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Solving Trigonometric Equations MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 J. Robert Buchanan Solving Trigonometric Equations Objectives In this lesson we will learn to: use standard algebraic techniques to solve trigonometric equations, solve trigonometric equations of quadratic type, solve trigonometric equations involving multiple angles, use inverse trigonometric functions to solve trigonometric equations. J. Robert Buchanan Solving Trigonometric Equations Examples Solve the following trigonometric equations: 2 sin x + 1 = 0 3 cot2 x − 1 = 0 tan2 3x = 3 2 sin2 x + 3 sin x + 1 = 0 cos x + sin x tan x = 2 sin x − 2 = cos x − 2 √ 3 sin 2x = − 2 tan2 x − tan x − 2 = 0 J. Robert Buchanan Solving Trigonometric Equations Equation (1 of 8) 2 sin x + 1 = 0 sin x x= = − 1 2 7π 11π + 2nπ or x = + 2nπ 6 6 where n is any integer. J. Robert Buchanan Solving Trigonometric Equations Equation (2 of 8) cot 2Π y -1 = 3 3 3 cot2 x − 1 = 0 1 cot2 x = 3 cot 1.0 1 Π = 3 3 0.5 1 -1.0 = ±√ 3 π 2π x = + nπ or x = + nπ 3 3 0.5 -0.5 cot x 1.0 -0.5 cot J. Robert Buchanan 4Π 1 = 3 -1.0 cot 3 Solving Trigonometric Equations 5Π -1 = 3 3 x Equation (3 of 8) tan2 3x = 3 √ tan 3x = ± 3 π 2π 3x = + nπ or 3x = + nπ 3 3 π nπ 2π nπ x= + or x = + 9 3 9 3 J. Robert Buchanan Solving Trigonometric Equations Equation (4 of 8) 2 sin2 x + 3 sin x + 1 = 0 (2 sin x + 1)(sin x + 1) = 0 2 sin x + 1 = 0 or sin x + 1 = 0 1 or sin x = −1 sin x = − 2 7π 3π x= + 2nπ or x = + 2nπ 6 2 J. Robert Buchanan Solving Trigonometric Equations Equation (5 of 8) cos x + sin x tan x sin2 x cos x + cos x 2 cos x + sin2 x = 2 = 2 = 2 cos x 1 = 2 cos x 1 cos x = 2 π 5π x = + 2nπ or x = + 2nπ 3 3 J. Robert Buchanan Solving Trigonometric Equations Equation (6 of 8) y 1.0 tan Π 4 =1 0.5 sin x − 2 = cos x − 2 sin x = cos x tan x = 1 π = + nπ 4 x -1.0 0.5 -0.5 -0.5 tan J. Robert Buchanan 5Π 4 =1 -1.0 Solving Trigonometric Equations 1.0 x Equation (7 of 8) √ sin 2x = − 3 2 4π 5π + 2nπ or 2x = + 2nπ 3 3 2π 5π x= + nπ or x = + nπ 3 6 2x = J. Robert Buchanan Solving Trigonometric Equations Equation (8 of 8) tan2 x − tan x − 2 = 0 (tan x − 2)(tan x + 1) = 0 tan x − 2 = 0 or tan x + 1 = 0 tan x = 2 or tan x = −1 3π x = tan−1 (2) + nπ or x = + nπ 4 where n is an integer. J. Robert Buchanan Solving Trigonometric Equations Homework Read Section 5.3. Exercises: 1, 5, 9, 13, . . . , 69, 73 J. Robert Buchanan Solving Trigonometric Equations