Question

Find the principal values of each of the following:
(i) ${\tan }^{-1}(-\sqrt{3})$
(ii) ${\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)$
(iii) ${\mathrm{cosec}}^{-1}(-\sqrt{2})$
(iv) ${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
(v) ${\sin }^{-1}\left(-\frac{1}{2}\right)$
(vi) ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
(vii) ${\sec }^{-1}(-\sqrt{2})$
(viii) ${\cot }^{-1}(-\sqrt{3})$
(ix) ${\sec }^{-1}\left(2\right)$
(x) ${\mathrm{cosec}}^{-1}\left(\frac{2}{\sqrt{3}}\right)$

Answer 1

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

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

(iii) ​ Let ${\mathrm{cosec}}^{-1}\left(-\sqrt{2}\right)=y$
Then,
$\mathrm{cosec}y=-\sqrt{2}$
We know that the range of the principal value branch is $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]-\left\{0\right\}$.
Thus,
$$\begin{gathered} \mathrm{cosec}y=-\sqrt{2}=\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{4}\right) \\ y=-\frac{\mathrm{\pi }}{4}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right],y\ne 0 \end{gathered}$$
Hence, the principal value of ${\mathrm{cosec}}^{-1}\left(-\sqrt{2}\right)\mathrm{is}-\frac{\mathrm{\pi }}{4}$.

(iv) ​ ​ Let ${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)=y$
Then,
$\cos y=-\frac{\sqrt{3}}{2}$
We know that the range of the principal value branch is $\left[0,\mathrm{\pi }\right]$.
Thus,
$$\begin{gathered} \cos y=-\frac{\sqrt{3}}{2}=\cos \left(\frac{5\mathrm{\pi }}{6}\right) \\ \Rightarrow y=\frac{5\mathrm{\pi }}{6}\in \left[0,\mathrm{\pi }\right] \end{gathered}$$
Hence, the principal value of ${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\mathrm{is}\frac{5\mathrm{\pi }}{6}$.

(v) Let ${\sin }^{-1}\left(-\frac{1}{2}\right)=y$
Then,
$\sin y=-\frac{1}{2}$
We know that the range of the principal value branch is $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$.
Thus,
$$\begin{gathered} \sin y=-\frac{1}{2}=\sin \left(-\frac{\mathrm{\pi }}{6}\right) \\ \Rightarrow y=-\frac{\mathrm{\pi }}{6}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right] \end{gathered}$$
Hence, the principal value of ${\sin }^{-1}\left(-\frac{1}{2}\right)\mathrm{is}-\frac{\mathrm{\pi }}{6}$.

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

(vii) Let ${\sec }^{-1}\left(-\sqrt{2}\right)=y$
Then,
$\sec y=-\sqrt{2}$
We know that the range of the principal value branch is $\left[0,\mathrm{\pi }\right]-\left\{\frac{\mathrm{\pi }}{2}\right\}$.
Thus,
$$\begin{gathered} \sec y=-\sqrt{2}=\sec \left(\frac{3\mathrm{\pi }}{4}\right) \\ \Rightarrow y=\frac{3\mathrm{\pi }}{4}\in \left[0,\mathrm{\pi }\right],y\ne \frac{\mathrm{\pi }}{2} \end{gathered}$$
Hence, the principal value of ${\sec }^{-1}\left(-\sqrt{2}\right)\mathrm{is}\frac{3\mathrm{\pi }}{4}$.

(viii) 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 ${\cot }^{-1}\left(-\sqrt{3}\right)\mathrm{is}\frac{5\mathrm{\pi }}{6}$.

(ix) Let ${\sec }^{-1}\left(2\right)=y$
Then,
$\sec y=2$
We know that the range of the principal value branch is $\left[0,\mathrm{\pi }\right]-\left\{\frac{\mathrm{\pi }}{2}\right\}$.
Thus,
$$\begin{gathered} \sec y=2=\sec \left(\frac{\mathrm{\pi }}{3}\right) \\ \Rightarrow y=\frac{\mathrm{\pi }}{3}\in \left[0,\mathrm{\pi }\right],y\ne \frac{\mathrm{\pi }}{2} \end{gathered}$$
Hence, the principal value of ${\sec }^{-1}\left(2\right)\mathrm{is}\frac{\mathrm{\pi }}{3}$.

(x) Let ${\mathrm{cosec}}^{-1}\left(\frac{2}{\sqrt{3}}\right)=y$
Then,
$\mathrm{cosec}y=\frac{2}{\sqrt{3}}$
We know that the range of the principal value branch is $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]-\left\{0\right\}$.
Thus,
$$\begin{gathered} \mathrm{cosec}y=\frac{2}{\sqrt{3}}=\mathrm{cosec}\left(\frac{\mathrm{\pi }}{3}\right) \\ \Rightarrow y=\frac{\mathrm{\pi }}{3}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right],y\ne 0 \end{gathered}$$
Hence, the principal value of ${\mathrm{cosec}}^{-1}\left(\frac{2}{\sqrt{3}}\right)\mathrm{is}\frac{\mathrm{\pi }}{3}$.