Question

Write the principal value of ${\sin }^{-1}\left\{\cos \left({\sin }^{-1}\frac{1}{2}\right)\right\}$.

Answer 1

Given, ${\sin }^{-1}\left\{\cos \left({\sin }^{-1}\frac{1}{2}\right)\right\}$

$={\sin }^{-1}\left\{\cos \frac{\pi }{6}\right\}$ $[\because \sin \frac{\pi }{6}=\frac{1}{2}]$

$={\sin }^{-1}\frac{\sqrt{3}}{2}$ $[\because \sin \frac{\pi }{6}=\frac{\sqrt{3}}{2}]$

$=\frac{\pi }{3}$ $[\because \sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}]$