Question

Find the general solution of the equation $\cos 3x+\cos x-\cos 2x=0$

Answer 1

Simplification

Given: $\cos 3x+\cos x-\cos 2x=0$

$\Rightarrow (\cos 3x+\cos x)-\cos 2x=0$

$\Rightarrow 2\cos \left(\frac{3x+x}{2}\right)\cos \left(\frac{3x-x}{2}\right)-\cos 2x=0$

$\Rightarrow 2\cos \left(\frac{4x}{2}\right)\cos \left(\frac{2x}{2}\right)-\cos 2x=0$

$\Rightarrow 2\cos 2x\cos x-\cos 2x=0$

$\cos 2x=0$ or $2\cos x-1=0$

$\cos 2x=0$ or $\cos x=\frac{1}{2}$

General solution for $\cos 2x=0$

The general solution is

$2x=(2n+1)\frac{\pi }{2}$

Or $x=(2n+1)\frac{\pi }{4}$ where $n\in Z$

General solution for $\cos x=\frac{1}{2}$

$\Rightarrow \cos x=\cos \frac{\pi }{3}$

We know that general solution for $\cos x=\cos y$ is $x=2n\pi \pm y,n\in Z$

Put $y=\frac{\pi }{3}$

$x=2n\pi \pm \frac{\pi }{3}$ where $n\in Z$
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