Discussion
The value of \(cos^{-1}\left(cos\frac{2\pi}{3}\right) + sin^{-1}\left(sin\frac{2\pi}{3}\right)\) is equal to
Answer: E
The value is \(\pi\).
The principal ranges are \([0, \pi]\) for \(\cos^{-1}\) and \([-\frac{\pi}{2}, \frac{\pi}{2}]\) for \(\sin^{-1}\).
\[ \begin{aligned} \cos^{-1}\left(\cos\tfrac{2\pi}{3}\right) &= \tfrac{2\pi}{3} \\ \sin^{-1}\left(\sin\tfrac{2\pi}{3}\right) &= \sin^{-1}\left(\sin\tfrac{\pi}{3}\right) = \tfrac{\pi}{3} \\ \text{sum} &= \pi \end{aligned} \]
\(\frac{2\pi}{3}\) lies outside the range of \(\sin^{-1}\), so the second term is \(\pi - \frac{2\pi}{3}\). Taking it as \(\frac{2\pi}{3}\) would give \(\frac{4\pi}{3}\), which is not offered.
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