Trigonometry: Definitions and Formulas Unit Circle Definitions: x 2  y 2  1

Right Triangle Definitions: a 2  b 2  c 2

opposite a  hypotenuse c adjacent b cos A   hypotenuse c opposite a tan A   adjacent b

hypotenuse c  opposite a hypotenuse c sec A   adjacent b adjacent b cot A   opposite a

sin A 

Radians:  radians 

csc A 

cos  

x x 1

sin  

y y  y tan   1 x

s r

Conversions:   180

2  360 s  r 1 Area of Sector: As  r 2 2

Arc-length:



s t t where v  r  . (Use radians for all formulas above.) Circular Motion:  

and v 

Reciprocal Identities: 1 1 sin   cos   csc  sec  1 1 csc   sec   sin  cos 

1 cot  1 cot   tan  tan  

Pythagorean Identities: sin 2   cos 2   1

tan 2   1  sec2 

1  cot 2   csc2 

Quotient Identities: sin  tan   cos 

cot  

Odd Functions: sin      sin 

tan      tan  Even Function: cos     cos 

cos  sin 

Sum and Difference Formulas: sin  A  B   sin A cos B  cos A sin B

sin  A  B   sin A cos B  cos A sin B cos  A  B   cos A cos B  sin A sin B cos  A  B   cos A cos B  sin A sin B Double Angle Formulas:

tan  A  B  

tan A  tan B 1  tan A tan B

tan  A  B  

tan A  tan B 1  tan A tan B

Half Angle Formulas: 1  cos    sin     2 2

sin  2   2sin  cos 

cos  2   cos 2   sin 2 

1  cos    cos     2 2

 2 cos 2   1  1  2sin 2 

   1  cos  tan    sin  2

2 tan  tan  2   1  tan 2 

or

sin    tan     2  1  cos 

1 Area of a Triangle given SAS (use radians): Area  bc sin A 2 Heron’s Formula given SSS: Area  s  s  a  s  b  s  c  where s  Law of Sines: AAS, ASA, or ASS

Law of Cosines: SAS or SSS

sin A sin B sin C   a b c Graphing:

c 2  a 2  b2  2ab cos C

y  A sin  x     k

y  A tan  x     k

y  A cos  x     k

Period:

Amplitude: A Period:

2



Phase Shift:

abc . 2

 

 

Asymptotes:

Vertical Translation: k

Phase Shift:

 1  period  2

Vertical Translation: k

DeMoive’s Theorem:  r  cos   i sin     r n  cos n  i sin n  n

 

Graphs of the Six Trigonometric Functions (One Cycle) y  sin x

y  csc x

y  cos x

y  sec x

y  tan x

y  cot x

Inverse Trigonometric Functions Domain : 1  x  1 1

y  sin x

( y  arcsin x )

Range : 

 2

 y

 2

------------------------------------------------------------------------------------------------------

y  cos 1 x

( y  arccos x )

Domain : 1  x  1 Range : 0  y  

-----------------------------------------------------------------------------------------------------Domain :  1

y  tan x

( y  arctan x )

Range : 

 2

 y

 2