Question

Solve : ${\sin }^{-1}\left\{\frac{\sin x+\cos x}{\sqrt{2}}\right\},-\frac{3\pi }{4}<x<\frac{\pi }{4}$

Answer 1

${\sin }^{-1}\left\{\frac{\sin x+\cos x}{\sqrt{2}}\right\}$

$={\sin }^{-1}\{\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x\}$

$={\sin }^{-1}\{\cos \frac{\pi }{4}\sin x+\sin \frac{\pi }{4}\cos x\}$

$={\sin }^{-1}\left\{\sin (x+\frac{\pi }{4})\right\}$

$=x+\frac{\pi }{4}$

It is given that,

$\frac{-3\pi }{4}<x<\frac{\pi }{4}$

So,

$\frac{-3\pi }{4}+\frac{\pi }{4}<x+\frac{\pi }{4}<\frac{\pi }{4}+\frac{\pi }{4}$

$\frac{-\pi }{2}<x+\frac{\pi }{4}<\frac{\pi }{2}$