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Simple Trigonometric Equations Examples for trigonometric equations 2 cos x 3 cos x 8 0 sin x cos x 1 2 tan x cot x 3 Recall cos x sin x 1 2 2 1 tan x sec x 2 2 1 cot x csc x sin x tan x cos x 2 2 sin cos tan sin cos tan 180 sin -cos -tan 180+ -sin -cos tan 180 S A T C 180 360 360 -sin cos -tan Example Solve the equation cos x=1/2. Solution: 1 cos x 2 x 60or 360 60 x 60or 300 360 where = 60 Example Solve the equation sin x = -0.7 for 0<x<2. Solution: sin x 0.7 x 0.775or 2 0.775 x 3.917or 5.508 2 where 0.775 Class Practice Solve the following equations for 0360. 1 1. sin x = 2 2. tan x = 1 3. cos x = -0.1 Answers 1. X=30 or 150. 2. X=45 or 225 . 3. X=95.74 or 275.74 . Procedure in solving trigonometric equations Example Solve the equation Solution: 2 cos 2 x 3 sin x 0 , for 0 x 360 2 cos 2 x 3 sin x 0 2(1 sin 2 x) 3 sin x 0 2 sin 2 x 3 sin x 2 0 (2 sin x 1)(sin x 2) 0 2 sin x 1 0or sin x 2 0 1 sin x , or sin x 2(rejected ) 2 x 180 30, or 360 30 x 210, or 330 180 360 Example Solve the equation for 0 360 . sec 2 1 3 tan Solution: sec 2 1 3 tan (tan 2 1) 1 3 tan tan 2 3 tan 2 0 (tan 2)(tan 1) 0 tan 2, or tan 1 180 63.43,180 63.43, or 45, or180 45 63.43,243.43, or 45,225 Example Solve the equation 5 cos for 0 x < . 2 x 4 sin x cos x 2 sin 2 x 2 Solution: 5 cos 2 x 4 sin x cos x 2 sin 2 x 2(sin 2 x cos 2 x) 3 cos 2 x 4 sin x cos x 4 sin 2 x 0 cos 2 x 4 sin x cos x 4 sin 2 x 3( ) 0 cos 2 x cos 2 x cos 2 x 4 tan 2 x 4 tan x 3 0 (2 tan x 3)( 2 tan x 1) 0 3 1 , or tan x 2 2 x 0.98, 0.98, or 0.46,2 0.46 tan x x 0.98,4.12.or 2.68,5.82 2 Class Practice Textbook Ex.5C : 1 - 5