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Question

Mathematics Question on Properties of Inverse Trigonometric Functions

The value of cos⁡−1 (sin⁡ 7π6){{\cos }^{-1}}\,\left( \sin \,\frac{7\pi }{6} \right) is equal to

A

π3\frac{\pi }{3}

B

π6\frac{\pi }{6}

C

2π3\frac{2\pi }{3}

D

−π3-\frac{\pi }{3}

Answer

2π3\frac{2\pi }{3}

Explanation

Solution

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