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

The principal value of sin⁡−1(−32)+cos⁡−1(cos⁡7π6)\sin^{- 1}\left( - \frac{\sqrt{3}}{2} \right) + \cos^{- 1}\left( \cos\frac{7\pi}{6} \right) is

A

5π6\frac{5\pi}{6}

B

π2\frac{\pi}{2}

C

3π2\frac{3\pi}{2}

D

None of these

Answer

π2\frac{\pi}{2}

Explanation

Solution

sin⁡−1(−32)=−sin⁡−1(32)=−π3\sin^{- 1}\left( - \frac{\sqrt{3}}{2} \right) = - \sin^{- 1}\left( \frac{\sqrt{3}}{2} \right) = - \frac{\pi}{3} and cos⁡−1(cos⁡7π6)\cos^{- 1}\left( \cos\frac{7\pi}{6} \right)

= cos⁡−1[cos⁡(2π−5π6)]=cos⁡−1(cos⁡5π6)=5π6\cos^{- 1}\left\lbrack \cos\left( 2\pi - \frac{5\pi}{6} \right) \right\rbrack = \cos^{- 1}\left( \cos\frac{5\pi}{6} \right) = \frac{5\pi}{6}so that

sin⁡−1(32)+cos⁡−1(cos⁡7π6)=−π3+5π6=π2\sin^{- 1}\left( \frac{\sqrt{3}}{2} \right) + \cos^{- 1}\left( \cos\frac{7\pi}{6} \right) = - \frac{\pi}{3} + \frac{5\pi}{6} = \frac{\pi}{2}