Question

$\sin \left[{\sin }^{-1}\left(\frac{2}{3}\right)+2{\cos }^{-1}\frac{2}{3}\right]$

• A. $\frac{3}{2}$
• B. $\frac{1}{2}$
• C. $\frac{2}{3}$
• D. none of these

Answer 1

Answer: (C)

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