Find the principal value of cosec-1(-2). - Vidyadip

Question

Find the principal value of cosec^-1$(-\sqrt{2})$.

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Answer

Let cosec^-1$(-\sqrt{2})=y$. Then, cosec y = $-\sqrt{2}=-cosec \left(\frac{\pi}{4}\right)=cosec \left(-\frac{\pi}{4}\right)$.
We know that the range of the principal value branch of
cosec^-1 is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}$ and $cosec \left(-\frac{\pi}{4}\right)=-\sqrt{2}$.
Therefore, the principal value of cosec^-1$(-\sqrt{2})$ is $-\frac{\pi}{4}$.

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