9.4 Evaluate Inverse Trigonometric Functions

How are inverse Trigonometric functions used?How much information must be given about side lengths in a right triangle in order for you to be able to find the measures

of its acute angles?

Inverse Trig Functions•

x

y

Inverse Trig Functions•

x

y

0

Inverse Trig Functions•

x

y

Evaluate the expression in both radians and degrees.

a. cos–1 32

√

SOLUTION

a. When 0 θ π or 0° 180°, the angle whose cosine is

≤ ≤ ≤ θ ≤32

√

cos–1 32

√θ =π6

= cos–1 32

√θ = = 30°

0°360°180°

90°

270°

45°135°

225° 315°

30°

60°120°

150°

210°

240° 300°

330°

x

y

Evaluate the expression in both radians and degrees.

b. sin–1 2

SOLUTION

sin–1b. There is no angle whose sine is 2. So, is undefined.

2

Evaluate the expression in both radians and degrees.

3 ( – )c. tan–1 √

SOLUTION

c. When – < θ < , or – 90° < θ < 90°, the angle whose tangent is – is:

π2

π2

√ 3

( – )tan–1 3√θ =π3

–= ( – )tan–1 3√θ = –60° =

Evaluate the expression in both radians and degrees.

1. sin–1 22

√

ANSWERπ4

, 45°

2. cos–1 12

ANSWER π3

, 60°

3. tan–1 (–1)

ANSWER π4

, –45°–

4. sin–1 (– )12

π6

, –30°–ANSWER

Solve the equation sin θ = – where 180° < θ < 270°.

58

SOLUTIONSTEP 1

sine is – is sin–1 – 38.7°. This58

58

–

Use a calculator to determine that in theinterval –90° θ 90°, the angle whose≤ ≤

angle is in Quadrant IV, as shown.

STEP 2 Find the angle in Quadrant III (where180° < θ < 270°) that has the same sinevalue as the angle in Step 1. The angle is:

θ 180° + 38.7° = 218.7°CHECK : Use a calculator to check the answer.

58sin

218.7°– 0.625=–

Solve a Trigonometric Equation

Solve the equation for

270° < θ < 360°5. cos θ = 0.4;

ANSWER about 293.6°

180° < θ < 270°6. tan θ = 2.1;

ANSWER about 244.5°

270° < θ < 360°7. sin θ = –0.23;

ANSWER about 346.7°

6.2934.66360

5.2441805.64

7.3463.13360

180° < θ < 270°8. tan θ = 4.7;

ANSWER about 258.0°

90° < θ < 180°9. sin θ = 0.62;

ANSWER about 141.7°

180° < θ < 270°10. cos θ = –0.39;

ANSWER about 247.0°

Solve the equation for

25818078

7.1413.38180

247113360

SOLUTION

In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse, so use the inverse cosine function to solve for θ.

cos θ =adjhyp =

611

cos – 1θ = 611

56.9°

The correct answer is C.ANSWER

Monster Trucks

A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle θ of the ramp?

http://www.youtube.com/watch?v=SrzXaDFZcAo

http://www.youtube.com/watch?v=7SjX7A_FR6g

SOLUTION

STEP 1 Draw: a triangle that represents the ramp.

STEP 2 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length.

tan θ =oppadj =

820

STEP 3 Use: a calculator to find the measure of θ.

tan–1θ = 820

21.8°

The angle of the ramp is about 22°.

ANSWER

Find the measure of the angle θ.

11.

SOLUTION

In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse. So, use the inverse cosine function to solve for θ.

cos θ =adjhyp = 4

9= 63.6°θ cos–1 4

9

Find the measure of the angle θ.

SOLUTION

In the right triangle, you are given the lengths of the side opposite to θ and the side adjacent. So, use the inverse tan function to solve for θ.

12.

tan θ =oppadj =

108

θ 51.3°= tan–1 108

Find the measure of the angle θ.

SOLUTION

In the right triangle, you are given the lengths of the side opposite to θ and the hypotenuse. So, use the inverse sin function to solve for θ.

13.

sin θ =opphyp = 5

1224.6°θ = sin–1 5

12

9.4 AssignmentPage 582, 3-29 odd