Question

​Find the principal values of each of the following:

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

(iv) ${\cos }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right)$

Answer 1

(i) 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 ${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\mathrm{is}\frac{5\mathrm{\pi }}{6}$.

(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 ${\cos }^{-1}\left(\sin \frac{4\mathrm{\pi }}{3}\right)=y$
Then,
$\cos y=\sin \frac{4\mathrm{\pi }}{3}$
We know that the range of the principal value branch is $\left[0,\mathrm{\pi }\right]$.
Thus,
$$\begin{gathered} \cos y=\sin \frac{4\mathrm{\pi }}{3}=-\frac{\sqrt{3}}{2}=\cos \left(\frac{5\mathrm{\pi }}{6}\right) \\ \Rightarrow y=\frac{5\mathrm{\pi }}{6}\in \left[0,\mathrm{\pi }\right] \end{gathered}$$
Hence, the principal value of ${\cos }^{-1}\left(\sin \frac{4\mathrm{\pi }}{3}\right)\mathrm{is}\frac{5\mathrm{\pi }}{6}$.

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