The principal value of {sin ^{ - 1}}left[ {si | vedclass

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(D) The principal value branch of ${\sin^{-1}}(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Since $\frac{2\pi}{3}$ does not lie in the interval $[-\frac{\

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$\frac{\pi}{3}$

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(D) The principal value branch of ${\sin^{-1}}(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Since $\frac{2\pi}{3}$ does not lie in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$,we simplify the expression.We know that $\sin(\pi - 	heta) = \sin(	heta)$.Therefore,$\sin(\frac{2\pi}{3}) = \sin(\pi - \frac{\pi}{3}) = \sin(\frac{\pi}{3})$.Now,${\sin^{-1}}[\sin(\frac{2\pi}{3})] = {\sin^{-1}}[\sin(\frac{\pi}{3})]$.Since $\frac{\pi}{3} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,the value is $\frac{\pi}{3}$.

The principal value of ${\sin ^{ - 1}}\left[ {\sin \left( {\frac{{2\pi }}{3}} \right)} \right]$ is

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