The principal value of sin^{-1} left( sin fra | vedclass

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(C) We know that the range of the principal value branch of $\sin^{-1} x$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Given expression is $\sin^{-1} \left( \

Vedclass · Accepted Answer

$-\frac{\pi}{3}$

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(C) We know that the range of the principal value branch of $\sin^{-1} x$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Given expression is $\sin^{-1} \left( \sin \frac{5\pi}{3} \right)$.First,simplify the argument: $\sin \frac{5\pi}{3} = \sin \left( 2\pi - \frac{\pi}{3} \right) = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}$.Now,$\sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) = -\sin^{-1} \left( \frac{\sqrt{3}}{2} \right) = -\frac{\pi}{3}$.Since $-\frac{\pi}{3} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,the principal value is $-\frac{\pi}{3}$.

The principal value of $\sin^{-1} \left( \sin \frac{5\pi}{3} \right)$ is

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