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

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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.**

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` .**

To solve the given trig equation, transform it into a product of 3 basic trig equations. Solve: cos 3x - cos x + sin 2x = 0.

First, transform (cos 3x - cos x), by using trig identiy #27 (cos a - cos b). See book titled "Solving trig equations and inequalities" (Amazon e-book 2010).

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

(-2sin 2x.sin x) + sin 2x = 0

sin 2x (1 - 2sin x) = 0 Factor out sin 2x

2sin x.cos x (1-2sin x) = 0 Identity: sin 2a = 2sin a. cos a

Next, solve the 3 basic trig equations within the period (0 , 2Pi), using the trig unit circle and trig conversion table (or calculator):

1. sin x = 0 --> x = 0 ; x = Pi ; x = 2Pi

2. cos x = 0 --> x = Pi/2 and x = 3Pi/2

3. sin x = 1/2 --> x= Pi/6 and x = 5Pi/6.

Finally, if the end points 0 and 2Pi are included, the answers are:

0 ; Pi/6 ; Pi/2 ; 5Pi/6 ; Pi ; 3Pi/2 ; 2Pi, or in degrees:

0 ; 30 ; 90 ; 150 ; 180 ; 270 ; and 360.

You can check these answers, using a graphing calculator, and by graphing the trig function F(x) = cos 3x - cos x + sin 2x. Remember that the values of x (real roots) will be given in decimals. For example Pi is given by the value 3.14