Warm Up

Convert each measure from degrees to radians.

1. 120° 2. 180° 3. 225° 4. –30°

Warm Up

Find the exact value of each trigonometric function.

5. 6.

7. 8.

Evaluate inverse trigonometric functions.

Use trigonometric equations and inverse trigonometric functions to solve problems.

Objectives

You have evaluated trigonometric functions for a given angle. You can also find the measure of angles given the value of a trigonometric function by using an inverse trigonometric relation.

The expression sin-1 is read as “the inverse sine.” In this notation,-1 indicates the inverse of the sine function, NOT the reciprocal of the sine function.

Reading Math

The inverses of the trigonometric

functions are not functions

themselves because there are

many values of θ for a particular

value of a. For example, suppose

that you want to find cos-1 .

Based on the unit circle, angles

that measure and radians

have a cosine of . So do all

angles that are coterminal with

these angles.

Example 1: Finding Trigonometric Inverses

Find all possible values of cos-1 .

Step 1 Find the values between 0 and 2 radians for which cos θ is equal to .

Use the x-coordinates of points on the unit circle.

Example 1 Continued

Step 2 Find the angles that are coterminal with angles measuring and radians.

Add integer multiples of 2 radians, where n is an integer

Find all possible values of cos-1 .

Find all possible values of tan-11.

Step 1 Find the values between 0 and 2 radians for which tan θ is equal to 1.

Use the x and y-coordinates of points on the unit circle.

Add integer multiples of 2 radians, where n is an integer

Check It Out! Example 1 Continued

Find all possible values of tan-11.

Step 2 Find the angles that are coterminal with angles measuring and radians.

Because more than one value of θ produces the same output value for a given trigonometric function, it is necessary to restrict the domain of each trigonometric function in order to define the inverse trigonometric functions.

Trigonometric functions with restricted domains are indicated with a capital letter. The domains of the Sine, Cosine, and Tangent functions are restricted as follows.

Sinθ = sinθ for {θ| }

θ is restricted to Quadrants I and IV.

Cosθ = cosθ for {θ| }

θ is restricted to Quadrants I and II.

Tanθ = tanθ for {θ| }

θ is restricted to Quadrants I and IV.

These functions can be used to define the inverse trigonometric functions. For each value of a in the domain of the inverse trigonometric functions, there is only one value of θ. Therefore, even though tan-1 has many values, Tan-11 has only one value.

The inverse trigonometric functions are also called the arcsine, arccosine, and arctangent functions.

Reading Math

Example 2A: Evaluating Inverse Trigonometric Functions

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

Find value of θ for

or whose

Cosine .

Use x-coordinates of points on the unit circle.

Example 2B: Evaluating Inverse Trigonometric Functions

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

The domain of the inverse sine function is {a|1 = –1 ≤ a ≤ 1}. Because is outside this domain. Sin-1 is undefined.

Check It Out! Example 2a

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

Find value of θ for

or whose

Sine is .

Use y-coordinates of points on the unit circle.

Check It Out! Example 2b

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

Find value of θ for or whose

Cosine is 0.

Use x-coordinates of points on the unit circle.

0 = Cos θ

(0, 1)

Example 3: Safety Application

A painter needs to lean a 30 ft ladder against a wall. Safety guidelines recommend that the distance between the base of the ladder and the wall should be of the length of the ladder. To the nearest degree, what acute angle should the ladder make with the ground?

Example 3 Continued

θ7.5

Step 1 Draw a diagram. The base of the ladder should be (30) = 7.5 ft from the wall. The angle between the ladder and the ground θ is the measure of an acute angle of a right triangle.

Example 3 Continued

Step 2 Find the value of θ.

Use the cosine ratio.

Substitute 7.5 for adj. and 30 for hyp. Then simplify.

The angle between the ladderand the ground should be about 76°

A group of hikers wants to walk form a lake to an unusual rock formation.

The formation is 1 mile east and 0.75 mile north of the lake. To the nearest degree, in what direction should the hikers head from the lake to reach the rock formation?

Step 1 Draw a diagram. The base of the triangle should be 1 mile. The angle North from that point to the rock is 0.75 miles. θ is the measure of an acute angle of a right triangle.

Lake

θ

Rock

0.75 mi

1 mi

Check It Out! Example 3 Continued

Step 2 Find the value of θ

Use the tangent ratio.

Substitute 0.75 for opp. and 1 for adj. Then simplify.

The angle the hikers should take is about 37° north of east.

Example 4A: Solving Trigonometric Equations

Solve each equation to the nearest tenth. Use the given restrictions.

sin θ = 0.4, for – 90° ≤ θ ≤ 90°

The restrictions on θ are the same as those for the inverse sine function.

= Sin-1(0.4) ≈ 23.6°Use the inverse sine

function on your calculator.

Example 4B: Solving Trigonometric Equations

Solve each equation to the nearest tenth. Use the given restrictions.

sin θ = 0.4, for 90° ≤ θ ≤ 270°

The terminal side of θ is restricted to Quadrants ll and lll. Since sin θ > 0, find the angle in Quadrant ll that has the same sine value as 23.6°.

θ ≈ 180° –23.6° ≈ 156.4°

θ has a reference angle of 23.6°, and 90° < θ < 180°.

Solve each equations to the nearest tenth. Use the given restrictions.

tan θ = –2, for –90° < θ < 90°

The restrictions on θ are the same for those of the inverse tangent function.

θ = Tan-1 –2 ≈ –63.4° Use the inverse tangent function on your calculator.

Solve each equations to the nearest tenth. Use the given restrictions.

tan θ = –2, for 90° < θ < 180°

The terminal side of θ is restricted to Quadrant II. Since tan θ < 0, find the angle in Quadrant II that has the same value as –63.4°.

θ ≈ 180° – 63.4° ≈ 116.6°

116.6°–63.4°

Lesson Quiz: Part I

1. Find all possible values of cos-1(–1).

2. Evaluate Sin-1 Give your answer in both radians and degrees.

3. A road has a 5% grade, which means that there is a 5 ft rise for 100 ft of horizontal distance. At what angle does the road rise from the horizontal? Round to the nearest tenth of a degree.

2.9°

Lesson Quiz: Part II

Solve each equation to the nearest tenth. Use the given restrictions.

4. cos θ = 0.3, for 0° ≤ θ ≤ 180°

5. cos θ = 0.3, for 270° < θ < 360° θ ≈ 287.5°

θ ≈ 72.5°