1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved Chapter 6 Trigonometric Identities and Equations

OBJECTIVES Trigonometric Equations II SECTION Solve trigonometric equations involving multiple angles. Solve equations using sum-to-product identities. Solve equations containing inverse trigonometric functions. 3

3 © 2011 Pearson Education, Inc. All rights reserved EQUATIONS INVOLVING MULTIPLE ANGLES If x is the measure of an angle, then for any real number k, the number kx is a multiple angle of x.

4 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Trigonometric Equation Containing Multiple Angles Find all solutions of the equation in the interval [0, 2π). Solution Recall that Since cos θ > 0 in QI and QIV, The period of cos x is 2π. Replace θ with 3x. So or for any integer n.

5 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solution continued or n = –1 n = 0 n = 1 Solving a Trigonometric Equation Containing Multiple Angles  To find solutions in the interval [0, 2π), try the following:

6 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solution continued Values resulting from n = –1 are too small. n = 2 n = 3 Values resulting from n = 3 are too large. The solution set corresponding to n = 0, 1, and 2 is Solving a Trigonometric Equation Containing Multiple Angles  

7 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 3 The Phases of the Moon The moon orbits the earth in 29.5 days. The portion F of the moon visible from the earth x days after the new moon is given by the equation Find x when 75% of the moon is visible from the earth. Solution Find x when F = 75% = 0.75.

8 © 2011 Pearson Education, Inc. All rights reserved Thus, a solution of is EXAMPLE 3 The Phases of the Moon Solution continued 0.75

9 © 2011 Pearson Education, Inc. All rights reserved Since and sin θ > 0 in QI and QII, EXAMPLE 3 The Phases of the Moon Solution continued or So, 75% of the moon is visible 10 days and 20 days after the new moon.

10 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solving an Equation Using a Sum-to-Product Identity Solve cos 2x + cos 3x = 0 over the interval [0, 2π). Solution Use So

11 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solving an Equation Using a Sum-to-Product Identity Solution continued We know if cos θ = 0, then for any integer n, Replace θ with and find the values of x that lie in the interval [0, 2π).

12 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solving an Equation Using a Sum-to-Product Identity Solution continued The solution set to the equation is

13 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 7 Solving an Equation Containing an Inverse Tangent Function Solve Solution