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Chapter 5 Analytic Trigonometry © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved 1 SECTION 5.2 Trigonometric Equations OBJECTIVES 1 2 3 4 5 Solve trigonometric equations of the form a sin (x − c) = k, a cos (x − c) = k, and a sin (x − c) = k Solve trigonometric equations involving multiple angles. Solve trigonometric equations by using the zeroproduct property. Solve trigonometric equations that contain more than one trigonometric function. Solve trigonometric equations by squaring both sides. © 2010 Pearson Education, Inc. All rights reserved 2 TRIGONOMETRIC EQUATIONS A trigonometric equation is an equation that contains a trigonometric function with a variable. Equations that are true for all values in the domain of the variable are called identities. Solving a trigonometric equation means to find its solution set. © 2010 Pearson Education, Inc. All rights reserved 3 EXAMPLE 1 Solving a Trigonometric Equation Find all solutions of each equation. Express all solutions in radians. 2 a. sin x 2 3 b. cos 2 c. tan x 3 © 2010 Pearson Education, Inc. All rights reserved 4 EXAMPLE 1 Solving a Trigonometric Equation 2 a. sin x Solution 2 a. First find all solutions in [0, 2π). We know and sin x > 0 only in quadrants I and II. QI and QII angles with reference angles of are: and © 2010 Pearson Education, Inc. All rights reserved 5 EXAMPLE 1 Solving a Trigonometric Equation Solution continued Since sin x has a period of 2π, all solutions of the equation are given by or for any integer n. © 2010 Pearson Education, Inc. All rights reserved 6 EXAMPLE 1 Solving a Trigonometric Equation 3 b. cos 2 Solution a. First find all solutions in [0, 2π). 3 We know cos and cos x < 0 only in 6 2 quadrants II and III. QII and QIII angles with reference angles of 6 7 5 are: and 6 6 6 6 © 2010 Pearson Education, Inc. All rights reserved 7 EXAMPLE 1 Solving a Trigonometric Equation Solution continued Since cos x has a period of 2π, all solutions of the equation are given by 5 7 2n or 2n 6 6 for any integer n. © 2010 Pearson Education, Inc. All rights reserved 8 EXAMPLE 1 Solving a Trigonometric Equation c. tan x 3 Solution a. Because tan x has a period of π, first find all solutions in [0, π). We know tan 3 and tan x < 0 only in 3 quadrant II. The QII angle with a reference angle of is: 3 2 3 3 © 2010 Pearson Education, Inc. All rights reserved 9 EXAMPLE 1 Solving a Trigonometric Equation Solution continued Since tan x has a period of π, all solutions of the equation are given by 2 n 3 for any integer n. © 2010 Pearson Education, Inc. All rights reserved 10 EXAMPLE 3 Solving a Linear Trigonometric Equation Find all solutions in the interval [0, 2π) of the equation: 2sin x 1 2 4 Solution Replace x by in the given equation. 4 1 2sin 1 2 We know sin 6 2 2sin 1 sin > 0 in Q I and II 1 5 sin , 2 6 6 © 2010 Pearson Education, Inc. All rights reserved 11 EXAMPLE 3 Solving a Linear Trigonometric Equation Solution continued 5 or 6 6 5 x x 4 6 4 6 5 x x 6 4 6 4 10 3 13 2 3 5 x x 12 12 12 12 12 12 5 13 Solution set in [0, 2π) is , . 12 12 © 2010 Pearson Education, Inc. All rights reserved 12 Solving a Trigonometric Equation Containing Multiple Angles EXAMPLE 4 1 Find all solutions of the equation cos 3x in 2 the interval [0, 2π). Solution 1 Recall cos . cos > 0 in Q I and IV, 3 2 5 so , 3 3 The period of cos x is 2π. Replace with 3x. 5 3x 2n 3x 2n So or 3 3 © 2010 Pearson Education, Inc. All rights reserved 13 Solving a Trigonometric Equation Containing Multiple Angles EXAMPLE 4 Solution continued 2n 5 2n x x Or or 9 3 9 3 To find solutions in the interval [0, 2π), try: n = –1 n=0 n=1 2 5 x 9 3 9 x 9 2 7 x 9 3 9 5 2 x 9 3 9 5 x 9 5 2 11 x 9 3 9 © 2010 Pearson Education, Inc. All rights reserved 14 EXAMPLE 4 Solving a Trigonometric Equation Containing Multiple Angles Solution continued n=2 n=3 4 13 x 9 3 9 19 x 2 9 9 5 4 17 x 9 3 9 5 23 x 2 9 9 Values resulting from n = –1 are too small. Values resulting from n = 3 are too large. Solutions we want correspond to n = 0, 1, and 2. 5 7 11 13 17 , , , . Solution set is , , 9 9 9 9 9 9 © 2010 Pearson Education, Inc. All rights reserved 15 EXAMPLE 7 Solving a Quadratic Trigonometric Equation Find all solutions of the equation 2 2sin 5sin 2 0. Express the solutions in radians. Solution Factor 2sin 5sin 2 0. 2sin 1 sin 2 0 2sin 1 0 or sin 2 0 1 sin 2 sin 2 No solution because 5 or –1 ≤ sin ≤ 1. 6 6 2 © 2010 Pearson Education, Inc. All rights reserved 16 EXAMPLE 7 Solving a Quadratic Trigonometric Equation Solution continued 5 are the only two and So, 6 6 solutions in the interval [0, 2π). Since sin has a period of 2π, the solutions are 6 2n 5 or 2n , 6 for any integer n. © 2010 Pearson Education, Inc. All rights reserved 17 EXAMPLE 8 Solving a Trigonometric Equation Using Identities Find all the solutions of the equation 2sin 2 3 cos 1 0 in the interval [0, 2π). Solution Use the Pythagorean identity to rewrite the equation in terms of cosine only. 2 sin 2 3 cos 1 0 2 1 cos 3 cos 1 0 2 2 2cos 2 3 cos 1 0 3 2cos 2 3 cos 0 © 2010 Pearson Education, Inc. All rights reserved 18 Solving a Trigonometric Equation Using Identities EXAMPLE 8 Solution continued 2 cos 3 cos 3 0 2 Use the quadratic formula to solve this equation. cos cos 3 3 2 4 2 3 2 2 3 3 24 4 3 27 4 © 2010 Pearson Education, Inc. All rights reserved 33 3 4 19 EXAMPLE 8 Solving a Trigonometric Equation Using Identities Solution continued So, 33 3 cos 4 4 3 cos 4 cos 3 1 No solution because –1 ≤ cos ≤ 1. or 33 3 cos 4 2 3 cos 4 3 cos 2 cos < 0 in QII, QIII © 2010 Pearson Education, Inc. All rights reserved 20 EXAMPLE 8 Solving a Trigonometric Equation Using Identities Solution continued 3 cos when 2 6 5 6 6 7 6 6 5 7 Solution set in the interval [0, 2π) is , . 6 6 © 2010 Pearson Education, Inc. All rights reserved 21 Solving a Trigonometric Equation by Squaring EXAMPLE 9 Find all the solutions in the interval [0, 2π) to the equation 3 cos x sin x 1. Solution Square both sides and use identities to convert to an equation containing only sin x. 3 cos x sin x 1. 3 cos x 2 sin x 1 2 3 cos 2 x sin 2 x 2 sin x 1 3 1 sin x sin x 2 sin x 1 2 2 © 2010 Pearson Education, Inc. All rights reserved 22 EXAMPLE 9 Solving a Trigonometric Equation by Squaring Solution continued 3 3sin 2 x sin 2 x 2sin x 1 2 4 sin x 2sin x 2 0 2sin 2 x sin x 1 0 2sin x 1 sin x 1 0 2sin x 1 0 1 sin x 2 or sin x 1 0 5 x or 6 6 © 2010 Pearson Education, Inc. All rights reserved sin x 1 3 x 2 23 EXAMPLE 9 Solving a Trigonometric Equation by Squaring Solution continued Possible solutions are: x 6 5 x 6 3 x 2 3 cos 6 5 3 cos 6 3 3 cos 2 ? sin 1 6 ? 5 sin 1 6 ? 3 sin 1 2 3 3 2 2 3 3 2 2 00 3 Solution set in the interval [0, 2π) is , . 6 2 © 2010 Pearson Education, Inc. All rights reserved 24