Question:
The value of $\sin ^{-1}\left(\cos \frac{33 \pi}{5}\right)$ is
(a) $\frac{3 \pi}{5}$
(b) $-\frac{7 \pi}{5}$
(c) $\frac{\pi}{10}$
(d) $-\frac{\pi}{10}$
Solution:
$\sin ^{-1}\left(\cos \frac{33 \pi}{5}\right)$
$=\sin ^{-1}\left[\cos \left(6 \pi+\frac{3 \pi}{5}\right)\right]$
$=\sin ^{-1}\left(\cos \frac{3 \pi}{5}\right)$
$=\sin ^{-1}\left[\sin \left(\frac{\pi}{2}-\frac{3 \pi}{5}\right)\right]$
$=\sin ^{-1}\left[\sin \left(-\frac{\pi}{10}\right)\right]$
$=-\frac{\pi}{10}$
Thus, the value of $\sin ^{-1}\left(\cos \frac{33 \pi}{5}\right)$ is $-\frac{\pi}{10}$.
Hence, the correct answer is option (d).
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