Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 6 Trigonometric Identities and Equations © 2011 Pearson Education, Inc. All rights reserved © 2010 2011 Pearson Education, Inc. All rights reserved 1 SECTION 6.5 Trigonometric Equations I OBJECTIVES 1 2 3 4 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. 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. © 2011 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 © 2011 Pearson Education, Inc. All rights reserved 4 EXAMPLE 1 Solution Solving a Trigonometric Equation 2 a. sin x 2 a. First find all solutions in [0, 2π). We know I and II. and sin x > 0 in quadrants QI and QII angles with reference angles of are and . © 2011 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. © 2011 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 θ < 0 in 6 2 quadrants II and III. QII and QIII angles with reference angles of 6 7 5 are and . 6 6 © 2011 Pearson Education, Inc. All rights reserved 6 6 7 EXAMPLE 1 Solving a Trigonometric Equation Solution continued Since cos θ has a period of 2π, all solutions of the equation are given by 5 7 2n or 2n 6 6 for any integer n. © 2011 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 in 3 quadrant II. The QII angle with a reference angle of is 3 2 . x 3 3 © 2011 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 x n 3 for any integer n. © 2011 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 4 2sin 1 2 2sin 1 1 sin 2 with θ in the given equation. The reference angle is 6 1 . 6 2 In QI and QII, sin θ > 0. because sin © 2011 Pearson Education, Inc. All rights reserved 11 EXAMPLE 3 Solving a Linear Trigonometric Equation Solution continued 5 or 6 6 5 or 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 The solution set in the interval [0, 2π) is , . 12 12 © 2011 Pearson Education, Inc. All rights reserved 12 EXAMPLE 6 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 5 No solution or 6 6 2 © 2011 Pearson Education, Inc. All rights reserved 13 EXAMPLE 6 Solving a Quadratic Trigonometric Equation Solution continued 5 are the only solutions and So, 6 6 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. © 2011 Pearson Education, Inc. All rights reserved 14 EXAMPLE 8 Solving a Trigonometric Equation by Squaring 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 © 2011 Pearson Education, Inc. All rights reserved 15 EXAMPLE 8 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 sin x 1 5 x or 6 6 © 2011 Pearson Education, Inc. All rights reserved 3 x 2 16 EXAMPLE 8 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 The solution set in the interval [0, 2π) is , . 6 2 © 2011 Pearson Education, Inc. All rights reserved 17