Question

The principal value of ${\cos }^{-1}\left[\sin \right({\cos }^{-1}\frac{1}{2}\left)\right]$.

• A. $\frac{\pi }{5}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{12}$

Answer 1

The correct option is C $\frac{\pi }{6}$

We know, ${\cos }^{-1}\frac{1}{2}=\frac{\pi }{3}$

Therefore,

${\cos }^{-1}\left[\sin \right({\cos }^{-1}\frac{1}{2}\left)\right]$

$={\cos }^{-1}\left(\sin \frac{\pi }{3}\right)$

$={\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi }{6}$