Question

The value of the expression :
$\sin (\frac{\pi }{3}-{\sin }^{-1}(-\frac{1}{2}))$ is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $1$
• D. $\frac{1}{3}$

Answer 1

The correct option is C $1$

Given:
$\sin (\frac{\pi }{3}-{\sin }^{-1}(-\frac{1}{2}))$
$=\sin (\frac{\pi }{3}+{\sin }^{-1}\left(\frac{1}{2}\right))$
We know that the range of the principal value branch of ${\sin }^{-1}$ is $[-\frac{\pi }{2},\frac{\pi }{2}]$.
$\Rightarrow \sin (\frac{\pi }{3}-{\sin }^{-1}(-\frac{1}{2}))=\sin (\frac{\pi }{3}+\frac{\pi }{6})$
$=\sin \left(\frac{\pi }{2}\right)=1$
Hence, the correct option is (D).