1. Copyright © 2005 Pearson Education, Inc.

2. Chapter 6 Inverse Circular Functions and Trigonometric Equations

3. Copyright © 2005 Pearson Education, Inc. 6.1 Inverse Circular Functions

4. Copyright © 2005 Pearson Education, Inc. Slide 6-4 Horizontal Line Test Any horizontal line will intersect the graph of a one-to-one function in at most one point. The inverse function of the one-to-one function f is defined as

5. Copyright © 2005 Pearson Education, Inc. Slide 6-5 Inverse Functions Review 1. In a one-to-one function, each x-value corresponds to only one y-value and each y-value corresponds to only one x-value. 2. If a function f is one-to-one, then f has an inverse function f -1. 3. The domain of f is the range of f -1 and the range of f is the domain of f -1. 4. The graphs of f and f -1 are reflections of each other about the line y = x. 5. To find f -1 (x) from f(x), follow these steps.  Replace f(x) with y and interchange x and y.  Solve for y.  Replace y with f -1 (x).

6. Copyright © 2005 Pearson Education, Inc. Slide 6-6 Inverse Sine Function means that x = sin y, for Example: Find

7. Copyright © 2005 Pearson Education, Inc. Slide 6-7 Examples sin  1 2

8. Copyright © 2005 Pearson Education, Inc. Slide 6-8

9. Copyright © 2005 Pearson Education, Inc. Slide 6-9 Inverse Cosine Function means that x = cos y, for Example: Find

10. Copyright © 2005 Pearson Education, Inc. Slide 6-10

11. Copyright © 2005 Pearson Education, Inc. Slide 6-11 Inverse Tangent Function means that x = tan y, for

12. Copyright © 2005 Pearson Education, Inc. Slide 6-12

13. Copyright © 2005 Pearson Education, Inc. Slide 6-13 Other Inverse Functions

14. Copyright © 2005 Pearson Education, Inc. Slide 6-14 Examples Find the degree measure of  in the following. a)  = arctan 1b)  = sec  1 2

15. Copyright © 2005 Pearson Education, Inc. Slide 6-15 Example Find y in radians if y = arctan(  6.24).  Calculator in radian mode  Enter tan  1 (  6.24) y   1.411891065 Find y in radians if y = arcsec(  6.24). Find y in radians if y = arccos 2.  Calculator in radian mode  Enter cos  1 (2) (error message since the domain of the inverse cosine function is [  1, 1].

16. Copyright © 2005 Pearson Education, Inc. Slide 6-16 Example: Evaluate the expression without using a calculator. Let The inverse tangent function yields values only in quadrants I and IV, since 3/2 is positive,  is in quadrant I.

17. Copyright © 2005 Pearson Education, Inc. Slide 6-17 Example Evaluate the expression without using a calculator.

18. Copyright © 2005 Pearson Education, Inc. Slide 6-18 92

19. Copyright © 2005 Pearson Education, Inc. 6.2 Trigonometric Equations I

20. Copyright © 2005 Pearson Education, Inc. Slide 6-20 Example: Linear Method Solve 2 cos 2 x  1 = 0

21. Copyright © 2005 Pearson Education, Inc. Slide 6-21 Example: Factoring Solve 2 cos x + sec x = 0

22. Copyright © 2005 Pearson Education, Inc. Slide 6-22 Example: Factoring Solve 2 sin 2 x + sin x  1 = 0

23. Copyright © 2005 Pearson Education, Inc. Slide 6-23 Example: Squaring Solve cos x + 1 = sin x [0, 2  ]

24. Copyright © 2005 Pearson Education, Inc. Slide 6-24 42, 52, 56

25. Copyright © 2005 Pearson Education, Inc. Slide 6-25 Solving a Trigonometric Equation Decide whether the equation is linear or quadratic in form, so you can determine the solution method. If only one trigonometric function is present, first solve the equation for that function. If more than one trigonometric function is present, rearrange the equation so that one side equals 0. Then try to factor and set each factor equal to 0 to solve.

26. Copyright © 2005 Pearson Education, Inc. Slide 6-26 Solving a Trigonometric Equation continued If the equation is quadratic in form, but not factorable, use the quadratic formula. Check that solutions are in the desired interval. Try using identities to change the form of the equation. If may be helpful to square both sides of the equation first. If this is done, check for extraneous solutions.

27. Copyright © 2005 Pearson Education, Inc. 6.3 Trigonometric Equations II

28. Copyright © 2005 Pearson Education, Inc. Slide 6-28 Example: Using a Half-Angle Identity a) over the interval [0, 2  ), and b) give all solutions

29. Copyright © 2005 Pearson Education, Inc. Slide 6-29 Example: Double Angle Solve cos 2x = cos x over the interval [0, 2  ).

30. Copyright © 2005 Pearson Education, Inc. Slide 6-30 Example: Multiple-Angle Identity Solve over the interval [0, 360  ).

31. Copyright © 2005 Pearson Education, Inc. Slide 6-31 Example: Multiple Angle Solve tan 3x + sec 3x = 2 over the interval [0, 2  ). Tangent and secant are related so use the identity

32. Copyright © 2005 Pearson Education, Inc. Slide 6-32 Example: Multiple Angle continued Use a calculator and the fact that cosine is positive in quadrants I and IV, Since both sides of the equation were squared, each proposed solution must be checked. The solution set is {.2145, 2.3089, 4.4033}.

33. Copyright © 2005 Pearson Education, Inc. 6.4 Equations Involving Inverse Trigonometric Functions

34. Copyright © 2005 Pearson Education, Inc. Slide 6-34 Solving for x in Terms of y Using Inverse Function Example: y = 3 cos 2x for x.

35. Copyright © 2005 Pearson Education, Inc. Slide 6-35 Solving an Equation Involving an Inverse Trigonometric Function Example: Solve 2 arcsin

36. Copyright © 2005 Pearson Education, Inc. Slide 6-36 Solving an Equation Involving Inverse Trigonometric Functions Example: Solve

37. Copyright © 2005 Pearson Education, Inc. Slide 6-37 Solving an Inverse Trigonometric Equation Using an Identity Example: Solve