13-4

1. Inverses of Trigonometric Functions 13-4 Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

2. Objectives Evaluate inverse trigonometric functions. Use trigonometric equations and inverse trigonometric functions to solve problems.

3. Vocabulary inverse sine functions inverse cosine function inverse tangent function

4. 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.

5. Reading Math 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.

6. The inverses of the trigonometric functions are not functions themselves because there are many values of θfor 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.

7. Step 1 Find the values between 0 and 2 radians for which cos θis equal to . Example 1: Finding Trigonometric Inverses Find all possible values of cos-1 . Use the x-coordinates of points on the unit circle.

8. Step 2 Find the angles that are coterminal with angles measuring and radians. Example 1 Continued Find all possible values of cos-1 . Add integer multiples of 2 radians, where n is an integer

9. Check It Out! Example 1 Find all possible values of tan-11.

10. Because more than one value of θproduces 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.

11. Sinθ = sinθfor{θ| } Cosθ = cosθfor {θ| } Tanθ = tanθfor {θ| } Trigonometric functions with restricted domains are indicated with a capital letter. The domains of the Sine, Cosine, and Tangent functions are restricted as follows. θ is restricted to Quadrants I and IV. θ is restricted to Quadrants I and II. θ is restricted to Quadrants I and IV.

12. 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.

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

15.  Find value of θ for or whose Cosine . Example 2A: Evaluating Inverse Trigonometric Functions Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. Use x-coordinates of points on the unit circle.

16. The domain of the inverse sine function is {a|1 = –1 ≤ a ≤ 1}. Because is outside this domain. Sin-1 is undefined. Example 2B: Evaluating Inverse Trigonometric Functions Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

17. Check It Out! Example 2a Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

18. Check It Out! Example 2b Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

19. 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?

20. 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 θis the measure of an acute angle of a right triangle. θ 7.5 Example 3 Continued

21. 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 ladder and the ground should be about 76°

22. Rock 0.75 mi θ 1 mi Lake Check It Out! Example 3 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?

23. 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 θare the same as those for the inverse sine function. Use the inverse sine function on your calculator. = Sin-1(0.4) ≈ 23.6°

24. 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 θis 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 has a reference angle of 23.6°, and 90° < θ < 180°. θ ≈ 180° –23.6° ≈ 156.4°

25. Check It Out! Example 4a Solve each equations to the nearest tenth. Use the given restrictions. tan θ = –2, for –90° < θ < 90°

26. Check It Out! Example 4b Solve each equations to the nearest tenth. Use the given restrictions. tan θ = –2, for 90° < θ < 180°