Question

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

Answer 1

Step 1: Simplification
Given : $\sin 2x+\cos x=0$
$\Rightarrow 2\sin x\cos x+\cos x=0$
$\Rightarrow \cos x=0$ or $2\sin x+1=0$
$\cos x=0$ or $\sin x=-\frac{1}{2}$

Step 2: General solution for $\cos x=0$
The general solution is $x=(2n+1)\frac{\pi }{2}$ where $n\in Z$

Step 3: General solution for $\sin x=-\frac{1}{2}$
$\sin x=-\frac{1}{2}$
$\Rightarrow \sin x=\sin \frac{7\pi }{6}$
$x=n\pi +(-1{)}^n\frac{7\pi }{6},$ where $n\in Z$