The value of cos ^{-1}left(cos left(frac{7 pi | vedclass

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Question

(A) We know that the range of the principal value branch of $\cos ^{-1} x$ is $[0, \pi]$.Given expression is $\cos ^{-1}\left(\cos \left(\frac{7 \pi}{

Vedclass · Accepted Answer

$\frac{5 \pi}{6}$

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(A) We know that the range of the principal value branch of $\cos ^{-1} x$ is $[0, \pi]$.Given expression is $\cos ^{-1}\left(\cos \left(\frac{7 \pi}{6}ight)ight)$.Since $\frac{7 \pi}{6} > \pi$,we cannot directly write $\cos ^{-1}(\cos 	heta) = 	heta$.We use the property $\cos(2 \pi - 	heta) = \cos 	heta$.$\cos \left(\frac{7 \pi}{6}ight) = \cos \left(2 \pi - \frac{5 \pi}{6}ight) = \cos \left(\frac{5 \pi}{6}ight)$.Now,$\cos ^{-1}\left(\cos \left(\frac{5 \pi}{6}ight)ight) = \frac{5 \pi}{6}$,which lies in the interval $[0, \pi]$.Thus,the correct option is $A$.

The value of $\cos ^{-1}\left(\cos \left(\frac{7 \pi}{6}\right)\right)$ is

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