# Solve the trig equation: cos 3x - cos x + sin 2x = 0 for (0 < x < 2pi)

The equation to be solved is cos 3x - cos x + sin 2x = 0

cos 3x - cos x + sin 2x = 0

=> `4*cos^3 x - 3*cos x - cos x + 2*cos x*sin x = 0`

=> `cos x(4*cos^2 x - 4 + 2*sin x) = 0`

=> `cos x(4 - 4*sin^2 x + 2*sin x - 4) = 0`

=> `cos x(4*sin^2 x - 2*sin x) = 0`

=> `cos x = 0` and `4*sin^2 x - 2*sin x = 0`

=> `cos x = 0` and `sin x = 1/2`

=> x = 0 and x = 270 and x = 30 and x = 150

The solution of the equation are x = 0, x = 30 and x = 150  and x = 270 degrees.

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You need to write the equation in terms of one whole angle such that:

`4cos^3 x - 3 cos x - cos x + 2 sin x*cos x = 0`

Collecting like terms yields:

`4cos^3 x - 4 cos x + 2 sin x* cos x = 0`

Factoring out cos x yields:

`cos x*(4 cos^2 x - 4 + 2 sin x) = 0`

You need to solve for x the equation `cos x = 0 =gt x = pi/2; x = 3pi/2`

You need to solve

`4 cos^2 x - 4 + 2 sin x = 0`

You need to use the basic rigonometric formula to write the equation  `4 cos^2 x - 4 + 2 sin x = 0`  in terms of sin x only such that:

`4(1 - sin^2 x) - 4 + 2 sin x = 0`

Opening the brackets yields:

`-4sin^2 x + 2 sin x = 0`

Factoring out sin x yields:

`sin x(-4 sin x + 2) = 0`

`` `sin x = 0 =gt x = pi`

`-4sin x + 2 = 0 =gt -4sin x = -2 =gt sin x = 1/2`

`x = pi/6 ; x = 5pi/6 ; x = 7pi/6 ; x = 11pi/6`

Hence,evaluating the solutions to the given equation, for `x in (0,2pi)`  yields `x = pi/6 ; x = pi/2 ; x = 5pi/6 ; x = pi ; 7pi/6 ; x = 3pi/2 ; x = 11pi/6` .

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