Question

Evaluate each of the following:

(i) ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{\mathrm{\pi }}{4}\right)$
(ii) ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{3\mathrm{\pi }}{4}\right)$
(iii) ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{6\mathrm{\pi }}{5}\right)$
(iv) ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{11\mathrm{\pi }}{6}\right)$
(v) ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{13\mathrm{\pi }}{6}\right)$
(vi) ${\mathrm{cosec}}^{-1}\left\{\mathrm{cosec}\left(-\frac{9\mathrm{\pi }}{4}\right)\right\}$

Answer 1

We know that

${\mathrm{cosec}}^{-1}\left(\mathrm{cosec\theta }\right)=\mathrm{\theta }$, [−π/2, 0) ⋃ ( 0, π/2]

(i) ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{\mathrm{\pi }}{4}\right)=\frac{\mathrm{\pi }}{4}$
(ii)
$$\begin{gathered} {\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{3\mathrm{\pi }}{4}\right)={\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(\mathrm{\pi }-\frac{\mathrm{\pi }}{4}\right)\right] \\ ={\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{\mathrm{\pi }}{4}\right) \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$
(iii)
$$\begin{gathered} {\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{6\mathrm{\pi }}{5}\right)={\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(\mathrm{\pi }+\frac{\mathrm{\pi }}{5}\right)\right] \\ ={\mathrm{cosec}}^{-1}\left(\mathrm{cosec}-\frac{\mathrm{\pi }}{5}\right) \\ =-\frac{\mathrm{\pi }}{5} \end{gathered}$$
(iv)
$$\begin{gathered} {\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{11\mathrm{\pi }}{6}\right)={\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(2\mathrm{\pi }-\frac{\mathrm{\pi }}{6}\right)\right] \\ ={\mathrm{cosec}}^{-1}\left(\mathrm{cosec}-\frac{\mathrm{\pi }}{6}\right) \\ =-\frac{\mathrm{\pi }}{6} \end{gathered}$$
(v)
$$\begin{gathered} {\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{13\mathrm{\pi }}{6}\right)={\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(2\mathrm{\pi }+\frac{\mathrm{\pi }}{6}\right)\right] \\ ={\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{\mathrm{\pi }}{6}\right) \\ =\frac{\mathrm{\pi }}{6} \end{gathered}$$
(vi)
$$\begin{gathered} {\mathrm{cosec}}^{-1}\left\{\mathrm{cosec}\left(-\frac{9\mathrm{\pi }}{4}\right)\right\}={\mathrm{cosec}}^{-1}\left[-\mathrm{cosec}\left(2\mathrm{\pi }+\frac{\mathrm{\pi }}{4}\right)\right] \\ ={\mathrm{cosec}}^{-1}\left(-\mathrm{cosec}\frac{\mathrm{\pi }}{4}\right) \\ ={\mathrm{cosec}}^{-1}\left(\mathrm{cosec}-\frac{\mathrm{\pi }}{4}\right) \\ =-\frac{\mathrm{\pi }}{4} \end{gathered}$$