Question

Solve : ${\cos }^{-1}\left(\sin \right({\cos }^{-1}x\left)\right)=\frac{\pi }{6}$.

Answer 1

Given : ${\cos }^{-1}\left(\sin \right({\cos }^{-1}x\left)\right)=\frac{\pi }{6}$
$\Rightarrow \sin \left({\cos }^{-1}x\right)=\cos \frac{\pi }{6}$
$\Rightarrow \sin \left({\cos }^{-1}x\right)=\sin \frac{\pi }{3}$ $\left[\begin{array}{c}\because \cos \frac{\pi }{6}=\cos \left(\begin{array}{c}\frac{\pi }{2}-\frac{\pi }{3}\end{array}\right)=\sin \frac{\pi }{3}\end{array}\right]$
$\Rightarrow {\cos }^{-1}x=\frac{\pi }{3}$
$\Rightarrow x=\cos \frac{\pi }{3}$
Hence, $x=\frac{1}{2}$.