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# Evaluate the following: ${\cos ^{ - 1}}(\cos 5)$  Answer Verified
Hint: We are going to solve the given problem using $\cos^{-1} ({\cos }\theta ) = \theta$ if $\left( {0 \leqslant \theta \leqslant \pi } \right)$

$\because$5 >$\pi$(radian measure), we have
${\cos ^{ - 1}}\left( {\cos 5} \right) = {\cos ^{ - 1}}\left\{ {\cos \left( {2\pi - 5} \right)} \right\}$
[ $\because \cos (2\pi - \theta ) = \cos \theta$ ]
${\cos ^{ - 1}}\left( {\cos 5} \right) = 2\pi - 5$
$\therefore$ The value of ${\cos ^{ - 1}}\left( {\cos 5} \right) = 2\pi - 5$

Note:
The measure of 5 radians lie in the fourth quadrant.
We have $\cos^{-1} ({\cos }\theta ) = \theta$ only for $\left( {0 \leqslant \theta \leqslant \pi } \right)$
So we converted that as $\cos 5 = cos\left( {2\pi - 5} \right)$. The value of $\left( {2\pi - 5} \right)$ is less than $\pi$.
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CBSE Class 12 Maths Chapter-2 Inverse Trigonometric Functions Formula  Inverse Trigonometric Functions  Important Properties of Inverse Trigonometric Functions  Graphical Representation of Inverse Trigonometric Functions  Cos Inverse Formula  Inverse Functions  Trigonometric Functions  Integration of Trigonometric Functions  CBSE Class 11 Maths Chapter 3 - Trigonometric Functions Formulas  Composition of Functions and Inverse of a Function  