Question

The number of real solutions for $x,y$ if $y=\left|\sin x\right|$ and $y={\sin }^{-1}\left(\sin x\right)$ where $xϵ[-2\pi ,2\pi ]$ is:

• A. $4$
• B. $5$
• C. $3$
• D. $2$

Answer 1

The correct option is B $5$

$y=\mid \sin x\mid$
and $y={\sin }^{-1}\left(\sin x\right)=2\pi +x$ if $-2\pi \le x\le -\frac{3\pi }{2}$
$=-\pi -x$ if $-\frac{3\pi }{2}<x\le -\frac{\pi }{2}$
$=x$ if $-\frac{\pi }{2}<x\le \frac{\pi }{2}$
$=\pi -x$ if $\frac{\pi }{2}<x\le \frac{3\pi }{2}$
$=-2\pi +x$ if $\frac{3\pi }{2}<x\le 2\pi$
Plotting both the curve it can be easily observed that both curve intersect at $5$ points in the given interval, these points are $x=-2\pi ,-\pi ,0,\pi ,2\pi$