Question

Evaluate each of the following:

(i) ${\tan }^{-1}\left(\tan \frac{\mathrm{\pi }}{3}\right)$
(ii) ${\tan }^{-1}\left(\tan \frac{6\mathrm{\pi }}{7}\right)$
(iii) ${\tan }^{-1}\left(\tan \frac{7\mathrm{\pi }}{6}\right)$
(iv) ${\tan }^{-1}\left(\tan \frac{9\mathrm{\pi }}{4}\right)$
(v) ${\tan }^{-1}\left(\tan 1\right)$
(v) ${\tan }^{-1}\left(\tan 2\right)$
(v) ${\tan }^{-1}\left(\tan 4\right)$
(v) ${\tan }^{-1}\left(\tan 12\right)$

Answer 1

We know that

${\tan }^1\left(\tan \theta \right)=\theta ,-\frac{\mathrm{\pi }}{2}<\theta <\frac{\mathrm{\pi }}{2}$

(i) We have
${\tan }^{-1}\left(\tan \frac{\mathrm{\pi }}{3}\right)=\frac{\mathrm{\pi }}{3}$
(ii) We have
$$\begin{gathered} {\tan }^{-1}\left(\tan \frac{6\mathrm{\pi }}{7}\right)={\tan }^{-1}\left[\tan \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{7}\right)\right] \\ ={\tan }^{-1}\left[\tan \left(-\frac{\mathrm{\pi }}{7}\right)\right] \\ =-\frac{\mathrm{\pi }}{7} \end{gathered}$$
(iii) We have

$$\begin{gathered} {\tan }^{-1}\left(\tan \frac{7\mathrm{\pi }}{6}\right)={\tan }^{-1}\left[\tan \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{6}\right)\right] \\ ={\tan }^{-1}\left[\tan \left(\frac{\mathrm{\pi }}{6}\right)\right] \\ =\frac{\mathrm{\pi }}{6} \end{gathered}$$
(iv) We have

$$\begin{gathered} {\tan }^{-1}\left(\tan \frac{9\mathrm{\pi }}{4}\right)={\tan }^{-1}\left[\tan \left(2\mathrm{\pi }+\frac{\mathrm{\pi }}{4}\right)\right] \\ ={\tan }^{-1}\left[\tan \left(\frac{\mathrm{\pi }}{4}\right)\right] \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$
(v) We have
${\tan }^{-1}\left(\tan 1\right)=1$
(vi) We have

$$\begin{gathered} {\tan }^{-1}\left(\tan 2\right)={\tan }^{-1}\left[\tan \left(-\mathrm{\pi }+2\right)\right] \\ =2-\mathrm{\pi } \end{gathered}$$

(vii) We have

$$\begin{gathered} {\tan }^{-1}\left(\tan 4\right)={\tan }^{-1}\left[\tan \left(-\mathrm{\pi }+4\right)\right] \\ =4-\mathrm{\pi } \end{gathered}$$

(viii) We have

$$\begin{gathered} {\tan }^{-1}\left(\tan 12\right)={\tan }^{-1}\left[\tan \left(-4\mathrm{\pi }+12\right)\right] \\ =12-4\mathrm{\pi } \end{gathered}$$