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In this section, you 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 Two basic techniques for solving trigonometric equations are factoring and applying known identities. a) Rewrite the equation with a single trigonometric function. b) Remember only to factor after you have set the equation equal to zero. Collect Like Terms/Take the Square Root: 1) Solve 2tan x 1 3 and find all solutions. 2 tan x 1 3 2 tan x 2 tan x 1 3 7 x and between 0,2 4 4 3 x n 4 Collect Like Terms/Take the Square Root: 2) Solve sec x 2 0 in the interval 0,2 . 2 sec x 2 0 2 sec x 2 2 sec x 2 (Solve for extraneous roots.) 2 3 5 7 cos x x , , , 2 4 4 4 4 3 5 7 x , , , 4 4 4 4 Use identities to solve: 3) Solve sin 2 x cos 2 x for all solutions. 2 2 sin x cos x Set the equation to zero tofactor. sin x cos x 0 2 2 Substitute in known identities. sin x 1 sin x 0 Simplify and solve. 2 2 2sin x 1 0 2 Find all solutions for x. 1 sin x sin x 2 x n 2 4 2 2 2 Use identities to solve: 4) Solve 2sin 2 x cos x 1 0 for the interval 0,2 . 2sin 2 x cos x 1 0 Substitute in known identities. 2 1 cos x cos x 1 0 Simplify and combine like terms. 2 2cos x cos x 1 0 Simplify and combine like terms. 2cos2 x cos x 1 0 Multiply by 1. 2cos2 x cos x 1 0 Factor and solve. 2 2 2cos x 1 cos x 1 0 1 cos x and cos x 1 2 5 x , 3 3 Find all solutions for x. 5 x x , , 3 3 Factor to solve: 5) Solve sin x 3sin x 4 0 for the interval 0,2 . 4 2 Factor to solve: sin 4 x 3sin 2 x 4 0 2 2 sin x 4 sin x 1 0 sin x 4 Factor Solve separately 2 sin x 1 sin x 2 sin x 1 2 x no solution since sine oscillates between 1 and 1 Solving of Multiple Angles: 6) Solve 2sin 4 x 3 0 for all solutions. 2sin 4 x 3 0 3 sin 4 x 2 4x x x 3 12 12 2 4x 3 and x and 2 n, 6 6 2 n Homework Page 376-379 35-47 odd, 49-53 odd, 61-71 odd