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Solving Trig Equations Keeper 19 Accelerated Pre-Calculus HELPFUL HINTS FOR SOLVING TRIGONOMETRIC EQUATIONS •Try to get equations in terms of one trig function by using identities. •Be on the look-out for ways to substitute using identities •Try to get trig functions of the same angle. If one term is cos2 and another is cos for example, use the double angle formula to express first term in terms of just instead of 2 •Get one side equals zero and factor out any common trig functions •See if equation is quadratic in form and will factor. (replace the trig function with x to see how it factors if that helps) •If the angle you are solving for is a multiple of , don't forget to add 2 to your answer for each multiple of since will still be less than 2 when solved for. Example Solve the equation. 2 cos x 1. *** This is called a general solution because it shows a solution for all possible values of x. However most problems will state the interval in which the solution is located; usually Between 0º, 360º or 0,2𝜋 . 0,2𝜋 will be the specified interval for the remainder of the Lesson unless otherwise stated. 2sinx 1 0 1 sinx 2 7 11 x , 6 6 2cos2 x 1 0 1 2 cos x 2 2 2 3 5 7 x , , , 4 4 4 4 cos x 2cos x 3 0 3 cos x 2 11 x , 6 6 tanx 1 5 x , 4 4 3 sinx 2 2 x , 3 3 sinx cos x sinx cos x cos x cos x sinx sinx cos x sinx sinx cos x 0 sinx 1 cos x 0 tanx 1 x sinx 0 5 , 4 4 x 0, cos x 1 x sinx tanx sinx sinx 0 cos x 1 sinx 1 0 cos x sinx 1 sec x 0 sinx 0 x 0, sec x 1 x0 cos2x cos x 2cos2 x 1 cos x 2cos2 x cos x 1 0 2cos x 1 cos x 1 0 1 cos x cos x 1 2 2 4 x , 3 3 x0 2A2 A 1 2A 1 A 1 cos2x cos x 1 0 2cos2 x 1 cos x 1 0 2cos2 x cos x 0 cos x 2cos x 1 0 cos x 0 1 cos x 2 3 x , 2 2 2 4 x , 3 3 cos2 x 3sin2 x cos2 x 3sin2 x 2 sin x sin2 x cot 2 x 3 cot x 3 5 7 11 x , , , 6 6 6 6 2sin2x 1 0 1 sin2x 2 7 11 19 23 2x , , , 6 6 6 6 7 11 19 23 x , , , 12 12 12 12 sin3x 1 0 sin3x 1 3 7 11 3x , , 2 2 2 3 7 11 x , , 6 6 6 cos2x 7sinx 3 0 1 2sin2 x 7sinx 3 0 2sin2 x 7sinx 4 0 2sinx 1 sinx 4 0 1 sinx 4 2 5 7 x , 6 6 sinx 2A2 7A 4 2A 1 A 4 cos 2x 2cos2 x 0 2cos2 x 1 2cos2 x 0 4cos2 x 1 0 1 4 1 cos x 2 2 4 5 x , , , 3 3 3 3 cos2 x sin2x sinx cos x 2sinx cos x sinx cos x 2sin2 cos x cos x 0 cos x 2sin2 x 1 0 1 cos x 0 sin x 2 2 cos x 0 sinx 2 3 3 5 7 x , x , , , 2 2 4 4 4 4 2 csc 2 x cot x 1 cot 2 x 1 cot x 1 cot 2 x cot x cot 2 x cot x 0 cot x cot x 1 0 cot x 0 cot x 1 3 x , 2 2 5 x , 4 4 Example Solve cos sin sin 0 2 2 Example When we don't have squared trig functions, we can't use the Pythagorean identities. If you have two terms with different trig functions you can try squaring both sides. cos sin 0 Example 3tan 2x 3 in the interval 0,2 . Example 1 cos 1 1.2108 in [0º,360º ). 2 Example 2 2 cos u 1 cosu in 0º, 360º . Solve Example 2 10sin x 12sin x 7 0 in 0º, 360º . Solve