Question

$2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$ is equal to:

• A. $\frac{7 \pi}{4}$
• B. $\frac{5 \pi}{4}$
• C. $\frac{3 \pi}{2}$
• D. $\frac{\pi}{2}$

Answer 1

Answer: (C)

$2 \pi-\left(\sin ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)+\sin ^{-1}\left(\frac{16}{65}\right)\right)$
$=2 \pi-\left(\tan ^{-1}\left(\frac{4}{3}\right)+\tan ^{-1}\left(\frac{5}{12}\right)+\tan ^{-1}\left(\frac{16}{63}\right)\right)$
$=2 \pi-\left(\tan ^{-1}\left(\frac{63}{16}\right)+\tan ^{-1}\left(\frac{16}{63}\right)\right)$
$=2 \pi-\frac{\pi}{2}=\frac{3 \pi}{2}$