1. 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

2. OBJECTIVES Trigonometric Equations I SECTION 6.5 1 2 Solve trigonometric equations of the form a sin ( x – c ) = k, a cos ( x – c ) = k, and a tan ( x – c ) = k. Solve trigonometric equations by using the zero-product property. Solve trigonometric equations that contain more than one trigonometric function. Solve trigonometric equations by squaring both sides. 3 4

3. 3 © 2011 Pearson Education, Inc. All rights reserved TRIGONOMETRIC EQUATIONS A trigonometric equation is a conditional equation that contains a trigonometric function with a variable. An identity is an equation that is true for all values in the domain of the variable. Solving a trigonometric equation means finding its solution set.

4. 4 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Trigonometric Equation Find all solutions of each equation. Express all solutions in radians.

5. 5 © 2011 Pearson Education, Inc. All rights reserved a. First find all solutions in [0, 2π). We know and sin x > 0 in quadrants I and II. QI and QII angles with reference angles of are and. EXAMPLE 1 Solving a Trigonometric Equation Solution

6. 6 © 2011 Pearson Education, Inc. All rights reserved Since sin x has a period of 2π, all solutions of the equation are given by or for any integer n. EXAMPLE 1 Solving a Trigonometric Equation Solution continued

7. 7 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Trigonometric Equation Solution a. First find all solutions in [0, 2π). We know and cos θ < 0 in quadrants II and III. QII and QIII angles with reference angles of are and.

8. 8 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Trigonometric Equation Solution continued Since cos θ has a period of 2π, all solutions of the equation are given by or for any integer n.

9. 9 © 2011 Pearson Education, Inc. All rights reserved The QII angle with a reference angle of is. We know and tan x < 0 in quadrant II. EXAMPLE 1 Solving a Trigonometric Equation Solution a. Because tan x has a period of π, first find all solutions in [0, π).

10. 10 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Trigonometric Equation Solution continued Since tan x has a period of π, all solutions of the equation are given by for any integer n.

11. 11 © 2011 Pearson Education, Inc. All rights reserved The reference angle is because In QI and QII, sin θ > 0. EXAMPLE 3 Solving a Linear Trigonometric Equation Find all solutions in the interval [0, 2π) of the equation. Solution Replace with θ in the given equation.

12. 12 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 3 Solution continued Solving a Linear Trigonometric Equation or The solution set in the interval [0, 2π) is

13. 13 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solving a Quadratic Trigonometric Equation Find all solutions of the equation Express the solutions in radians. Solution Factor No solution

14. 14 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solution continued So, Since sin  has a period of 2π, the solutions are for any integer n. are the only solutions in the interval [0, 2π). Solving a Quadratic Trigonometric Equation

15. 15 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Solution Square both sides and use identities to convert to an equation containing only sin x. Find all the solutions in the interval [0, 2π) to the equation Solving a Trigonometric Equation by Squaring

16. 16 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Solution continued Solving a Trigonometric Equation by Squaring

17. 17 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Solution continued Possible solutions are: Solving a Trigonometric Equation by Squaring The solution set in the interval [0, 2π) is