Question

Find the principal values of each of the following:

(i) ${\cot }^{-1}(-\sqrt{3})$
(ii) ${\cot }^{-1}\left(\sqrt{3}\right)$
(iii) ${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$
(iv) ${\cot }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right)$

Answer 1

(i) Let ${\cot }^{-1}\left(-\sqrt{3}\right)=y$
Then,
$\cot y=-\sqrt{3}$
We know that the range of the principal value branch is $\left(0,\mathrm{\pi }\right)$.
Thus,
$$\begin{gathered} \cot y=-\sqrt{3}=\cot \left(\frac{5\mathrm{\pi }}{6}\right) \\ \Rightarrow y=\frac{5\mathrm{\pi }}{6}\in \left(0,\mathrm{\pi }\right) \end{gathered}$$
Hence, the principal value of ${\cot }^{-1}\left(-\sqrt{3}\right)\mathrm{is}\frac{5\mathrm{\pi }}{6}$.

(ii) Let ${\cot }^{-1}\left(\sqrt{3}\right)=y$
Then,
$\cot y=\sqrt{3}$
We know that the range of the principal value branch is $\left(0,\mathrm{\pi }\right)$.
Thus,
$$\begin{gathered} \cot y=\sqrt{3}=\cot \left(\frac{\mathrm{\pi }}{6}\right) \\ \Rightarrow y=\frac{\mathrm{\pi }}{6}\in \left(0,\mathrm{\pi }\right) \end{gathered}$$
Hence, the principal value of ${\cot }^{-1}\left(\sqrt{3}\right)\mathrm{is}\frac{\mathrm{\pi }}{6}$.
(iii) Let ${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)=y$
Then,
$\cot y=-\frac{1}{\sqrt{3}}$
We know that the range of the principal value branch is $\left(0,\mathrm{\pi }\right)$.
Thus,
$$\begin{gathered} \cot y=-\frac{1}{\sqrt{3}}=\cot \left(\frac{2\mathrm{\pi }}{3}\right) \\ \Rightarrow y=\frac{2\mathrm{\pi }}{3}\in \left(0,\mathrm{\pi }\right) \end{gathered}$$
Hence, the principal value of ${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)\mathrm{is}\frac{2\mathrm{\pi }}{3}$.
(iv)
Let ${\cot }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right)=y$
Then,
$\cot y=\tan \frac{3\mathrm{\pi }}{4}$
We know that the range of the principal value branch is $\left(0,\mathrm{\pi }\right)$.
Thus,
$$\begin{gathered} \cot y=\tan \frac{3\mathrm{\pi }}{4}=-1=\cot \left(\frac{3\mathrm{\pi }}{4}\right) \\ \Rightarrow y=\frac{3\mathrm{\pi }}{4}\in \left(0,\mathrm{\pi }\right) \end{gathered}$$
Hence, the principal value of ${\cot }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right)\mathrm{is}\frac{3\mathrm{\pi }}{4}$.