Question

Find the general solution for the following equation:
sin 2x + cos x = 0

Answer 1

sin 2x + cos x = 0

$\Rightarrow$ 2 sin x cos x + cos x = 0

$\Rightarrow$ cos x (2 sin x + 1) = 0

$\Rightarrow$ Either cos x = 0 or 2sin x + 1 = 0

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

or sin x = -$\frac{1}{2}$ = - sin $\frac{\pi }{6}$ = sin $(-\frac{\pi }{6})$, n $ϵ$ Z

x = (2n+1) $\frac{\pi }{2}$ or x = $n\pi +(-1{)}^n(-\frac{\pi }{6})$

x = (2n+1) $\frac{\pi }{2}$ or x = $n\pi +(-1{)}^{n+1}\left(\frac{\pi }{6}\right)$

or x = n$\pi +(-1{)}^n\frac{7\pi }{6}$

$[\because sin(\pi +\frac{\pi }{6})=-sin\frac{\pi }{6}]$

$\Rightarrow$ x = (2n+1) $\frac{\pi }{2}$ or x = $n\pi +(-1{)}^n\frac{7\pi }{6}$, n $ϵ$ Z