What is the principal value of ^ -1 (- 3 2 ) - Vidyadip

Question

What is the principal value of $\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)$

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Answer

$\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)$ $-\sin^{-1}\Big(\frac{\sqrt3}{2}\Big)$ $\Big\{\text{Since},\sin^{-1}(-\theta)=-\sin^{-1}(\theta)\Big\} $$=-\frac{\pi}{3} $ $\Big\{\text{Since},\sin^{-1}\text{x}=\text{An angle in }\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]\text{ whose sine is x}\Big\}$
Hence, $\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)=-\frac{\pi}{3}$

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