The value of ^ -1 ( 3 2 )+2 ^ -1 ( 3 2 ) : - Vidyadip

MCQ

The value of $\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)+2 \cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ :

A
$\frac{\pi}{2}$
B
$\frac{\pi}{3}$
✓
$\frac{2 \pi}{3}$
D
$\pi$

✓

Answer

Correct option: C.

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

(C)
The range of principal value branch of $\sin ^{-1} x$ is $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ and the range of principal value branch of $\cos ^{-1} x$ is $[0, \pi]$.
$\therefore \sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3}$ and $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{6}$
Hence, $\frac{\pi}{3}+2 \times \frac{\pi}{6}=\frac{\pi}{3}+\frac{\pi}{3}=\frac{2 \pi}{3}$
Hence correct option is (C)

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