Question

Evaluate each of the following:

(i) ${\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{3}\right)$
(ii) ${\cot }^{-1}\left(\cot \frac{4\mathrm{\pi }}{3}\right)$
(iii) ${\cot }^{-1}\left(\cot \frac{9\mathrm{\pi }}{4}\right)$
(iv) ${\cot }^{-1}\left(\cot \frac{19\mathrm{\pi }}{6}\right)$
(v) ${\cot }^{-1}\left\{\cot \left(-\frac{8\mathrm{\pi }}{3}\right)\right\}$
(vi) ${\cot }^{-1}\left\{\cot \left(\frac{21\mathrm{\pi }}{4}\right)\right\}$

Answer 1

We know that

${\cot }^{-1}\left(\mathrm{cot\theta }\right)=\mathrm{\theta }$, ( 0, π)

(i) We have
${\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{3}\right)=\frac{\mathrm{\pi }}{3}$
(ii) We have
$$\begin{gathered} {\cot }^{-1}\left(\cot \frac{4\mathrm{\pi }}{3}\right)={\cot }^{-1}\left[\cot \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{3}\right)\right] \\ ={\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{3}\right) \\ =\frac{\mathrm{\pi }}{3} \end{gathered}$$
(iii) We have
$$\begin{gathered} {\cot }^{-1}\left(\cot \frac{9\mathrm{\pi }}{4}\right)={\cot }^{-1}\left[\cot \left(2\mathrm{\pi }+\frac{\mathrm{\pi }}{4}\right)\right] \\ ={\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{4}\right) \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$
(iv) We have
$$\begin{gathered} {\cot }^{-1}\left(\cot \frac{19\mathrm{\pi }}{6}\right)={\cot }^{-1}\left[\cot \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{6}\right)\right] \\ ={\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{6}\right) \\ =\frac{\mathrm{\pi }}{6} \end{gathered}$$
(v) We have
$$\begin{gathered} {\cot }^{-1}\left[\cot \left(-\frac{8\mathrm{\pi }}{3}\right)\right]={\cot }^{-1}\left[-\cot \left(\frac{8\mathrm{\pi }}{3}\right)\right] \\ ={\cot }^{-1}\left[-\cot \left(3\mathrm{\pi }-\frac{\mathrm{\pi }}{3}\right)\right] \\ ={\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{3}\right) \\ =\frac{\mathrm{\pi }}{3} \end{gathered}$$
(vi) We have
$$\begin{gathered} {\cot }^{-1}\left(\cot \frac{21\mathrm{\pi }}{4}\right)={\cot }^{-1}\left[\cot \left(5\mathrm{\pi }+\frac{\mathrm{\pi }}{4}\right)\right] \\ ={\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{4}\right) \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$