Question

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

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

Answer 1

The correct option is B $\frac{3\pi }{2}$

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

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

$=2\pi -{\tan }^{-1}\left(\frac{48+15}{36-20}\right)-{\tan }^{-1}\left(\frac{16}{63}\right)$

$=2\pi -[{\tan }^{-1}\left(\frac{63}{16}\right)+{\cot }^{-1}\left(\frac{63}{16}\right)]$

$=2\pi -\frac{\pi }{2}$ $(\because {\tan }^{-1}+{\cot }^{-1}x=\frac{\pi }{2})$

$=\frac{3\pi }{2}$