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

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

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

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(B) We know that the range of the principal value branch of ${\sin ^{ - 1}}x$ is $\left[ { - \frac{\pi }{2}, \frac{\pi }{2}} \right]$.Since ${\sin ^{ - 1}}( - x) = - {\sin ^{ - 1}}x$ for all $x \in [ - 1, 1]$,we have:${\sin ^{ - 1}}\left( { - \frac{{\sqrt 3 }}{2}} \right) = - {\sin ^{ - 1}}\left( {\frac{{\sqrt 3 }}{2}} \right)$.We know that $\sin \left( \frac{\pi }{3} \right) = \frac{{\sqrt 3 }}{2}$,therefore ${\sin ^{ - 1}}\left( \frac{{\sqrt 3 }}{2} \right) = \frac{\pi }{3}$.Thus,${\sin ^{ - 1}}\left( { - \frac{{\sqrt 3 }}{2}} \right) = - \frac{\pi }{3}$.

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

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