The value of sin^{-1}left(sin frac{5pi}{3}rig | vedclass

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Question

(A) We know that 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

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) We know that the principal value branch of $\sin^{-1}(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Given expression is $\sin^{-1}\left(\sin \frac{5\pi}{3}ight)$.First,simplify the angle inside the sine function:$\frac{5\pi}{3} = 2\pi - \frac{\pi}{3}$.Since $\sin(2\pi - 	heta) = -\sin(	heta)$,we have:$\sin\left(\frac{5\pi}{3}ight) = \sin\left(2\pi - \frac{\pi}{3}ight) = -\sin\left(\frac{\pi}{3}ight) = \sin\left(-\frac{\pi}{3}ight)$.Now,substitute this back into the expression:$\sin^{-1}\left(\sin\left(-\frac{\pi}{3}ight)ight)$.Since $-\frac{\pi}{3} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,the value is $-\frac{\pi}{3}$.Therefore,the correct option is $A$.

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

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