1. Slide 1-1 6 Inverse Trigonometric Functions Y. Ath

2. Slide 1-2 6.1 Inverse Circular Functions 6.2 Trigonometric Equations I 6.3 Trigonometric Equations II 6.4 Equations Involving Inverse Trigonometric Functions

3. Slide 1-3 Vertical Line Test Horizontal Line Test If a function f is one-to-one on its domain, then f has an inverse function

4. Slide 1-4 Inverse Function The inverse function of the one-to-one function f is defined as follows.

5. Slide 1-5 Caution The –1 in f –1 is not an exponent.

6. Slide 1-6

7. Slide 1-7 Inverse Sine Function Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test. f(x) = sin x does not pass the Horizontal Line Test and must be restricted to find its inverse.

8. Slide 1-8

9. Slide 1-9

10. Slide 1-10 Find y in each equation. Example FINDING INVERSE SINE VALUES

11. Slide 1-11 Example FINDING INVERSE SINE VALUES (cont.)

12. Slide 1-12 Example FINDING INVERSE SINE VALUES (cont.) –2 is not in the domain of the inverse sine function, [–1, 1], so does not exist.

13. Slide 1-13 Inverse Cosine Function Cos x has an inverse function on this interval. f(x) = cos x must be restricted to find its inverse. y x y = cos x

14. Slide 1-14

15. Slide 1-15 Inverse Tangent Function f(x) = tan x must be restricted to find its inverse. Tan x has an inverse function on this interval. y x y = tan x

16. Slide 1-16 Inverse Tangent Function The inverse tangent function is defined by y = arctan x if and only if tan y = x. The domain of y = arctan x is. The range of y = arctan x is [–  /2,  /2].

17. Slide 1-17 Graphing Utility: Graphs of Inverse Functions Graphing Utility: Graph the following inverse functions. a. y = arcsin x b. y = arccos x c. y = arctan x –1.5 1.5 ––  –1.5 1.5 22 –– –3 3  –– Set calculator to radian mode.

18. Slide 1-18 Graphing Utility: Inverse Functions Graphing Utility: Approximate the value of each expression. a. cos – 1 0.75b. arcsin 0.19 c. arctan 1.32d. arcsin 2.5 Set calculator to radian mode.

19. Slide 1-19 Composition of Functions Example: a. sin –1 (sin (–  /2)) = –  /2 does not lie in the range of the arcsine function, –  /2  y   /2. y x However, it is coterminal with which does lie in the range of the arcsine function.

20. Slide 1-20 Example: Evaluating Composition of Functions Example: x y 3 2 u

21. Slide 1-21 Inverse Function Values

22. Slide 1-22 Trigonometric Equations I 6.2 Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities

23. Slide 1-23 Example

24. Slide 1-24 Example 1(b) SOLVING A TRIGONOMETRIC EQUATION BY LINEAR METHODS Solve the equation 2 sinθ + 1 = 0 for all solutions.

25. Slide 1-25 Example SOLVING A TRIGONOMETRIC EQUATION BY FACTORING Subtract sin θ. Factor out sin θ. Zero-factor property Solution set: {0°, 45°, 180°, 225°}

26. Slide 1-26 Trigonometric Equations II 6.3 Equations with Half-Angles ▪ Equations with Multiple Angles

27. Slide 1-27 Example (a) over the interval and (b) for all solutions.

28. Slide 1-28 In-class exercises (pp 270-271) Solution set: {30°, 60°, 210°, 240°} Solution set, where 180º represents the period of sin2θ: {30° + 180°n, 60° + 180°n, where n is any integer} (1) (2) (3) Solve tan 3x + sec 3x = 2 over the interval (4) Solution set: {0.2145, 2.3089, 4.4033}

29. Slide 1-29 Equations Involving Inverse Trigonometric Functions 6.4 Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations

30. Slide 1-30 Example

31. Slide 1-31 Example:

32. Slide 1-32 Example

33. Slide 1-33 In class exercise