Question

The number of solution(s) of $\cos x=|1+\sin x|$ in $[0,3\pi ]$ is

Answer 1

$\cos x=|1+\sin x|,0\le x\le 3\pi$
As $1+\sin x\ge 0$
$\cos x=1+\sin x\Rightarrow \cos x-\sin x=1\Rightarrow \frac{1}{\sqrt{2}}\cos x-\frac{1}{\sqrt{2}}\sin x=\frac{1}{\sqrt{2}}\Rightarrow \cos (\frac{\pi }{4}+x)=\frac{1}{\sqrt{2}}\Rightarrow \frac{\pi }{4}+x=2n\pi \pm \frac{\pi }{4}\Rightarrow x=2n\pi ,2n\pi -\frac{\pi }{2}\therefore x=0,2\pi ,\frac{3\pi }{2}$

Hence, the number of solution is $3$.