Question

Find the value of each of the following:

$$\begin{gathered} \left(i\right){\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right) \\ \left(ii\right){\tan }^{-1}\left(\frac{-1}{\sqrt{3}}\right) \\ \left(iii\right){\tan }^{-1}\left(\cos \left(\frac{\mathrm{\pi }}{2}\right)\right) \\ \left(iv\right){\tan }^{-1}\left(2\cos \left(\frac{2\mathrm{\pi }}{3}\right)\right) \end{gathered}$$

Answer 1

Solve for the required values

(i) ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Let ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)=\theta$

$\begin{array}{cccl}\Rightarrow & \theta & =& {\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)\\ & & =& {\tan }^{-1}\left(\tan \left(\frac{\pi }{6}\right)\right)\\ \Rightarrow & \theta & =& \frac{\pi }{6}\end{array}$

(ii) ${\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

Let ${\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)=\theta$

$\begin{array}{cccl}\Rightarrow & \theta & =& {\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)\\ & & =& {\tan }^{-1}\left(\tan \left(-\frac{\pi }{6}\right)\right)\\ \Rightarrow & \theta & =& -\frac{\pi }{6}\end{array}$

(iii) ${\tan }^{-1}\left(\cos \left(\frac{\mathrm{\pi }}{2}\right)\right)$

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

$\begin{array}{cccl}\Rightarrow & \theta & =& {\tan }^{-1}\left(\cos \left(\frac{\mathrm{\pi }}{2}\right)\right)\\ & & =& {\tan }^{-1}\left(0\right)\\ \Rightarrow & \theta & =& 0\end{array}$

(iv) ${\tan }^{-1}\left(2\cos \left(\frac{2\mathrm{\pi }}{3}\right)\right)$

Let ${\tan }^{-1}\left(2\cos \left(\frac{2\mathrm{\pi }}{3}\right)\right)=\theta$

$\begin{array}{cccl}\Rightarrow & \theta & =& {\tan }^{-1}\left(2\cos \left(\frac{2\mathrm{\pi }}{3}\right)\right)\\ & & =& {\tan }^{-1}\left(2\cos \left(\pi -\frac{\pi }{3}\right)\right)\\ & & =& {\tan }^{-1}\left(-2\cos \left(\frac{\pi }{3}\right)\right)\\ & & =& {\tan }^{-1}\left(-2\times \frac{1}{2}\right)\\ & & =& {\tan }^{-1}\left(-1\right)\\ \Rightarrow & \theta & =& -\frac{\pi }{4}\end{array}$