1. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 6 Inverse Circular Functions and Trigonometric Equations

2. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2 6.1 Inverse Circular Functions 6.2 Trigonometric Equations I 6.3 Trigonometric Equations II 6.4 Equations Involving Inverse Trigonometric Functions 6 Inverse Circular Functions and Trigonometric Equations

3. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Trigonometric Equations II 6.3 Equations with Half-Angles ▪ Equations with Multiple Angles

4. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Example 1 SOLVING AN EQUATION WITH A HALF- ANGLE The two numbers over the interval with sine value (a) over the interval and (b) for all solutions.

5. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Example 2 SOLVING AN EQUATION USING A DOUBLE ANGLE IDENTITY or Factor.

6. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Caution In Example 2, because 2 is not a factor of cos 2x, The only way to change cos 2x to a trigonometric function of x is by using one of the identities for cos 2x.

7. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 SOLVING AN EQUATION USING A MULTIPLE-ANGLE IDENTITY Solution set: {30°, 60°, 210°, 240°} From the given interval 0 ° ≤ θ < 360°, the interval for 2θ is 0 ° ≤ 2θ < 720°. Example 3a

8. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 SOLVING AN EQUATION USING A MULTIPLE-ANGLE IDENTITY Solution set, where 180º represents the period of sin2θ: {30° + 180°n, 60° + 180°n, where n is any integer} All angles 2θ that are solutions of the equation are found by adding integer multiples of 360° to the basic solution angles, 60° and 120°. Solve for all solutions. Example 3b

9. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE Solve tan 3x + sec 3x = 2 over the interval One way to begin is to express everything in terms of secant. Square both sides.

10. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE (continued) Multiply each term of the inequality by 3 to find the interval for 3x: Using a calculator and the fact that cosine is positive in quadrants I and IV, we have

11. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE (continued) Since the solution was found by squaring both sides of an equation, we must check that each proposed solution is a solution of the original equation. Solution set: {0.2145, 2.3089, 4.4033}