Question

Write the principal value of \[\cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{2\pi}{3} \right)\]

Answer 1

\[\text{We have}, \cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{2\pi}{3} \right) = \cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left\{ \sin\left( \pi - \frac{\pi}{3} \right) \right\} \left[ \because \left( \pi - \frac{2\pi}{3} \right) \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \right]\]
\[ = \cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left\{ \sin\left( \frac{\pi}{3} \right) \right\}\]
\[ = \frac{2\pi}{3} + \frac{\pi}{3}\]
\[ = \pi\]
∴ \[\cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{2\pi}{3} \right) = \pi\]