Question

The value of $\sin^{-1}\left ( \cos\, \frac { 53 \pi}{5}\right)$ is

• A. $\frac {3 \pi}{5}$
• B. $\frac {-3 \pi}{5}$
• C. $\frac {\pi}{10}$
• D. $\frac {- \pi}{10}$

Answer 1

Answer: (D)

We have, $\sin ^{-1}\left(\cos \frac{53 \pi}{5}\right)=\sin ^{-1}\left[\cos \left(10 \pi+\frac{3 \pi}{5}\right)\right]$
$=\sin ^{-1}\left[\cos \left(\frac{3 \pi}{5}\right)\right]$
$[\because \cos (2 n \pi \pm \theta)=\cos \theta]$
$=\sin ^{-1}\left[\sin \left(\frac{\pi}{2}-\frac{3 \pi}{5}\right)\right]$
$=\sin ^{-1}\left[\sin \left(\frac{5 \pi-6 \pi}{10}\right)\right]$
$=\sin ^{-1}\left[\sin \left(\frac{-\pi}{10}\right)\right]=\frac{-\pi}{10}$