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Question
The value of \[\cos^{- 1} \left( \cos\frac{5\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{5\pi}{3} \right)\] is
Options
`pi/2`
`(5pi)/3`
`(10pi)/3`
0
Solution
(d) 0
We have
\[\cos^{- 1} \left( \cos\frac{5\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{5\pi}{3} \right) = \cos^{- 1} \left\{ \cos\left( 2\pi - \frac{\pi}{3} \right) \right\} + \sin^{- 1} \left\{ \sin\left( 2\pi - \frac{\pi}{3} \right) \right\}\]
\[ = \cos^{- 1} \left\{ \cos\left( \frac{\pi}{3} \right) \right\} + \sin^{- 1} \left\{ - \sin\left( \frac{\pi}{3} \right) \right\}\]
\[ = \cos^{- 1} \left\{ \cos\left( \frac{\pi}{3} \right) \right\} - \sin^{- 1} \left\{ \sin\left( \frac{\pi}{3} \right) \right\}\]
\[ = \frac{\pi}{3} - \frac{\pi}{3}\]
\[ = 0\]
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