1. 1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved Chapter 5 Trigonometric Functions

2. OBJECTIVES Inverse Trigonometric Functions SECTION 5.6 1 2 Graph and apply the inverse sine function. Graph and apply the inverse cosine function. Graph and apply the inverse tangent function. Evaluate inverse trigonometric functions using a calculator. Find exact values of composite functions involving the inverse trigonometric functions. 3 5 4

3. 3 © 2011 Pearson Education, Inc. All rights reserved INVERSE SINE FUNCTION If we restrict the domain of y = sin x to the interval, then it is a one- to-one function and its inverse is also a function.

4. 4 © 2011 Pearson Education, Inc. All rights reserved INVERSE SINE FUNCTION The inverse function for y = sin x,, is called the inverse sine, or arcsine, function. The graph is obtained by reflecting the graph of y = sin x, for in the line y = x.,

5. 5 © 2011 Pearson Education, Inc. All rights reserved INVERSE SINE FUNCTION The equation y = sin –1 x means that sin y = x, where –1 ≤ x ≤ 1 and Read y = sin –1 x as “y equals inverse sine at x.” The domain of y = sin –1 x is [–1, 1]. The range of y = sin –1 x is

6. 6 © 2011 Pearson Education, Inc. All rights reserved INVERSE COSINE FUNCTION If we restrict the domain of y = cos x to the interval [0, π], then it is a one- to-one function and its inverse is also a function.

7. 7 © 2011 Pearson Education, Inc. All rights reserved INVERSE COSINE FUNCTION The inverse function for y = cos x,, is called the inverse cosine, or arccosine, function. The graph is obtained by reflecting the graph of y = cos x, with, in the line y = x.

8. 8 © 2011 Pearson Education, Inc. All rights reserved INVERSE COSINE FUNCTION The equation y = cos –1 x means that cos y = x, where –1 ≤ x ≤ 1 and Read y = cos –1 x as “y equals inverse cosine at x.” The domain of y = cos –1 x is [–1, 1]. The range of y = cos –1 x is

9. 9 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding the Exact Value for cos –1 x Find the exact values of y.

10. 10 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding the Exact Value for cos –1 x Solution continued

11. 11 © 2011 Pearson Education, Inc. All rights reserved INVERSE TANGENT FUNCTION If we restrict the domain of y = tan x to the interval, then it is a one-to- one function and its inverse is also a function.

12. 12 © 2011 Pearson Education, Inc. All rights reserved INVERSE TANGENT FUNCTION The inverse function for y = tan x,, is called the inverse tangent, or arctangent, function. The graph is obtained by reflecting the graph of y = tan x, with line y = x. in the

13. 13 © 2011 Pearson Education, Inc. All rights reserved INVERSE TANGENT FUNCTION The equation y = tan –1 x means that tan y = x, where –∞ ≤ x ≤ ∞ and Read y = tan –1 x as “y equals inverse tangent at x.” The domain of y = tan –1 x is [–∞, ∞]. The range of y = tan –1 x is

14. 14 © 2011 Pearson Education, Inc. All rights reserved y = csc –1 x means that csc y = x, where |x| ≥ 1 and INVERSE COTANGENT FUNCTION y = cot –1 x means that cot y = x, where –∞ ≤ x ≤ ∞ and 0 < y < π. INVERSE COSECANT FUNCTION INVERSE SECANT FUNCTION y = sec –1 x means that sec y = x, where |x| ≥ 1 and

15. 15 © 2011 Pearson Education, Inc. All rights reserved Since and we have y = csc −1 2 = EXAMPLE 4 Finding the Exact Value for csc –1 x Find the exact value for y = csc −1 2. Solution

16. 16 © 2011 Pearson Education, Inc. All rights reserved USING A CALCULATOR WITH INVERSE TRIGONOMETRIC FUNCTIONS To find csc –1 x find If x ≥ 0, this is the correct value. To find sec –1 x find To find cot –1 x start by finding If x < 0, add π to get the correct value.

17. 17 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 5 Using a Calculator to Find the Values of Inverse Functions Use a calculator to find the value of y in radians rounded to four decimal places. Solution Set the calculator to Radian mode. c. y = cot −1 (−2.3) c. y = cot −1 (−2.3) = π + tan −1 ≈ 2.7315

18. 18 © 2011 Pearson Education, Inc. All rights reserved COMPOSITION OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS

19. 19 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Exact Value of sin − 1 (sin x ) and cos − 1 (cos x ) Find the exact value of Solution a. Because we have

20. 20 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Exact Value of sin − 1 (sin x ) and cos − 1 (cos x ) Solution continued b. is not in the interval [0,π], but cos. So,

21. 21 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 9 Finding the Rotation Angle for a Security Camera A security camera is to be installed 20 feet away from the center of a jewelry counter. The counter is 30 feet long. What angle, to the nearest degree, should the camera rotate through so that it scans the entire counter?

22. 22 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 9 Finding the Rotation Angle for a Security Camera The counter center C, the camera A, and a counter end B form a right triangle. Solution The angle at vertex A is where θ is the angle through which the camera rotates.

23. 23 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 9 Finding the Rotation Angle for a Security Camera Set the camera to rotate through 74º to scan the entire counter. Solution continued