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

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(A) We know that the range of the principal value branch of $\sin ^{-1} x$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Since $\frac{3 \pi}{4}$ does not lie i

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$\frac{\pi}{4}$

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(A) We know that the range of the principal value branch of $\sin ^{-1} x$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Since $\frac{3 \pi}{4}$ does not lie in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$,we simplify the expression:$\sin ^{-1}\left(\sin \frac{3 \pi}{4}ight) = \sin ^{-1}\left[\sin \left(\pi - \frac{\pi}{4}ight)ight]$Using the identity $\sin(\pi - 	heta) = \sin 	heta$,we get:$= \sin ^{-1}\left(\sin \frac{\pi}{4}ight)$Since $\frac{\pi}{4} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,the expression simplifies to:$= \frac{\pi}{4}$

The principal value of $\sin ^{-1}\left(\sin \frac{3 \pi}{4}\right)$ is

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