Question:
Mark the tick against the correct answer in the following:
$\sin \left\{\frac{\pi}{3}-\sin ^{-1}\left(\frac{-1}{2}\right)\right\}=?$
A. 1
B. 0
C. $\frac{-1}{2}$
D. none of these
Solution:
To Find: The value of of $\sin \left\{\frac{\pi}{3}-\sin ^{-1}\left(\frac{-1}{2}\right)\right\}$
Let, $x=\sin \left\{\frac{\pi}{3}-\sin ^{-1}\left(\frac{-1}{2}\right)\right\}$
$\Rightarrow x=\sin \left\{\frac{\pi}{3}-\left(-\sin ^{-1} \frac{1}{2}\right)\right\}\left(\because \sin ^{-1}(-\theta)=-\sin \theta\right)$
$\Rightarrow x=\sin \left(\frac{\pi}{3}+\frac{\pi}{6}\right)$
$\Rightarrow x=\sin \left(\frac{3 \pi}{6}\right)=\sin \left(\frac{\pi}{2}\right)=1$