MCQ
$\sin \left( {\frac{\pi }{3} - {{\sin }^{ - 1}}\left( { - \frac{1}{2}} \right)} \right) = $ ............. .
- A$\frac{1}{2}$
- B$\frac{1}{3}$
- ✓$1$
- D$\frac{1}{4}$
Then $, \sin x=\frac{-1}{2}=-\sin \frac{\pi}{6}=\sin \left(\frac{-\pi}{6}\right)$
We know that the range of the principal value branch of $\sin ^{-1}$ is $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ $\sin ^{-1}\left(\frac{-1}{2}\right)=\frac{\pi}{6}$
$\therefore \sin \left(\frac{\pi}{3}-\sin ^{-1}\left(\frac{-1}{2}\right)\right)=\sin \left(\frac{\pi}{3}+\frac{\pi}{6}\right)$$=\sin \left(\frac{3 \pi}{6}\right)=\sin \left(\frac{\pi}{2}\right)=1$
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