1 | S e c t i o n 8 . 1

Chapter 8.1: Trigonometric Functions and Right Triangle Trigonometry 8.1.1 Introduction to Trigonometry Trigonometry is the study of the relationship between angles and lengths. It is often associated with right triangles. Trigonometry is used in architecture and astronomy, where sides and angles determine distance. It is also used in such disparate fields as engineering and music.

The study of trigonometry dates back many centuries, through many cultures. The image below shows a facsimile of the cover page from a text on quadratic equations and trigonometry by the Persian Muḥammad ibn Mūsā al-Khwārizmī, written nearly 2000 years ago.

This is a facsimile, not the real page from the text, since the original no longer exists.

8.1.1 Example Here is an illustration of what we can find using the ideas of trigonometry and right triangles:

In a classroom with a tiled floor, suppose you know each floor tile is 12 inches on each side, or 1 foot. If you want to know the height of the ceiling in the classroom, you can use the floor tiles: Walk across the floor until it looks like there is a 45-degree angle from your shoes to the ceiling. Next, count how many tiles there are from your shoes to the edge of the wall.

2 | S e c t i o n 8 . 1

If there are 11 tiles from your shoes to the edge of the room,

and the angle to the ceiling is 45 degrees,

and you know the distance from your shoes to the wall is 11 (by counting the tiles).

Then you know that height of the ceiling is also 11 feet, because in a 45, 45, 90 triangle, the two legs of the triangle are always equal.

Using trigonometry, if we know one side and one angle of a right triangle, we can find the other sides and angles. 8.1.2 The Sine Function The sine function is defined to be the ratio of two sides of a right triangle:

sin θ =𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜=𝑂𝑂𝐻𝐻

=𝑦𝑦𝑟𝑟

8.1.2 Example 1

θ (theta) is the angle to the x-axis.

The letter O is the side that is opposite that angle. H is the hypotenuse (opposite the 90° angle). It is also the radius of the circle.

Find the coordinates of the point (x,y). These are the legs of your triangle. x = ? y = ? How can you figure out the hypotenuse?

θ

(x,y)

90°

11 feet

3 | S e c t i o n 8 . 1

Replace a and b with 3: 𝑎𝑎2 + 𝑏𝑏2 = 𝑐𝑐2

32 + 32 = 𝑐𝑐2

9 + 9 = 𝑐𝑐2

18 = 𝑐𝑐2 To find c, take the square root of both sides.

√18 = √𝑐𝑐2 This gives c = ±√18. We take the positive value since we have a hypotenuse (more on this later!).

We can rewrite √18 in simplified radical form as 3√2 or in decimal form, approximately 4.243 (rounded to three decimal places).

Since the sine is defined as sin θ = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

, we have sin θ = 33√2

= 1√2

.

This can be rewritten as 1(√2)√2(√2)

= √2√4

= √22

or as a decimal, √2÷2 ≈ 0.707. 8.1.2 Example 2 Now let’s look further out along the same angle.

For the new, larger circle, find the coordinates of the point (x,y). Use this point to find the legs of the new, larger triangle. Next, find the new hypotenuse, and use the opposite side and the hypotenuse to find the sine of θ.

Try it on your own before you go to the next page.

3 3

The point is (3,3). Each leg of the triangle is also 3, because the x and y values tell us how far across (the base of the triangle) and how far up (the height). To find the hypotenuse, because we have a right triangle (with a 90° angle), we can use the Pythagorean Theorem:

𝑎𝑎2 + 𝑏𝑏2 = 𝑐𝑐2

where a and b are the legs of the triangle (the base and height) and c is the hypotenuse.

(?,?)

θ

θ

(3,3)

4 | S e c t i o n 8 . 1

The new points is (5,5). This tells us that the base and height of the triangle (also called the legs of the triangle) are both 5. Replace a and b with 5: 𝑎𝑎2 + 𝑏𝑏2 = 𝑐𝑐2

52 + 52 = 𝑐𝑐2

25 + 25 = 𝑐𝑐2

50 = 𝑐𝑐2 To find c, take the square root of both sides.

√50 = √𝑐𝑐2

This gives c = ±√50. We take the positive value, since we have the

hypotenuse, so we get √50 = 5√2

sin θ = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

, sin θ = 55√2

= 1√2

= √22

≈ 0.707 Does this look familiar? It’s the same thing we got before! We get the same answer because our angle is the same. In fact, this angle happens to be a 45-degree angle. We can tell it is 45° because the two sides are always the same, and also because, visually, the angle cuts the 90° angle exactly in half: Find sin (45°) on your calculator. You should get 1.414…! (Make sure your calculator is set to degrees, not radians). The sine of 45 degrees will always be √2

2≈ 0.707.

The sine function tells you the ratio of two sides of a triangle, for a given degree angle. For a 45° – 45° – 90° triangle, the ratio of the side to the hypotenuse will always be 1.414…. There are many more trigonometric functions! But for now, let’s practice with just the sine function to see how we can use it.

θ

(5,5)

90° 45°

5 | S e c t i o n 8 . 1

8.1.3 Opposite and Adjacent Sides 8.1.3 Example 1 θ is the angle, and O means the side opposite to angle θ. The hypotenuse is the side opposite the 90° angle. The adjacent side is the side next to angle θ. Alternatively, you can think of it as the side that is not opposite to angle θ, and not the hypotenuse. 8.1.3 Example 2 Before you go to the next page, see if you can identify O, A, and H for each of the two triangles below.

θ

O

θ

O

A

H

θ

θ

O

H

θ

6 | S e c t i o n 8 . 1

Example 2, continued

8.1.4 Finding the Value of the Sine of an Angle 8.1.4 Example 1 Find sin θ for the following triangle:

sin θ = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

, and in this example, the side opposite θ is 3. The hypotenuse is 5.

So, we get sinθ = 35. You can leave this in fraction from, or write as a decimal, 0.600.

🙄🙄 😎😎

Okaaay.

θ 5

Wait, why are there those extra zeros? Why didn’t you

just write 0.6?

3

4

θ

θ

O

O

A

A

H

H

Usually in trigonometry, we keep at least three

decimal places while we do our calculations. I’m

showing that I have three decimal places.

7 | S e c t i o n 8 . 1

8.1.4 Example 2 Find sin θ for the following triangle: sin θ = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜, and in this example, the side opposite θ is 8. But we don’t know

the hypotenuse. So far we have sin θ = 8𝐻𝐻

.

We use the Pythagorean theorem to find H. Replace a and b with 6 and 8 (it does not matter which is a and which is b):

𝑎𝑎2 + 𝑏𝑏2 = 𝑐𝑐2

62 + 82 = 𝑐𝑐2

36 + 64 = 𝑐𝑐2

100 = 𝑐𝑐2 To find c, take the square root of both sides.

√100 = √𝑐𝑐2 This gives c = ±10, but since c is the hypotenuse of a triangle, we use the positive value. sin θ = 8

𝐻𝐻= 8

10= 4

5= 0.800

8.1.5 Finding a Missing Side Using the Sine of an Angle If we know an angle in a triangle, and we know the hypotenuse or the opposite side, we can find the other sides of the triangle using the sine function. 8.1.5 Example 1

In the triangle on the left, what is the missing side, x? We know that sin A = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜. Plugging in the values

that we know, we have: sin 20° = 𝑥𝑥42

27 x

θ

20°

When we take the square root of both sides of a squared number, we could have either a positive or negative answer. We use only the positive value throughout this section. But we will return to possible negative values in later sections when we return to triangles on the unit circle in the coordinate plane.

8

6

8 | S e c t i o n 8 . 1

Now, we solve for x. Since x is being divided by 42, we multiply both sides of the equation by 42 to get x by itself, and we have: 42 • sin 20° = 𝑥𝑥

42 • 42

42(sin 20°) = x

So, our exact value for x is 42 sin200. We can find a decimal approximation for x by using our calculator. We find the sine of 20 degrees, (make sure you are in degree mode). Keep all those decimals on your calculator, do not hit clear! Or, copy down at least three of the decimal places. Then multiply that by 42.

x ≈ 14.4 Typically, we keep at least three decimal places while we are working, and then round off in the end to one decimal place, unless the instructions say otherwise.

Having trouble finding degree mode?

Not your calculator? Do a web search for your calculator model and “degree mode”

9 | S e c t i o n 8 . 1

8.1.6 The Remaining Trigonometric Functions

SOHCAHTOA! This mnemonic can help you remember the trigonometric ratios. SOHCAHTOA (pronounced so-cah-toe-a) stands for

Sine = opposite/hypotenuse (SOH) Cosine = adjacent/hypotenuse (CAH) Tangent = opposite/adjacent (TOA)

Sine Cosine sin θ = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜= 𝑂𝑂

𝐻𝐻= 𝑦𝑦

𝑟𝑟 cos θ = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜

ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜 = 𝐴𝐴

𝐻𝐻= 𝑥𝑥

𝑟𝑟

Tangent tan θ = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜= 𝑂𝑂

𝐴𝐴= 𝑦𝑦

𝑥𝑥

In addition, the three reciprocal ratios are Cosecant Secant csc𝜃𝜃 = 1

𝑜𝑜𝑜𝑜𝑦𝑦= H

O= 𝑟𝑟

𝑦𝑦 sec θ = 1

𝑎𝑎𝑜𝑜𝑜𝑜= 𝐻𝐻

𝐴𝐴= 𝑟𝑟

𝑥𝑥

Cotangent cot θ = 1

𝑜𝑜𝑎𝑎𝑦𝑦= 𝐴𝐴

𝑂𝑂= 𝑥𝑥

𝑦𝑦

8.1.6 Examples For the next examples, we will refer to right triangle ABC. This is the standard triangle that is used in textbooks and in homework sets so that the authors don’t have to keep drawing triangles! In this triangle, uppercase letters represent angles, and the lowercase letters represent the sides opposite those letters. Angle C is always the 90° angle, so letter c is always the hypotenuse.

8.1.6 Example 1 Find the six trigonometric functions of A, if a = 2 and b = 1. Begin by sketching a right triangle like the one below, with the sides labeled appropriately. Try this yourself before you go to the next page.

10 | S e c t i o n 8 . 1

c c or Either triangle is correct, as long as side a is opposite angle A and side b is opposite angle B, but you will probably find the first triangle, with A on top, to be the one most commonly drawn.

Notice that in this example, we only have two out of the three sides. In order for us to find all the trigonometric values, we need to have all the sides of the triangle. So, we begin by using the Pythagorean Theorem to find the hypotenuse.

𝑎𝑎2 + 𝑏𝑏2 = 𝑐𝑐2

12 + 22 = 𝑐𝑐2

1 + 4 = 𝑐𝑐2

5 = 𝑐𝑐2 This gives c = ±√5. Again, we take the positive value only.

Now we use the definition of the trigonometric functions to find the ratios. Since we are using A as our angle, each one of the sides are the sides in relation to angle A.

sin A = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

= 2√5

= 2√55

cos A = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

= 1√5

= √55

tan A = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜

= 21

= 2

To get the reciprocal trig functions, just take the reciprocal (the “flip”) of the functions above. csc A = ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜= √5

2 sec A =ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜= √5

1= √5

cot A = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

= 12

8.1.6 Example 2 If a = 3 and c = 6, find all six trigonometric functions of B. Begin by sketching the triangle on your own before going to the next page.

A

B C =90° a = 2

b = 1 B

b = 1

a = 2

11 | S e c t i o n 8 . 1

c = 6

For this triangle, we are missing side B. We will again use the Pythagorean Theorem to help us find this side, being careful to replace the correct variables, a and c, with 3 and 6:

𝑎𝑎2 + 𝑏𝑏2 = 𝑐𝑐2 Replace a and c with 3 and 6.

32 + 𝑏𝑏2 = 62

9 + 𝑏𝑏2 = 36 −9 − 9 Solve to get 𝑏𝑏2 alone by subtracting 9.

𝑏𝑏2 = 27 𝑏𝑏 = √27 = 3√3

Now we have:

Based on this, the trigonometric values of B are:

sin𝐵𝐵 = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜= 3√3

6= √3

2 cos𝐵𝐵 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜

ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜= 3

6= 1

2

tan𝐵𝐵 = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜= 3√3

3= √3

csc𝐵𝐵 = ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜= 6

3√3= 2

√3= 2√3

3 sec𝐵𝐵 = ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜= 6

3= 2

cot B = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜

𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜= 3

3√3= 1

√3= √3

3

8.1.7 Using Trigonometric Functions to Find the Side of a Triangle 8.1.7 Example In right triangle, ABC, B = 620, and a is 18, find b. See if you can draw the triangle and solve this before going to the next page.

B

A

a = 3 C

c = 6

A

C

6

3

3√3

12 | S e c t i o n 8 . 1

A 620 b C 18 B

We need to find side b, and we are given side a. We know that angle B is 620, and we have to find side b which is opposite to 620. We are also given that side a is 18, which is adjacent to 620. Which trigonometric function gives us information in terms of opposite and adjacent?

Tangent!

We know that tan A =𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝐴𝐴

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜 𝐴𝐴. Substituting in our values gives us the following

equation: tan 620 = 𝑏𝑏

18

Multiplying both sides of the equation by 18 to get b by itself yields:

18 tan 620 = b This is our exact solution. Using a calculator, we get an approximate solution of

b≈ 33.9

13 | S e c t i o n 8 . 1

8.1.8 Applications of Trigonometric Functions Surveyors use trigonometry to find the heights or depths of object they can’t measure directly, by finding angle to the top or bottom of that object.

Angle of elevation: the angle measured from the horizontal distance, going up

🙄🙄 Angle of depression: the angle measured from the horizontal, going down

🙃🙃 8.1.8 Example 1 You are standing 43 feet away from a tree. The angle of elevation from where you are standing to the top of the tree is 300. How tall is the tree?

We begin by sketching a picture of this scenario. See if you can draw the tree, the distance of 43 feet, and the angle of 30 before turning the page. Imagine the right triangle that is formed.

14 | S e c t i o n 8 . 1

This Photo by Unknown Author is licensed under CC BY-SA-NC

30°

---------------------- 43 feet -----------------------

We usually assume that the object standing (in this case, the tree) takes the 900 angle. We are standing 43 feet away from the tree, so we have a horizontal distance of 43 feet, and the angle of elevation is 300, which means that we are looking up to the top of the tree. So, which trigonometric function can we use here? Well, we know 300, and we need to find the side opposite of 300 (which is h), and we are given 43 which is the side adjacent to 300. Therefore, we would use tangent, since tan = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜

tan 300 = 𝑥𝑥43

Multiply both sides of the equation by 43 to get x by itself and we get:

43 tan 300 = x and our answer is approximately 24. 8 feet.

8.1.8 Example 2 A kite being blown by the wind makes an angle of elevation to the ground of 73 degrees. The kit string is 20 feet long. How high above the ground is the kite? See if you can make a picture on your own before you go to the next page.

Height = x

15 | S e c t i o n 8 . 1

73° Notice that the kite string is the hypotenuse of the triangle. It is not the same as the height of a triangle, because when you fly a kite, the wind usually pushes the kite at an angle to you.

Which trigonometric function can we use here? It should have the hypotenuse and the side opposite the angle, since the height is opposite the angle of 73°.

Therefore, we would use sine, since sin = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜ℎ𝑦𝑦𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦𝑜𝑜𝑜𝑜

.

sin 73° = 𝑥𝑥20

Multiply both sides of the equation by 43 to get h by itself and we get:

20 sin 73° = h and our answer is approximately 19.1 feet.

8.1.8 Example 3 A ship’s sonar locates a sunken treasure chest at a 20-degree angle of depression. A diver is lowered 35 feet directly down to the ocean floor. How far must the diver swim to get to the treasure? Again, see if you can make a picture on your own before you go to the next page.

Height = x

16 | S e c t i o n 8 . 1

20° 35 feet

💰💰 Notice that in this picture, the angle of 20° is outside the triangle that we see. We can fix that, keeping in mind that the sum of the angles in a triangle is 180°, and that the two angles up top add up to 90°.

This whole top angle of the boat to the diver (outlined with the thick black lines) is 90°, so the missing part is 70°.

See if you can find the rest of the angles inside the triangle before turning the page. See if you can use these angles to find the distance from the diver to the treasure.

70°

17 | S e c t i o n 8 . 1

x Notice that the angle of elevation from the treasure to the boat ends up also being 20 degrees.

(You can also use another geometric fact to find the angle, such as the fact that parallel lines (the ocean and the ocean floor) cut by a transversal have equal corresponding angles.)

Which trigonometric function can you use to find x, using the 20-degree angle and the side of 35?

Since 35 is opposite the 20° angle, and x is adjacent to the 20° angle, you can use the tangent.

tan(20) =𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑐𝑐𝑜𝑜𝑦𝑦𝑜𝑜 =

35𝑥𝑥

Multiply both sides by x to get the x out of the denominator of the fraction: (𝑥𝑥) tan(20) = 35

𝑥𝑥(𝑥𝑥) (𝑥𝑥) tan(20) = 35

Now divide both sides by tan 20. 𝑥𝑥 = 35

tan(20)≈ 96.2 feet

You could also have used the 70-degree angle: tan(70) = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑦𝑦𝑜𝑜= 𝑥𝑥

35 multiply by 35 to get: 35 tan(70) = 𝑥𝑥

We still get x ≈ 96.2.

70°

90° 20°

18 | S e c t i o n 8 . 1

8.1.9 Exact Values of Trigonometric Functions There are a few special angles that we can figure out the exact values for, based on what we know about sides and angles of triangles. We have already come across one of these, the 45-45-90 triangle. In this triangle, the sides are always equal to each other.