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  4. The value of cos -1 ( sin (7π /6) ) is equal to

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Q. The value of $ {{\cos }^{-1}}\,\left( \sin \,\frac{7\pi }{6} \right) $ is equal to

J & K CETJ & K CET 2009Inverse Trigonometric Functions

Solution:

$ {{\cos }^{-1}}\left( \sin \frac{7\pi }{6} \right)={{\cos }^{-1}}\left[ \sin \left( 2\pi -\frac{5\pi }{6} \right) \right] $
$ ={{\cos }^{-1}}\left[ -\sin \frac{5\pi }{6} \right] $
$ ={{\cos }^{-1}}\left[ -\sin \left( \frac{\pi }{2}+\frac{\pi }{3} \right) \right] $ $ ={{\cos }^{-1}}\left[ -\cos \frac{\pi }{3} \right] $
$ =\pi -\frac{\pi }{3}=\frac{2\pi }{3} $