Warm Up

Given the measure of one of the acute angles in a right triangle, find the measure of the other acute angle.

1. 45° 2. 60° 3. 24° 4. 38°

45° 30°

66° 52°

Warm Up Continued

Find the unknown length for each right triangle with legs a and b and hypotenuse c.

5. b = 12, c =13 6. a = 3, b = 3

a = 5

A trigonometric function is a function whose rule is given by a trigonometric ratio. A trigonometric ratio compares the lengths of two sides of a right triangle. The Greek letter theta θ is traditionally used to represent the measure of an acute angle in a right triangle. The values of trigonometric ratios depend upon θ.

Example 1: Finding Trigonometric Ratios

Find the value of the sine, cosine, and tangent functions for θ.

sin θ =

cos θ =

tan θ =

Check It Out! Example 1

Find the value of the sine, cosine, and tangent functions for θ.

sin θ =

cos θ =

tan θ =

Example 2: Finding Side Lengths of Special Right Triangles

Use a trigonometric function to find the value of x.

°

x = 37

The sine function relates the opposite leg and the hypotenuse.

Multiply both sides by 74 to solve for x.

Substitute for sin 30°.

Substitute 30° for θ, x for opp, and 74 for hyp.

Check It Out! Example 2

Use a trigonometric function to find the value of x.

The sine function relates the opposite leg and the hypotenuse.

Substitute 45 for θ, x for opp, and 20 for hyp.

°°

Substitute for sin 45°.

Multiply both sides by 20 to solve for x.

Example 3: Sports ApplicationIn a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp?

5 ≈ h

The height above the water is about 5 ft.

Substitute 15.1° for θ, h for opp., and 19 for hyp.

Multiply both sides by 19.

Use a calculator to simplify.

Make sure that your graphing calculator is set to interpret angle values as degrees. Press . Check that Degree and not Radian is highlighted in the third row.

Caution!

Check It Out! Example 3 A skateboard ramp will have a height of 12 in., and the angle between the ramp and the ground will be 17°. To the nearest inch, what will be the length l of the ramp?

l ≈ 41The length of the ramp is about 41 in.

Substitute 17° for θ, l for hyp., and 12 for opp.

Multiply both sides by l and divide by sin 17°.

Use a calculator to simplify.

When an object is above or below another object, you can find distances indirectly by using the angle of elevation or the angle of depression between the objects.

Example 4: Geology Application

A biologist whose eye level is 6 ft above the ground measures the angle of elevation to the top of a tree to be 38.7°. If the biologist is standing 180 ft from the tree’s base, what is the height of the tree to the nearest foot?

Step 1 Draw and label a diagram to represent the information given in the problem.

Example 4 Continued

Step 2 Let x represent the height of the tree compared with the biologist’s eye level. Determine the value of x.

Use the tangent function.

180(tan 38.7°) = x

Substitute 38.7 for θ, x for opp., and 180 for adj.

Multiply both sides by 180.

144 ≈ x Use a calculator to solve for x.

Example 4 Continued

Step 3 Determine the overall height of the tree.

x + 6 = 144 + 6

= 150

The height of the tree is about 150 ft.

Check It Out! Example 4

A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the highest hill on a roller coaster to be 60.7°. If the surveyor is standing 120 ft from the hill’s base, what is the height of the hill to the nearest foot?

Step 1 Draw and label a diagram to represent the information given in the problem.

120 ft

60.7°

Check It Out! Example 4 Continued

Use the tangent function.

120(tan 60.7°) = x

Substitute 60.7 for θ, x for opp., and 120 for adj.

Multiply both sides by 120.

Step 2 Let x represent the height of the hill compared with the surveyor’s eye level. Determine the value of x.

214 ≈ x Use a calculator to solve for x.

Check It Out! Example 4 Continued

Step 3 Determine the overall height of the roller coaster hill.

x + 6 = 214 + 6

= 220The height of the hill is about 220 ft.

The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent.

Example 5: Finding All Trigonometric Functions

Find the values of the six trigonometric functions for θ.

Step 1 Find the length of the hypotenuse.

70

24

θ

a2 + b2 = c2

c2 = 242 + 702

c2 = 5476

c = 74

Pythagorean Theorem.

Substitute 24 for a and 70 for b.

Simplify.

Solve for c. Eliminate the negative solution.

Example 5 Continued

Step 2 Find the function values.

In each reciprocal pair of trigonometric functions, there is exactly one “co”

Helpful Hint

Find the values of the six trigonometric functions for θ.

Step 1 Find the length of the hypotenuse.

80

18

θ

a2 + b2 = c2

c2 = 182 + 802

c2 = 6724

c = 82

Pythagorean Theorem.

Substitute 18 for a and 80 for b.

Simplify.

Solve for c. Eliminate the negative solution.

Check It Out! Example 5

Check It Out! Example 5 Continued

Step 2 Find the function values.

Lesson Quiz: Part I

Solve each equation. Check your answer.

1. Find the values of the six trigonometric functions for θ.

Lesson Quiz: Part II

2. Use a trigonometric function to find the value of x.

3. A helicopter’s altitude is 4500 ft, and a plane’s altitude is 12,000 ft. If the angle of depression from the plane to the helicopter is 27.6°, what is the distance between the two, to the nearest hundred feet?16,200 ft