1. Slide 5.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Inverse Trigonometric Functions Learn to use the inverse sine function. Learn to use the inverse cosine function. Learn to use the inverse tangent function. Learn to use a calculator to evaluate inverse trigonometric functions. Learn to find the exact values of composite functions involving the inverse trigonometric functions. Learn to solve right triangles. SECTION 5.6 1 2 3 4 5 6

3. Slide 5.2- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley INVERSE SINE FUNCTION We restrict the domain of y = sin x to the interval so it is a one- to-one function and its inverse is also a function.

4. Slide 5.2- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 y = x. in the line

5. Slide 5.2- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley INVERSE SINE FUNCTION y = sin –1 x means 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. Slide 5.2- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Finding the Exact Value for y = sin –1 x Find the exact values of y. Solution

7. Slide 5.2- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1Finding the Exact Value for y = sin –1 x Solution continued c. Since 3 is not in the domain of the inverse sine function, which is [–1, 1], sin –1 3 does not exist.

8. Slide 5.2- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley INVERSE COSINE FUNCTION We restrict the domain of y = cos x to the interval so it is a one- to-one function and its inverse is also a function.

9. Slide 5.2- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 y = x. in the line

10. Slide 5.2- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley INVERSE COSINE FUNCTION y = cos –1 x means 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

11. Slide 5.2- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Finding the Exact Value for y = cos –1 x Find the exact values of y. Solution

12. Slide 5.2- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2Finding the Exact Value for y = cos –1 x Solution continued

13. Slide 5.2- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley INVERSE TANGENT FUNCTION We restrict the domain of y = tan x to the interval so it is a one- to-one function and its inverse is also a function.

14. Slide 5.2- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

15. Slide 5.2- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley INVERSE TANGENT FUNCTION y = tan –1 x means 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

16. Slide 5.2- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Finding the Exact Value for y = tan –1 x Find the exact values of y. Solution

17. Slide 5.2- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3Finding the Exact Value for y = tan –1 x Solution continued

18. Slide 5.2- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley INVERSE COTANGENT FUNCTION y = cot –1 x means cot y = x, where –∞ ≤ x ≤ ∞ and INVERSE COSECANT FUNCTION y = csc –1 x means csc y = x, where |x| ≥ 1 and INVERSE SECANT FUNCTION y = sec –1 x means sec y = x, where |x| ≥ 1 and

19. Slide 5.2- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley USING A CALCULATOR WITH INVERSE TRIGONOMETRIC FUNCTIONS To find csc –1 x simply find If x ≥ 0, this is the correct value. To find sec –1 x simply findTo find cot –1 x start by finding If x < 0, add π to get the correct value.

20. Slide 5.2- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

21. Slide 5.2- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Exact Value of a Composite Trigonometric Expression Find the exact value of Solution a. Let  represent the radian measure of the angle in with Since tan  is positive,  must be positive,

22. Slide 5.2- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Exact Value of a Composite Trigonometric Expression Solution continued So x = 3 and y = 2.

23. Slide 5.2- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Exact Value of a Composite Trigonometric Expression Find the exact value of Solution b. Let  represent the radian measure of the angle in with Since cos  is negative, we have

24. Slide 5.2- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Exact Value of a Composite Trigonometric Expression Solution continued So x = –1 and r = 4.

25. Slide 5.2- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 10 Solving a Right Triangle Given Two Sides Solve right triangle ABC if a = 9.5 and b = 3.4. Solution Sketch triangle ABC. To find A:

26. Slide 5.2- 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 10 Solving a Right Triangle Given Two Sides Solution continued To find c : To find B: B = 90º – 70.3º = 19.7º