Question

For the principal values, evaluate the following:

(i) ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)+{\mathrm{cosec}}^{-1}\left(-\frac{2}{\sqrt{3}}\right)$
(ii) ${\sec }^{-1}\left(\sqrt{2}\right)+2{\mathrm{cosec}}^{-1}\left(-\sqrt{2}\right)$
(iii) ${\sin }^{-1}\left[\cos \left\{2{\mathrm{cosec}}^{-1}\left(-2\right)\right\}\right]$
(iv) ${\mathrm{cosec}}^{-1}\left(2\tan \frac{11\mathrm{\pi }}{6}\right)$

Answer 1

(i)
$$\begin{gathered} {\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)+{\mathrm{cosec}}^{-1}\left(-\frac{2}{\sqrt{3}}\right)=-{\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)+{\mathrm{cosec}}^{-1}\left(-\frac{2}{\sqrt{3}}\right) \\ =-{\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{3}\right)+{\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{3}\right)\right] \\ =-\frac{\mathrm{\pi }}{3}-\frac{\mathrm{\pi }}{3} \\ =-\frac{2\mathrm{\pi }}{3} \end{gathered}$$
(ii)
$$\begin{gathered} {\sec }^{-1}\left(\sqrt{2}\right)+2{\mathrm{cosec}}^{-1}\left(-\sqrt{2}\right)={\sec }^{-1}\left(\sec \frac{\mathrm{\pi }}{4}\right)+2{\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{4}\right)\right] \\ =\frac{\mathrm{\pi }}{4}-2\times \frac{\mathrm{\pi }}{4} \\ =\frac{\mathrm{\pi }}{4}-\frac{\mathrm{\pi }}{2} \\ =-\frac{\mathrm{\pi }}{4} \end{gathered}$$
(iii)

$$\begin{gathered} {\sin }^{-1}\left[\cos \left\{2{\mathrm{cosec}}^{-1}\left(-2\right)\right\}\right]={\sin }^{-1}\left[\cos \left\{2{\mathrm{cosec}}^{-1}\left(\mathrm{cosec}-\frac{\mathrm{\pi }}{6}\right)\right\}\right] \\ ={\sin }^{-1}\left[\cos \left\{-\frac{\mathrm{\pi }}{3}\right\}\right] \\ ={\sin }^{-1}\left[\cos \left(\frac{\mathrm{\pi }}{3}\right)\right] \\ ={\sin }^{-1}\left(\frac{1}{2}\right) \\ ={\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{6}\right) \\ =\frac{\mathrm{\pi }}{6} \end{gathered}$$
(iv)
$$\begin{gathered} {\mathrm{cosec}}^{-1}\left(2\tan \frac{11\mathrm{\pi }}{6}\right)={\mathrm{cosec}}^{-1}\left[2\times \left(-\frac{1}{\sqrt{3}}\right)\right] \\ ={\mathrm{cosec}}^{-1}\left[-\frac{2}{\sqrt{3}}\right] \\ ={\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{3}\right)\right] \\ =-\frac{\mathrm{\pi }}{3} \end{gathered}$$