc x thỏa mãn 90O  x  180O . Khi đó 4 4 3 3 A. cot x  . B. cosx  . C. sin x  . 3 5 5

Câu 17. Cho tan x 

4 . 5

D. sin x 

Hướng dẫn giải sin x  0 3  Do 90  x  180 nên  cosx  0 . Ta thấy sin x   0 nên chọn đáp án C. 5 cot x  0  O

O

3 và góc x thỏa mãn 90O  x  180O . Khi đó 5 4 4 3 A. cot x  . B. cosx  . C. tan x  . 3 5 4

Câu 18. Cho sin x 

D. cosx 

4 . 5

Hướng dẫn giải  tan x  0 4  Do 90  x  180 nên  cosx  0 . Ta thấy cosx   0 nên chọn đáp án D. 5 cot x  0  O

O

4 và góc x thỏa mãn 90O  x  180O . Khi đó 5 4 3 4 A. cot x  . B. sin x  . C. tan x  . 3 5 5

Câu 19. Cho cosx 

D. sinx 

3 . 5

Hướng dẫn giải  tan x  0 3  Do 90  x  180 nên  sinx  0 . Ta thấy sinx   0 nên chọn đáp án B. 5 cot x  0  O

O

3 và góc x thỏa mãn 0O  x  90O . Khi đó 4 4 3 4 A. tan x  . B. cosx  . C. sin x  . 3 5 5

Câu 20. Cho cotx 

D. sinx 

4 . 5

Hướng dẫn giải  tan x  0 4  Do 0  x  90 nên  sinx  0 . Ta thấy sinx   0 nên chọn đáp án C. 5  cosx  0  O

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O

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Câu 21. Gọi M  sin 2 10O  sin 2 20O  sin 2 30O  sin 2 40O  sin 2 50O  sin 2 60O  sin 2 70O  sin 2 80O thì

M bằng A. 0 .

B. 2 .

C. 4 .

D. 8 .

Hướng dẫn giải Do 10O  80O  20O  70O  30O  60O  40O  50O  90O nên các cung lượng giác tương ứng đôi một phụ nhau. Áp dụng công thức sin(90O  x)  cosx , ta được

M   sin 2 10O  cos 210O    sin 2 20O  cos 2 20O    sin 2 30O  cos 2 30O    sin 2 40O  cos 2 40O   1  1  1  1

Chọn đáp án C. Câu 22. Gọi M  cos 210O  cos2 20O  cos 2 30O  cos 2 40O  cos 2 50O  cos 2 60O  cos 2 70O  cos 2 80O thì

M bằng A. 0 .

B. 2 .

C. 4 .

D. 8 .

Hướng dẫn giải Do 10O  80O  20O  70O  30O  60O  40O  50O  90O nên các cung lượng giác tương ứng đôi một phụ nhau. Áp dụng công thức sin(90O  x)  cosx , ta được M   cos 210O  sin 210O    cos 2 20O  sin 2 20O    cos 2 30O  sin 2 30O    cos 2 40O  sin 2 40O   1111  4

Chọn đáp án C. Câu 23. Giá trị của biểu thức

M  cos2 230  cos2 270  cos 2 330  cos2 370  cos 2 430  cos 2 47 0  cos2 530  cos 2 570   cos 2 630  cos2 67 0 bằng: A. 1 .

B. 5 .

C. 10 .

D. Một kết quả khác với các kết quả đã

nêu. Hướng dẫn giải Áp dụng công thức cos   sin  90    , cos2   sin 2   1 ta có: 0

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M  cos 2 230  cos 2 27 0  cos 2 330  cos 2 370  cos 2 430  cos 2 470  cos 2 530  cos 2 57 0   cos 2 630  cos 2 670  sin 2 670  sin 2 630  sin 2 57 0  sin 2 530  sin 2 470  cos 2 47 0  cos 2 530  cos 2 57 0   cos 2 630  cos 2 670

   sin

  

 

 



 sin 2 670  cos 2 67 0  sin 2 630  cos 2 630  sin 2 570  cos 2 57 0  sin 2 530  cos 2 530  2

470  cos 2 470

M  5 Vậy chọn đáp án B .

Câu 24. Giá trị của biểu thức

M  cos 2 100  cos 2 200  cos2 300  cos2 400  cos 2 500  cos2 600  cos 2 700  cos 2 800   cos2 900  cos2 1000  cos 2 1100  cos 2 1200  cos 2 1300  cos 2 1400  cos2 1500  cos2 1600   cos2 1700  cos 2 1800 bằng: A. 0 .

B. 8 .

C. 9 .

D. 18 .

Hướng dẫn giải Áp dụng công thức cos   cos 1800    , cos2   sin 2   1 ta có: M  cos 2 100  cos2 200  cos 2 300  cos 2 400  cos2 500  cos2 600  cos2 700  cos2 800   cos 2 900  cos2 1000  cos 2 1100  cos 2 1200  cos 2 1300  cos 2 1400  cos2 1500  cos2 1600   cos 2 1700  cos 2 1800  cos 2 100  cos2 200  cos2 300  cos2 400  cos 2 500  cos 2 600  cos 2 700  cos2 800   cos 2 900  cos2 800  cos 2 700  cos2 600  cos2 500  cos2 400  cos 2 300  cos 2 200   cos 2 100  cos 2 900

  2  sin

 2 cos 2 100  cos 2 200  cos 2 300  cos 2 400  cos 2 500  cos 2 600  cos 2 700  cos 2 800  cos 2 900 2

800  sin 2 700  sin 2 600  sin 2 500  cos 2 500  cos 2 600  cos 2 700  cos 2 800  cos2 900



8 Vậy chọn đáp án B . Câu 25. Giá trị của biểu thức M  A.

2 . 7

B.

tan 2 300  sin 2 600  cos 2 450 bằng: cot 2 1200  cos 2 1500

1 . 7

C.

5 6 . 6 3

D.

7 . 13

Hướng dẫn giải

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

TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC 2

2

2

 3  3  2       tan 2 300  sin 2 600  cos 2 450  3   2   2  7 Ta có: M    2 2 2 0 2 0 cot 120  cos 150 13  3  3         3   2 

Vậy chọn đáp án D . Câu 26. Biết tan x  2 , giá trị của biểu thức M 

4 A.  . 9

B.

4 . 19

3sin x  2cos x bằng: 5cos x  7sin x 4 C.  . 19

D.

4 . 9

Hướng dẫn giải Cách 1: sin x 2 3.2  2 4 cos x Chia cả tử và mẫu của M cho cosx ta có: M    . sin x 5  7.2 19 57 cos x 3

Vậy chọn đáp án B . Cách 2: Ta có: tan x  2 

M

sin x  2  sin x  2cos x , thay sin x  2 cos x vào M : cos x

3.2 cos x  2cos x 4cos x 4   . 5cos x  7.2 cos x 19 cos x 19

Vậy chọn đáp án B . Câu 27. Biết tan x  A. 

8 . 13

1 2sin 2 x  3sin x.cos x  4 cos 2 x , giá trị của biểu thức M  bằng: 2 5 cos 2 x  sin 2 x 2 2 8 B. . C.  . D.  . 19 19 19 Hướng dẫn giải

Cách 1: Chia cả tử và mẫu của M cho cos2 x ta có:

2 M

sin 2 x sin x.cos x 1 1 3  4 2.  3.  4 2 2 8 cos x cos x 2  4  2 1 sin x 19 5 5 2 4 cos x

Vậy chọn đáp án D .

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Cách 2: Ta có: tan x 

M

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

1 sin x 1    cos x  2sin x , thay cos x  2 sin x vào M : 2 cos x 2

2sin 2 x  3sin x.2sin x  4.  2sin x  2

5.  2sin x   sin 2 x

2



8sin 2 x 8  2 19sin x 19

Vậy chọn đáp án D . Câu 28. Biết A, B, C là các góc của tam giác ABC , mệnh đề nào sau đây đúng: A. sin  A  C    sin B .

B. cos  A  C    cos B .

C. tan  A  C   tan B .

D. cot  A  C   cot B . Hướng dẫn giải

Ta có:

sin  A  C   sin   B   sin B cos  A  C   cos    B    cos B tan  A  C   tan   B    tan B cot  A  C   cot   B    cot B Vậy chọn đáp án B . Câu 29. Biết A, B, C là các góc của tam giác ABC , mệnh đề nào sau đây đúng: A. sin  A  C    sin B .

B. cos  A  C   cos B .

C. tan  A  C    tan B .

D. cot  A  C   cot B . Hướng dẫn giải

Ta có:

sin  A  C   sin   B   sin B cos  A  C   cos    B    cos B tan  A  C   tan   B    tan B cot  A  C   cot   B    cot B Vậy chọn đáp án C . Câu 30. Biết A, B , C là các góc của tam giác ABC , khi đó 144 | THBTN – CA

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A. sin C   sin  A  B  . B. cos C  cos  A  B  . C. tan C  tan  A  B  .

D. cot C   cot  A  B  . Hướng dẫn giải:

Vì A, B , C là các góc của tam giác ABC nên A  B  C  180o  C  180o   A  B  . Do đó C và  A  B  là 2 góc bù nhau

 sin C  sin  A  B  ; cos C   cos  a  b  ; tan C   tan  A  B  ; cot C  cot  A  B  . Câu 31. Biết A, B , C là các góc của tam giác ABC , khi đó A. sin C  sin  A  B  .

B. cos C  cos  A  B  .

C. tan C  tan  A  B  .

D. cot C   cot  A  B  . Hướng dẫn giải:

Vì A, B , C là các góc của tam giác ABC nên A  B  C  180o  C  180o   A  B  . Do đó C và  A  B  là 2 góc bù nhau

 sin C  sin  A  B  ; cos C   cos  a  b  ; tan C   tan  A  B  ; cot C  cot  A  B  . Câu 32. Biết A, B , C là các góc của tam giác ABC , khi đó

C  A B  A. sin    sin . 2  2 

C  A B  B. sin    cos . 2  2 

C  A B  C. tan    tan . 2  2 

C  A B  D. cot    cot . 2  2  Hướng dẫn giải:

Vì A, B , C là các góc của tam giác ABC nên A  B  C  180o  C  180o   A  B  

C A B C A B và là 2 góc phụ nhau.  90o  . Do đó 2 2 2 2

 sin

C A B C A B C A B C A B  cos ; cos  sin ; tan  cot ; cot  tan . 2 2 2 2 2 2 2 2

Câu 33. Biết A, B , C là các góc của tam giác ABC , khi đó

C  A B  A. cos    cos . 2  2 

C  A B  B. cos     cos . 2  2 

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C  A B  C. tan    cot . 2  2 

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

C  A B  D. cot    cot . 2  2  Hướng dẫn giải:

Vì A, B , C là các góc của tam giác ABC nên A  B  C  180o  C  180o   A  B  

C A B C A B và là 2 góc phụ nhau.  90o  . Do đó 2 2 2 2

 sin

C A B C A B C A B C A B  cos ; cos  sin ; tan  cot ; cot  tan . 2 2 2 2 2 2 2 2

Câu 34. Biết A, B , C là các góc của tam giác ABC , khi đó

C C  A B   A B  A. tan    tan . B. tan     tan . 2 2  2   2  C  A B  C. tan    cot . 2  2 

C  A B  D. tan     cot . 2  2  Hướng dẫn giải:

Vì A, B , C là các góc của tam giác ABC nên A  B  C  180o  C  180o   A  B  

C A B C A B và là 2 góc phụ nhau.  90o  . Do đó 2 2 2 2

 sin

C A B C A B C A B C A B  cos ; cos  sin ; tan  cot ; cot  tan . 2 2 2 2 2 2 2 2

Câu 35. Biết A, B , C là các góc của tam giác ABC , khi đó

C  A B  A. sin    sin . 2  2 

C  A B  B. sin     sin . 2  2 

C  A B  C. sin    cos . 2  2 

C  A B  D. sin     cos . 2  2  Hướng dẫn giải:

Vì A, B , C là các góc của tam giác ABC nên A  B  C  180o  C  180o   A  B  

C A B C A B và là 2 góc phụ nhau.  90o  . Do đó 2 2 2 2

 sin

C A B C A B C A B C A B  cos ; cos  sin ; tan  cot ; cot  tan . 2 2 2 2 2 2 2 2

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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

A. sin x  cos x  1.

B. sin 2 x  cos 2 x  1.

C. sin 3 x  cos 3 x  1.

D. sin 4 x  cos 4 x  1.

Câu 37. Với góc x bất kì. Khẳng định nào sau đây là khẳng định đúng?

 

 

A. sin 2 x  cos2 2 x  1 .

B. sin x 2  cos x 2  1 .

C. sin 2 x  cos 2 180  x   1 .

D. sin 2 x  cos 2 180  x   1 Hướng dẫn giải



Dựa vào công thức lượng giác cơ bản và cung liên quan đặc biệt.



Do cos 2 180  x   cos 2 x nên sin 2 x  cos 2 180  x   sin 2 x  cos 2 x  1 .



Chọn C.

Câu 38. Cho M  tan10.tan 20.tan 30.tan 40.tan 50.tan 60. tan 70.tan 80 . Giá trị của M bằng A. M  0 .

B. M  1 .

C. M  4 .

D. M  8 .

Hướng dẫn giải 

Ta có: tan x.tan  90  x   tan x.cot x  1 .



Vậy M  1 .



Chọn B.

Câu 39. Biết tan x  2 và M  A. M  1 .

2sin x  3cos x . Giá trị của M bằng 4sin x  7 cos x 1 1 B. M   C. M    15 15

2 D. M    9

Hướng dẫn giải   

sin x  sin x  tan x.cos x . cos x 2 tan x  3 1 Suy ra: M    4 tan x  7 15 Ta có: tan x 

Chọn B.

Câu 40. Biết tan x  2 và M  A. M 

9  13

2sin 2 x  3sin x.cos x  4 cos 2 x  Giá trị của M bằng 5sin 2 x  6 cos 2 x 9 9 24 B. M   C. M    D. M   65 65 29 Hướng dẫn giải



Ta có: tan x 

sin x  sin x  tan x.cos x . cos x



Suy ra: M 

2 tan 2 x  3 tan x  4 9   13 5 tan 2 x  6

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

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

Chọn A.

Câu 41. Biết tan x  3 và M  A. M 

31  47

2sin 2 x  3sin x.cos x  4 cos 2 x  Giá trị của M bằng 5 tan 2 x  6cot 2 x 93 93 31 B. M   C. M   D. M   137 1370 51 Hướng dẫn giải

sin x 1 1  sin x  tan x.cos x ; cos 2 x  và cot x   2 cos x tan x tan x  1



Ta có: tan x 



 2 tan Suy ra: M 

2

5 tan 2 x 





x  3 tan x  4 cos 2 x 6 tan 2 x



93  1370

Chọn C. 2

2

Câu 42. Cho M   sin x  cos x    sin x  cos x  . Biểu thức nào sau đây là biểu thức rút gọn của M ? A. M  1 .

B. M  2 .

C. M  4 .

D. M  4 sin x.cos x .

Hướng dẫn giải 

Ta có:

 sin x  cos x 2  sin 2 x  cos 2 x  2sin x.cos x  1  2sin x.cos x ;  sin x  cos x 2  sin 2 x  cos 2 x  2sin x.cos x  1  2sin x.cos x 

Suy ra: M  2 .



Chọn B. 2

2

Câu 43. Cho M   sin x  cos x    sin x  cos x  . Biểu thức nào sau đây là biểu thức rút gọn của M ? A. M  2 .

B. M  4 .

C. M  2 sin x.cos x . D. M  4 sin x.cos x . Hướng dẫn giải



Ta có:

 sin x  cos x 2  sin 2 x  cos 2 x  2sin x.cos x  1  2sin x.cos x ;  sin x  cos x 2  sin 2 x  cos 2 x  2sin x.cos x  1  2sin x.cos x

Câu 31.



Suy ra: M  4 sin x.cos x .



Chọn D. 2

Gọi M   tan x  cot x  , ta có

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A. M  2 .

C. M 

B. M 

2 . sin x.cos2 x

1 . sin x.cos2 x 2

D. M  4 .

2

Hướng dẫn giải 2

2

M   tan x  cot x 

2

2

2 2 1  sin x cos   sin x  cos x            .  cos x sin x   cos x.sin x   cos x.sin x 

Chọn B. Câu 32. Cho tan x  cot x  m , gọi M  tan 3 x  cot 3 x . Khi đó A. M  m3 .

B. M  m3  3m .

C. M  m3  3m .

D. M  m  m 2  1 .

Hướng dẫn giải 3

M  tan 3 x  cot 3 x   tan x  cot x   3 tan x.cot x  tan x  cot x   m3  3m

Chọn C. Câu 33. Cho sin x  cos x  m , gọi M  sin x  cos x . Khi đó A. M  2  m .

B. M  2  m2 .

C. M  m2  2 .

D. M  2  m 2 .

Hướng dẫn giải 2

Ta có: M 2   sin x  cos x   sin 2 x  2 sin x.cos x  cos 2 x  1  2sin x.cos x . 2

2

Mặt khác: M 2   sin x  cos x    sin x  cos x   4 sin x.cos x  m 2  4 sin x.cos x . Suy ra: 1  2 sin x.cos x  m 2  4sin x.cos x  sin x.cos x 

m2  1 . 2

Do đó: M 2  2  m 2  M  2  m 2 . Chọn D. Câu 34. Cho M  5  2sin 2 x . Khi đó giá trị lớn nhất của M là A. 3 .

B. 5 .

C. 6 .

D. 7 .

Hướng dẫn giải Ta có: 0  sin 2 x  1, x    0  2 sin 2 x  2, x    5  5  2 sin 2 x  3, x   . Gía trị lớn nhất là 5 . Chọn B. ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

Câu 35. Giá trị lớn nhất của biểu thức M  7 cos2 x  2sin 2 x là A. 2 .

B. 5 .

C. 7 .

D. 16 .

Hướng dẫn giải M  7 1  sin 2 x   2sin 2 x  7  9 sin 2 x .

Ta có: 0  sin 2 x  1, x    0  9 sin 2 x  9, x    7  7  2sin 2 x  2, x   . Gía trị lớn nhất là 7 . Chọn C. Câu 36. Cho M  6cos2 x  5sin 2 x . Khi đó giá trị lớn nhất của M là A. 1 .

B. 5 .

C. 6 .

D. 11 .

Hướng dẫn giải M  6 cos 2 x  5sin 2 x  6 1  sin 2 x   5sin 2 x  6  sin 2 x .

Ta có: 0  sin 2 x  1, x    0   sin 2 x  1, x    6  6  sin 2 x  5, x   . Gía trị lớn nhất là 6 . Chọn C. Câu 37. Cho M  3sin x  4 cos x . Chọn khẳng định đúng. A. M  5 .

B. 5  M .

C. M  5 .

D. 5  M  5 .

Hướng dẫn giải 4 3 4 3  M  5  sin x  cos x   5sin  x    với  cos  ,  sin  . 5 5 5 5  Ta có: 1  sin  x     1, x    5  5sin  x     5, x   . Chọn D. Câu 38. Giá trị lớn nhất của M  sin 4 x  cos 4 x bằng : A. 1.

B. 2.

C. 3.

D. 4.

Hướng dẫn giải

1 Ta có M  sin 4 x  cos4 x  1  sin 2 2 x 2 1 1 1 1 Vì 0  sin 2 2 x  1    sin 2 2 x  0   1  sin 2 2 x  1 2 2 2 2 Nên giá trị lớn nhất là 1. 150 | THBTN – CA

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Chọn đáp án A. Câu 39. Giá trị lớn nhất của N  sin 4 x  cos 4 x bằng : A. 0.

B. 1.

C. 2.

D. 3.

Hướng dẫn giải 4

4

2

Ta có N  sin x  cos x  sin x  cos 2 x   cos 2 x Vì 1  cos 2 x  1  1   cos 2 x  1 Nên giá trị lớn nhất là 1. Chọn đáp án B. Câu 40. Giá trị lớn nhất của Q  sin 6 x  cos 6 x bằng : A. 1.

B. 2.

C. 3.

D. 6.

Hướng dẫn giải

3 Ta có Q  sin 6 x  cos6 x  1  sin 2 2 x 4 3 3 1 3 Vì 0  sin 2 2 x  1    sin 2 2 x  0   1  sin 2 2 x  1 4 4 4 4 Nên giá trị lớn nhất là 1. Chọn đáp án A. Câu 41. Giá trị lớn nhất của M  sin 6 x  cos6 x bằng: A. 0.

B. 1.

C. 2.

D. 3.

Hướng dẫn giải Ta có M  sin 6 x  cos 6 x  (sin 2 x  cos 2 x )(sin 4 x  sin 2 x cos 2 x  cos 4 x) 1   cos 2 x(1  sin 2 x.cos 2 x)   cos 2 x (1  sin 2 2 x) 4 3 1  3 1 3 1   cos 2 x   cos 2 2 x    cos 2 2 x    1 4 4 4 4  4 4 (do  cos 2 x  1)

Nên giá trị lớn nhất là 1. Chọn đáp án B. Câu 42. Giá trị của biểu thức P  3(sin 4 x  cos 4 x)  2(sin 6 x  cos 6 x) là : A. 1.

B. 0.

C. 1.

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Hướng dẫn giải Ta có P  3(sin 4 x  cos 4 x )  2(sin 6 x  cos 6 x)  3(1  2sin 2 x cos 2 x)  2(1  3sin 2 x cos 2 x)  1 Chọn đáp án C. Câu 43. Biểu thức thu gọn của M  tan 2 x  sin 2 x là: A. M  tan 2 x.

B. M  sin 2 x.

C. M  tan 2 x.sin 2 x .

D. M  1.

Hướng dẫn giải  1  sin 2 x Ta có M  tan 2 x  sin 2 x   sin 2 x  sin 2 x  2  1  sin 2 x. tan 2 x 2 cos x  cos x  Chọn đáp án C. Câu 44. Biểu thức thu gọn của M  cot 2 x  cos 2 x là: A. M  cot 2 x.

B. M  cos 2 x.

C. M  1

D. M  cot 2 x.cos2 x.

Hướng dẫn giải  1  cos 2 x Ta có M  cot 2 x  cos2 x   cos 2 x  cos 2 x  2  1  cos 2 x.cot 2 x 2 sin x  sin x  Chọn đáp án D. Câu 45. Nếu M  A. tan 4 x .

cos 2 x  sin 2 x  , ( x  k , k   ) thì M bằng 2 2 cot x  tan x 4

B. sin 2 2 x. .

C.

1 2 sin 2 x. . 2

D.

1 2 sin 2 x. 4

Hướng dẫn giải cos 2 x  sin 2 x cos 2 x  sin 2 x  cot 2 x  tan 2 x cos 4 x  sin 4 x sin 2 x.cos 2 x 1  sin 2 x.cos 2 x  sin 2 2 x. 4

M

Chọn D. Câu 46. Giá trị của M  cos20.cos40.cos80 là A.

1 . 16

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B.

1 . 8

C.

1 . 4

D. 1

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Hướng dẫn giải

sin 20.M  sin 20.cos 20.cos 40.cos80 1  sin 40.cos 40.cos80 2 1  sin 80.cos80 4 1  sin160 8 1  sin 20. 8

1 Suy ra: M  . 8 Chọn B. Câu 47. Nếu M  sin4 x  cos4 x thì M bằng A. 1  2sin2 x.cos2 x .

B. 1  sin 2 2x .

C. 1  sin2 2x .

1 D. 1  sin 2 2 x . 2 Hướng dẫn giải

M  sin 4 x  cos4 x  (sin 2 x  cos 2 x )2  2sin 2 x.cos2 x 1  1  sin 2 2 x. 2 Chọn D. Câu 48. Nếu M  sin 6 x  cos6 x thì M bằng A. 1  3sin2 x.cos2 x .

B. 1  3sin2 x .

3 C. 1  sin 2 2 x . 2

3 D. 1  sin 2 2 x 4 Hướng dẫn giải

M  sin 6 x  cos6 x  (sin 2 x )3  (cos2 x)3  (sin 2 x  cos2 x )(sin 4 x  cos4 x  sin 2 x.cos2 x ) 1 1  1  sin 2 2 x  sin 2 2 x 2 4 3  1  sin 2 2 x. 4 ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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Chọn D. Câu 49. Giá trị nhỏ nhất của M  sin4 x  cos4 x là A. 0 .

B.

1 . 4

C.

1 . 2

D. 1 .

Hướng dẫn giải

M  sin 4 x  cos4 x 1  1  sin 2 2 x 2 1  1 2 1  . 2 Dấu bằng xảy ra khi x 

   k , k  . Chọn C. 4 2

Câu 50. Giá trị nhỏ nhất của M  sin 6 x  cos6 x là A. 0 .

B.

1 . 4

C.

1 . 2

D. 1 .

Hướng dẫn giải

M  sin 6 x  cos6 x 3  1  sin 2 2 x 4 3  1 4 1  . 4 Dấu bằng xảy ra khi x 

   k , k  . Chọn B. 4 2

Câu 76. Đề sai, đã sửa lại. 1  tan 3 x   Câu 51. Cho biểu thức M  , ( x    k , x   k , k   ) , mệnh đề nào trong 3 (1  tan x ) 4 2

các mệnh đề sau đúng? A. M  1 .

B. M  1

C. M 

1 . 4

D.

1  M  1. 4

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Ta có: M 

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

1 t3 t2  t  1   ( M  1)t 2  (2 M  1)t  M  1  0 . (*) 3 2 (1  t ) t  2t  1

Với M  1 thì (*) có nghiệm t  0. Với M  1 để (*) có nghiệm khác 1 thì 0  (2 M  1) 2  4(M  1) 2  0  12 M  3  0 1 M  . 4

Và

( M  1)(1)2  (2 M  1)( 1)  ( 1)  1  0  M  4. Chọn C. Câu 52. Tính sin1050 ta được : A.

6 2 . 4

B. 

6 2 . 4

C.

6 2 . 4

D. 

6 2 . 4

D. 

6 2 . 4

Hướng dẫn giải Có: sin1050  sin(60 0  450 )  sin 600.cos 450  cos 600.sin 450  sin1050 

3 2 1 2 .  .  2 2 2 2

6 2 . Chọn: C. 4

Câu 53. Tính cos1050 ta được : A.

6 2 . 4

B. 

6 2 . 4

C.

6 2 . 4

Hướng dẫn giải Có: cos1050  cos(600  450 )  cos 600.cos 450  sin 600.sin 450

1 2 3 2 6 2  .  . Chọn: B.  cos1050  . 2 2 2 2 4 Câu 54. Tính tan1050 ta được : A. (2  3) .

B. 2  3 .

C. 2  3 .

D. (2  3) .

Hướng dẫn giải 6 2 6 2 sin105 4 Có: tan1050     (2  3) . Chọn: A. 0 cos105 6 2 6 2  4 0

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Hoặc có thể làm theo cách sau:

1 3 tan 450  tan 600 tan105  tan(45  60 )    (2  3) . 0 0 1  tan 45 tan 60 1 3 0

0

0

Câu 55. Tính sin1650 ta được : A.

6 2 . 4

B. 

6 2 . 4

C.

6 2 . 4

D. 

6 2 . 4

Hướng dẫn giải 0

Có: sin165  sin(180  15 )  sin150  sin(450  30 0 ) 0

0

 sin1050  sin 450.cos 300  sin 300.cos 450 

2 3 1 2 6 2 .  .  . Chọn: A. 2 2 2 2 4

Câu 56. Tính cos1650 ta được : A.

6 2 . 4

B. 

6 2 . 4

C.

6 2 . 4

D. 

6 2 . 4

Hướng dẫn giải Có: cos1650  cos(1800  150 )   cos150   cos(450  30 0 )  cos1650  (cos 450.cos 30 0  sin 30 0.sin 450 )  (

2 3 1 2 6 2 .  . ) 2 2 2 2 4

Chọn : D. Câu 57. Tính tan1650 ta được : A. (2  3) .

B. 2  3 .

C. 2  3 .

D. (2  3) .

Hướng dẫn giải 6 2 6 2 sin165 4 Có: tan1650     (2  3) . Chọn D. 0 cos165 6 2 6 2  4 0

Hoặc có thể làm theo cách sau: 1 tan135  tan 30 3  (2  3) . tan1650  tan(1350  30 0 )   1  tan1350 tan 300 1  (1). 1 3 0

0

1 

Câu 58. Tính M  cos100 cos 200 cos 400 cos 800 ta được M là : 1 1 1 A. M  cos100 . B. M  cos100 . C. M  cos100 . 16 2 4

1 D. M  cos100 . 8

Hướng dẫn giải

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Do sin100  0 nên: M 

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

16 sin100 cos100 cos 200 cos 40 0 cos 800  16 sin100

8sin 200 cos 200 cos 40 0 cos 80 0 16 sin100

M 

4 sin 400 cos 400 cos800 2sin 800 cos800 sin1600   16 sin100 16sin100 16 sin100

M 

sin 200 2 sin100 cos100 1   cos100 . 16 sin100 16 sin100 8

Chọn: D. Câu 59. Gọi M  cos 4 15o  sin 4 15o thì: A. M  1.

B. M 

3 . 2

1 C. M  . 4

D. M  0.

Hướng dẫn giải 2

2

M  cos 4 15o  sin 4 15o   cos 2 15o    sin 2 15o    cos2 15o  sin 2 15o  cos2 15o  sin 2 15o   cos 2 15o  sin 2 15o  cos  2.15o   cos 30o 

3 . 2

Câu 60. Gọi M  cos6 15o  sin 6 15o thì: A. M  1.

1 B. M  . 2

1 C. M  . 4

D. M 

15 3 . 32

Hướng dẫn giải Ta có:

cos 6   sin 6    cos 2   sin 2   cos 4   cos 2  .sin 2   sin 4   2  cos 2 .  cos 2   sin 2    cos 2  .sin 2      1   cos 2 . 1  sin 2 2   4 

3  1 1  15 3  1  . 1  .   . Vậy M  cos 30o. 1  sin 2 30o   32  4  2  4 4 Câu 61. Gọi M   cos 4 15o  sin 4 15o    cos 2 15o  sin 2 15o  thì: A. M  1.

1 B. M  . 2

1 C. M  . 4

D. M  0.

Hướng dẫn giải Ta có: M   cos 4 15o  sin 4 15o    cos 2 15o  sin 2 15o    cos 2 15o  sin 2 15o  cos 2 15o  sin 2 15o    cos 2 15o  sin 2 15o 

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  cos 2 15o  sin 2 15o    cos 2 15o  sin 2 15o   0.

Câu 62. Gọi M   cos 4 15o  sin 4 15o    cos 2 15o  sin 2 15o  thì:

1 B. M  . 2

A. M  3.

1 C. M  . 4

D. M  0.

Hướng dẫn giải Ta có: M   cos 15  sin 15    cos 2 15o  sin 2 15o  4

o

4

o

  cos 2 15o  sin 2 15o  cos 2 15o  sin 2 15o    cos 2 15o  sin 2 15o    cos 2 15o  sin 2 15o    cos 2 15o  sin 2 15o   cos 30o  cos 30o  3.

Câu 63. Gọi M  1  sin 2 x  cos 2 x thì: A. M  2 cos x.  sin x  cos x  .

B. M  cos x.  sin x  cos x  .

  C. M  2 cos x.cos  x   . 4 

  D. M  2 2 cos x.cos  x   . 4  Hướng dẫn giải

Ta có: M  1  sin 2 x  cos 2 x  1  sin 2 x   cos 2 x 2

  sin x  cos x    cos x  sin x  cos x  sin x 

  sin x  cos x  sin x  cos x  cos x  sin x 

    sin x  cos x  .2cos x  2 cos  x   .2 cos x 4  Câu 64. Gọi M  cos x  cos 2 x  cos 3 x thì: A. M  2 cos 2 x  cos x  1 .

1  B. M  4cos 2 x.   cos x  . 2 

x  x  C. M  2cos 2 x.cos    .cos    . 2 6 2 6

x  x  D. M  4cos 2 x.cos    .cos    . 2 6 2 6

Hướng dẫn giải Ta có: M  cos x  cos 2 x  cos 3x   cos x  cos 3x   cos 2 x

1   2cos 2 x.cos x  cos 2 x  cos 2 x  2cos x  1  2cos 2 x  cos x   2 

  x  x   2cos 2 x  cos x  cos   2 cos 2 x.2cos    cos    3  2 6 2 6 158 | THBTN – CA

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Câu 65. Gọi M  tan x  tan y thì: A. M  tan  x  y  .

B. M 

sin  x  y  . cos x.cos y

sin  x  y  . cos x.cos y

D. M 

tan x  tan y . 1  tan x.tan y

C. M 

Hướng dẫn giải Ta có: M  tan x  tan y 

sin x sin y sin x cos y  cos x sin y sin  x  y     cos x cos y cos x cos y cos x cos y

Câu 66. Gọi M  tan x  tan y thì: A. M  tan x  tan y .

B. M 

sin  x  y  . cos x.cos y

C. M 

sin  x  y  . cos x.cos y

D. M 

tan x  tan y 1  tan x.tan y

. Hướng dẫn giải Ta có: M  tan x  tan y 

sin x siny sin x.cos y  siny.cos x sin  x  y     . cos x cos y cos x.cos y cos x.cos y

Câu 67. Gọi M  cot x  cot y thì: A. M  cot  x  y  .

B. M 

sin  x  y  . sin x.siny

C. M 

sin  y  x  . sin x.sin y

D. M 

tan x  tan y 1  tan x.tan y

. Hướng dẫn giải Ta có: M  cot x  cot y 

cos x cos y cos x.siny  sin x.cosy sin  y  x     . sin x siny sin x.siny sin x.siny

Câu 68. Gọi M  cot x  cot y thì: A. M  cot  x  y  .

B. M 

sin  x  y  . sin x.siny

C. M 

sin  y  x  . sin x.siny

D. M 

cot y.cot x  1 cot y  cot x

. Hướng dẫn giải Ta có: M  cot x  cot y 

cos x cos y cos x.siny  sin x.cosy sin  x  y     . sin x siny sin x.siny sin x.siny

1 1 1   thì: 0 0 0 0 cos10 .cos 20 cos 20 .cos 30 cos 30 .cos 400 1 A. M  B. M  tan 400  tan 200 0 0 sin 20 .cos 40

Câu 69. Gọi M 

0

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C. M 

1 2 cos100.cos 400

BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

D. M có kết quả khác với các kết quả nêu

trên. Hướng dẫn giải

sin 200  cos 400  cos 200  sin 200 Ta có: sin 20 .M   cos100.cos 200 cos 200.cos 300.cos 400 0



2sin100.cos100 2.sin 200.cos 30 0.cos10 0 2sin100 2.sin 200.cos10 0    cos100.cos 200 cos 200.cos 300.cos 40 0 cos 200 cos 200.cos 400



2sin100.cos 400  2.sin 200.cos10 0 sin 500  sin 300  sin 300  sin100  cos 200.cos 400 cos 200.cos 400

1 sin 500  sin100 2.sin 300.cos 200    . 0 0 0 0 cos 20 .cos 40 cos 20 .cos 40 cos 400 M 

1 . sin 20 .cos 400 0

Câu 70. Cho A , B , C là các góc của tam giác ABC thì: A. sin 2 A  sin 2 B  sin 2C  4 cos A.cos B.cos C . B. sin 2 A  sin 2 B  sin 2C  4 cos A.cos B.cos C . C. sin 2 A  sin 2 B  sin 2C  4 sin A.sin B.sin C . D. sin 2 A  sin 2 B  sin 2C  4 sin A.sin B.sin C . Hướng dẫn giải Ta có: sin 2 A  sin 2 B  sin 2C   sin 2 A  sin 2 B   sin 2C  2sin  A  B  .cos  A  B   2sin C.cosC

 2sin C.cos  A  B   2sin C.cosC  2sin C. cos  A  B   cosC   4sin C.cos  A  B  C  .cos  A  B  C   4sin C.cos

ABC ABC .cos 2 2

     4sin C.cos   A  .cos   B   4sin C .sin A.sin B . 2  2  Câu 71. Cho A , B , C là các góc của tam giác ABC (không phải tam giác vuông) thì: A B C A. tan A  tan B  tan C  tan .tan .tan . 2 2 2

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B. tan A  tan B  tan C   tan

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

A B C .tan . tan . 2 2 2

C. tan A  tan B  tan C   tan A.tan B.tan C . D. tan A  tan B  tan C  tan A.tan B.tan C . Hướng dẫn giải Ta có: tan A  tan B  tan C   tan A  tan B   tan C 

sin  A  B  sin C  cos A.cos B cos C

  cos  A  B   cos A.cos B  sin A.sin B.sin C  sin C.   cos A.cos B.cos C   cos A.cos B.cos C  tan A.tan B. tan C .

Câu 72. Cho A , B , C là các góc của tam giác ABC (không phải tam giác vuông) thì: A B C A B C A. cot  cot  cot  cot .cot .cot . 2 2 2 2 2 2 B. cot

A B C A B C  cot  cot   cot .cot .cot . 2 2 2 2 2 2

C. cot

A B C  cot  cot  cot A.cot B.cot C . 2 2 2

D. cot

A B C  cot  cot   cot A.cot B.cot C 2 2 2 Hướng dẫn giải

Ta có:  A B sin    cos C A B C  A B C 2 2 2 cot  cot  cot   cot  cot   cot  2 2 2  2 2 2 sin A .sin B sin C 2 2 2 A B  A B C A B cos     sin .sin  sin .sin C 2 2 2 2 2 2 2  cos C .  cos . C A B C A B 2 sin .sin .sin 2 sin .sin .sin 2 2 2 2 2 2 C B A cos .cos .cos 2 2 2  cot A .cot B .cot C  C A B 2 2 2 sin .sin .sin 2 2 2

sin

Câu 73. Cho A, B, C là các góc của tam giác ABC thì tan A. 1 .

A B B C C A .tan  tan . tan  tan . tan  2 2 2 2 2 2

B. 1 .

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2

A B C  C.  tan .tan .tan  . 2 2 2 

D. Một kết quả khác các kết quả đã nêu

trên. Hướng dẫn giải A B B C C A tan . tan  tan . tan  tan .tan 2 2 2 2 2 2

Ta có:

 tan

B  A C C A .  tan  tan   tan .tan 2  2 2 2 2

 tan

B A C C A A C  .tan    . 1  tan .tan   tan .tan 2 2 2 2 2 2 2

 tan

B A C C A  B  .tan    .1  tan .tan   tan .tan 2 2 2 2 2 2 2 

 tan

B B  A C C A .cot . 1  tan .tan   tan .tan 2 2  2 2 2 2

A C C A   1. 1  tan .tan   tan .tan  1. 2 2 2 2 

Chọn A. Câu 74. Cho A, B, C là các góc của tam giác ABC (không là tam giác vuông) thì cot A.cot B  cot B.cot C  cot C.cot A 

A. 1 .

B. 1 . 2

C.  cot A.cot B.cot C  .

D. Một kết quả khác các kết quả đã nêu

trên. Hướng dẫn giải Ta có : cot A.cot B  cot B.cot C  cot C .cot A



1 1 1 tan A  tan B  tan C    . tan A.tan B tan B.tan C tan C.tan A tan A.tan B.tan C

Mặt khác : tan A  tan B  tan C  tan  A  B 1  tan A.tan B   tan C

 tan   C 1  tan A.tan B   tan C   tan C 1  tan A. tan B   tan C  tan C tan A.tan B . Nên cot A.cot B  cot B.cot C  cot C.cot A  1 . Chọn A. Câu 75. Cho A, B, C là các góc của tam giác ABC thì: 162 | THBTN – CA

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A. cos A  cos B  cos C  1  4sin

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

A B C .sin .sin . 2 2 2

B. cos A  cos B  cos C  1  4sin

A B C .sin .sin . 2 2 2

C. cos A  cos B  cos C  1  4 cos

A B C .cos .cos 2 2 2

D. cos A  cos B  cos C  1  4 cos

A B C .cos .cos . 2 2 2

.

Hướng dẫn giải Ta có : cos A  cos B  cos C BC BC .cos 2 2 A B C  cos A  2 cos .cos 2 2 A A BC  1  2sin 2  2sin .cos 2 2 2 A  A BC   1  2sin .   sin  cos  2  2 2   cos A  2 cos

 1  2sin

  B  C A  B C  .   sin  cos  2  2 2 

A  BC B C  .   cos  cos  2  2 2  A B C  1  4sin .sin .sin . 2 2 2  1  2sin

Chọn B. Câu 76. Cho A, B, C là các góc của tam giác ABC thì A. sin 2 A  sin 2 B  2sin C .

B. sin 2 A  sin 2 B  2sin C .

C. sin 2 A  sin 2 B  2 sin C .

D. sin 2 A  sin 2 B  2 sin C . Hướng dẫn giải

Ta có: sin 2 A  sin 2 B  2sin  A  B  .cos  A  B   2sin   C  .cos  A  B   2sin C.cos  A  B   2sin C . Dấu đẳng thức xảy ra khi cos  A  B   1  A  B . ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM

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Chọn A. Câu 77. Gọi M  cos A. M  0

2 4 6  cos  cos thì: 7 7 7 1 B. M   . 2

C. M  1 .

D. M  2 .

Hướng dẫn giải  2  4  6  Ta có: 2M .sin  2cos .sin  cos .sin  cos .sin 7 7 7 7 7 7 7 3  5 3 7 5  sin  sin  sin  sin  sin 7 7 7 7 7 7     sin  sin    sin . 7 7  sin

1 Nên M   . Chọn B. 2 Câu 78. Gọi M  cos

2 4 6  cos  cos thì: 7 7 7 1 . 2

A. M  0 .

B. M 

C. M  1

D. Một kết quả khác các kết quả đã nêu

trên. Hướng dẫn giải Ta có: Câu 79. Gọi M  cos  a  b  .cos  a  b   sin  a  b  .sin  a  b  thì : A. M  1  2cos2 a

B. M  1  2sin 2 a .

C. M  cos 4a

D. M  sin 4a . Hướng dẫn giải

Ta có: M  cos  a  b  .cos  a  b   sin  a  b  .sin  a  b 

 cos  a  b  a  b   cos 2a  1  2sin 2 a . Chọn B. Câu 80. Gọi M  cos  a  b  .cos  a  b   sin  a  b  .sin  a  b  thì : A. M  1  2sin 2 b . 164 | THBTN – CA

B. M  1  2sin 2 b . FILE WORD LIÊN HỆ - THẦY TÀI: 0977.413.341

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C. M  cos 4b .

BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

D. M  sin 4b . Hướng dẫn giải

Ta có: M  cos  a  b  .cos  a  b   sin  a  b  .sin  a  b 

 cos   a  b   (a  b)   cos 2b  1  2sin 2 b . Chọn A.

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