Right Triangle Trigonometry In a right triangle we define sine, cosine, and tangent of an acute angle as follows. Draw a generic right triangle here! opp adj opp sin ( A ) = , cos ( A ) = , tan ( A ) = hyp hyp adj Many people use the pneumonic: SOHCAHTOA to remember this.

It is equally important to be able to find the ratio given the angle or the side lengths; the angle given the ratio or the side lengths; and a side given the angle, the ratio, and/or one side. For some reason I always draw my triangles with the angle of interest at the bottom left. Later in life, you’ll know why you do it. To solve a right triangle means to find all of the angles and side lengths. You will need to be very comfortable with your calculator in solving these problems. In particular watch out for order of operations and closing parentheses. Problem: Solve each of the following triangles. Try to use only the given information whenever possible. Be neat and organized. Box all of your final answers.

Trigonometry is extremely useful in solving many types of problems. The next three are examples of situations in which trigonometry can be used.

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Problem: OSHA Standard 1926.1053(b)(5)(i) states: “Non-self-supporting ladders shall be used at an angle such that the horizontal distance from the top support to the foot of the ladder is approximately one-quarter of the working length of the ladder (the distance along the ladder between the foot and the top support).” Assuming the ladder is against a vertical wall, find the angle of elevation of the ladder from the horizontal.

Problem: A 5 foot, 8 inch student who is standing 23 feet from the base of the school flag pole measures the angle of elevation from his eyes to the top of the pole to be 67.9° . Find the height of the pole to the nearest inch.

Problem: A surveyor stands at some unknown distance from a tall, inaccessible hill and measures the angle of elevation from her position to the top as 53° . She then walks back 400 feet in a straight line and re-measures the angle finding that it is now 27° . Calculate the height of the hill in feet. There are (at least) two different ways to solve this. I’d like you to understand both of them. One method involves finding a system of equations. Another method involves using the Law of Sines, which you’re not required to remember just yet, but you probably do.

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The Special Right Triangles Label the triangles below with the correct ratios of the sides. Let the smallest side be x. This should be review—these triangles are really important (SAT, SAT II, ACT, etc.).

Problem: Use triangle ABC above to answer each of the following. Use the relationships, not the Pythagorean Theorem.

AB

BC

AC

3 9

13 2 12 6 4 3 Problem: Use triangle DEF above to answer each of the following. Use the relationships, not the Pythagorean Theorem.

DE

EF

DF

34 9 17

8 3 8 2 6 6

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Sines, Cosines, and Tangents of Common Angles Using the special right triangles we can find the exact value of a couple of specific angles. Memorizing these exact values is—I assure you—far more important than you would guess at this point. (Of course you don’t need to guess…I’m telling you.) Problem: Use the general ratio of sides in a 30-60-90 triangle to find the sine, cosine, and tangent of both 30 and 60 degrees.

Problem: Repeat the process for 45° using a 45-45-90 triangle.

Problem: Fill in the table below with exact values. Start memorizing this now! sin (θ ) cos (θ ) tan (θ ) θ 30° 45° 60°

Reciprocal Functions (NOT INVERSE Functions) In right triangle trigonometry there’s no way for any side of a triangle to be 0 and so we can easily flip over each of the three ratios you are familiar with. In fact, this is such a common thing to do that the reciprocals of sine, cosine, and tangent have their own names: cosecant, secant, and cotangent, respectively.

sec (θ ) =

1 hyp = cos (θ ) adj

csc (θ ) =

1 hyp = sin (θ ) opp

cot (θ ) =

1 adj = tan (θ ) opp

Problem: Let’s pretend that your calculator is broken and the reciprocal trig functions aren’t working (or that you have some dinosaur calculator that lacks the built-in ability). Evaluate each of the following without using the reciprocal functions. Show the work, limited though it may be. a. csc ( 37.8° ) b. sec ( 48.2° ) c. cot ( 444.25° )

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Problem: Solve each of these on your calculator. Don’t write intermediate steps. Given Find Answer

sec (θ ) = 2.15

θ

cot ( t ) = 2.6

t

csc (α ) = 2.15

α

sec ( y ) = 4

tan ( y )

cot ( β ) = 6.24

sec ( β )

Problem: Fill in the table with exact values. Start memorizing this now! sin (θ ) cos (θ ) tan (θ ) csc (θ ) sec (θ ) θ

cot (θ )

30° 45° 60°

Problem: The lines y = 3x + 5 , y = −x 2 + 3 , and the x-axis form a triangle. Find each of the angles of the triangle to three decimal places. You do not need to find the intersection of the lines to solve this problem—so don’t! In this class we want to work hard but we also want to work smart.

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Practice Solving Right Triangles You should solve these on your own at some point. You need to practice showing the work as well as working quickly. These are the kinds of problems you want to become brainlessly competent at solving. 3 Problem 1: In QED , sin (Q ) = and ∠E = 90° . 7 a. Find the sides in most reduced form. b. Find the values of the other trig functions.



d. If QE = 21 , what are the other sides?

c. Find the angles of the triangle.

Problem 2: In

TGF , cos ( F ) = 20 and ∠G = 90° . 29

a. Find the sides in most reduced form.

b. Find the values of the other trig functions.

c. Find the angles of the triangle.

d. If FG = 10 , what are the other sides?

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Problem 3: In

STP , tan ( P ) = 53 and ∠S = 90° .

a. Find the sides in most reduced form.

b. Find the values of the other trig functions.

c. Find the angles of the triangle.

d. If SP = 20 , what are the other sides?

Problem 4: In

ACT , csc (C ) = 114 and ∠A = 90° .

a. Find the sides in most reduced form.

b. Find the values of the other trig functions.

c. Find the angles of the triangle.

d. If CT = 1.5 , what are the other sides?

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Problem 5: In

ASM , sec ( S ) = 135 and ∠M = 90° .

a. Find the sides in most reduced form.

b. Find the values of the other trig functions.

c. Find the angles of the triangle.

d. If AM = 1 / 3 , what are the other sides?

Problem 6: In

SAT , cot (T ) = 95

and ∠A = 90° .

a. Find the sides in most reduced form.

b. Find the values of the other trig functions.

c. Find the angles of the triangle.

d. If AT = 8 , what are the other sides?

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Cofunctions Problem: Draw right triangle ABC with a = 5 , b = 12 , c = 13 .

List the six trig functions of each of the acute angles. Ratio ∠A ∠B

Problem: Looking at the table, identify the pairs that are equal to each other.

Problem: What is the relationship between the acute angles of a right triangle? (Start with If A and B…)

Question: Why are some of these ratios equal? (Use words(!!!) and, possibly, SOHCAHTOA)

Problem: List the cofunction identities and explain their significance.

cos ( x ) =

sin ( x ) =

csc ( x ) =

sec ( x ) =

cot ( x ) =

tan ( x ) =

Problem: sin (15° ) ≈ 0.258819 , what do you automatically know?

Problem: cos ( x ) = .25 , what do you automatically know?

The cofunction relationships will be more significant when we get into more depth with all this trigonometry stuff. They’re also pretty important in calculus, but I understand how that feels far away to you right now. MA Notes 02

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Problem: Suppose that your calculator is extremely limited and only has a sine and a tangent button, with none of the others to be found. Show some work and then evaluate each of the following. (Check your answer by not pretending your calculator is broken.) a. cos ( 55.2° ) b. cos ( 218° )

c. csc ( 22.8° )

d. sec ( 22.5° )

e. cot ( 84.5° )

f.

sec ( 55.1° )

You and your calculator so far: • • • • • • •

You should have a document saved with your name as the title. You should be able to create new problems within a document. You should be able to define a function. You should be able to graph, find intersections, and store coordinates of intersection points. You should be able to change the window. You should be able to change quickly (very quickly) between degrees and radians. You should know how to put your calculator into test mode.

If you can’t do some of these things you need to go out of your way to learn. If you’re not good at using your calculator today and you don’t fix it, things are not going to go well for you in the long run. Question: What other things do you need to know how to do that I should add to the list above?

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