Find the value of sin ^{-1}left(sin frac{3 pi | vedclass

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(D) We know that the principal value branch of $\sin ^{-1} x$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Since $\frac{3 \pi}{5} 
otin [-\frac{\pi}{2}, \fra

Vedclass · Accepted Answer

$\frac{2 \pi}{5}$

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(D) We know that the principal value branch of $\sin ^{-1} x$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.Since $\frac{3 \pi}{5} 
otin [-\frac{\pi}{2}, \frac{\pi}{2}]$,we cannot directly write $\sin ^{-1}(\sin x) = x$.We use the property $\sin(\pi - 	heta) = \sin 	heta$.Thus,$\sin(\frac{3 \pi}{5}) = \sin(\pi - \frac{3 \pi}{5}) = \sin(\frac{2 \pi}{5})$.Since $\frac{2 \pi}{5} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,we have $\sin ^{-1}(\sin \frac{2 \pi}{5}) = \frac{2 \pi}{5}$.Therefore,$\sin ^{-1}(\sin \frac{3 \pi}{5}) = \frac{2 \pi}{5}$.

Find the value of $\sin ^{-1}\left(\sin \frac{3 \pi}{5}\right)$

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