Question

For the principal values, evaluate each of the following:

$\left(\mathrm{i}\right){\tan }^{-1}(-1)+{\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)$
(ii) ${\tan }^{-1}\left\{2\sin \left(4{\cos }^{-1}\frac{\sqrt{3}}{2}\right)\right\}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right){\tan }^{-1}\left(-1\right)+{\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right) \\ ={\tan }^{-1}\left\{\tan \left(-\frac{\mathrm{\pi }}{4}\right)\right\}+{\cos }^{-1}\left(\cos \frac{3\mathrm{\pi }}{4}\right)\left[\because \mathrm{Range}\mathrm{of}\tan \mathrm{is}\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right);-\frac{\mathrm{\pi }}{4}\in \left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right) \\ \mathrm{and}\mathrm{range}\mathrm{of}\mathrm{cosine}\mathrm{is}\left[0,\mathrm{\pi }\right];\frac{3\mathrm{\pi }}{4}\in \left[0,\mathrm{\pi }\right]\right] \\ =-\frac{\mathrm{\pi }}{4}+\frac{3\mathrm{\pi }}{4} \\ =\frac{\mathrm{\pi }}{2} \end{gathered}$$
$\therefore {\tan }^{-1}\left(-1\right)+{\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)=\frac{\mathrm{\pi }}{2}$

(ii)
$$\begin{gathered} {\tan }^{-1}\left\{2\sin \left(4{\cos }^{-1}\frac{\sqrt{3}}{2}\right)\right\}={\tan }^{-1}\left\{2\sin \left[4{\cos }^{-1}\left(\cos \frac{\mathrm{\pi }}{6}\right)\right]\right\} \\ ={\tan }^{-1}\left\{2\sin \left[4\times \frac{\mathrm{\pi }}{6}\right]\right\} \\ ={\tan }^{-1}\left(2\sin \frac{2\mathrm{\pi }}{3}\right) \\ ={\tan }^{-1}\left[2\times \left(\frac{\sqrt{3}}{2}\right)\right] \\ ={\tan }^{-1}\left(\sqrt{3}\right) \\ ={\tan }^{-1}\left[\tan \left(\frac{\mathrm{\pi }}{3}\right)\right] \\ =\frac{\mathrm{\pi }}{3} \end{gathered}$$