Question

​Find the principal values of each of the following:

(i) ${\mathrm{cosec}}^{-1}(-\sqrt{2})$
(ii) ${\mathrm{cosec}}^{-1}(-2)$
(iii) ${\mathrm{cosec}}^{-1}\left(\frac{2}{\sqrt{3}}\right)$
(iv) ${\mathrm{cosec}}^{-1}\left(2\cos \frac{2\mathrm{\pi }}{3}\right)$

Answer 1

(i) Let ${\mathrm{cosec}}^{-1}\left(-\sqrt{2}\right)=y$
Then,
$\mathrm{cosec}y=-\sqrt{2}$
We know that the range of the principal value branch is $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]-\left\{0\right\}$.
Thus,
$$\begin{gathered} \mathrm{cosec}y=-\sqrt{2}=\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{4}\right) \\ y=-\frac{\mathrm{\pi }}{4}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right],y\ne 0 \end{gathered}$$
Hence, the principal value of ${\mathrm{cosec}}^{-1}\left(-\sqrt{2}\right)\mathrm{is}-\frac{\mathrm{\pi }}{4}$.

(ii)
Let ${\mathrm{cosec}}^{-1}\left(-2\right)=y$
Then,
$\mathrm{cosec}y=-2$
We know that the range of the principal value branch is $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]-\left\{0\right\}$.
Thus,
$$\begin{gathered} \mathrm{cosec}y=-2=\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{6}\right) \\ y=-\frac{\mathrm{\pi }}{6}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right],y\ne 0 \end{gathered}$$
Hence, the principal value of ${\mathrm{cosec}}^{-1}\left(-2\right)\mathrm{is}-\frac{\mathrm{\pi }}{6}$.

(iii) Let ${\mathrm{cosec}}^{-1}\left(\frac{2}{\sqrt{3}}\right)=y$
Then,
$\mathrm{cosec}y=\frac{2}{\sqrt{3}}$
We know that the range of the principal value branch is $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]-\left\{0\right\}$.
Thus,
$$\begin{gathered} \mathrm{cosec}y=\frac{2}{\sqrt{3}}=\mathrm{cosec}\left(\frac{\mathrm{\pi }}{3}\right) \\ \Rightarrow y=\frac{\mathrm{\pi }}{3}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right],y\ne 0 \end{gathered}$$
Hence, the principal value of ${\mathrm{cosec}}^{-1}\left(\frac{2}{\sqrt{3}}\right)\mathrm{is}\frac{\mathrm{\pi }}{3}$.

(iv)
Let ${\mathrm{cosec}}^{-1}\left(2\cos \frac{2\mathrm{\pi }}{3}\right)=y$
Then,
$\mathrm{cosec}y=2\cos \frac{2\mathrm{\pi }}{3}$
We know that the range of the principal value branch is $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]-\left\{0\right\}$.
Thus,
$$\begin{gathered} \mathrm{cosec}y=2\cos \frac{2\mathrm{\pi }}{3}=2\times \frac{-1}{2}=-1=\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow y=-\frac{\mathrm{\pi }}{2}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right],y\ne 0 \end{gathered}$$
Hence, the principal value of ${\mathrm{cosec}}^{-1}\left(2\cos \frac{2\mathrm{\pi }}{3}\right)\mathrm{is}-\frac{\mathrm{\pi }}{2}$.