Question

How do I find the value of $\tan \left(\frac{5\mathrm{\pi }}{6}\right)$?

Answer 1

Find the value of $\tan \left(\frac{5\mathrm{\pi }}{6}\right)$:

Let us simplify $\tan \left(\frac{5\mathrm{\pi }}{6}\right)$ to get the required value:

$\begin{array}{rcl}\tan \left(\frac{5\pi }{6}\right)& =& \tan \left(\pi -\frac{\pi }{6}\right)\\ & =& \frac{\tan \pi -\tan \left(\frac{\pi }{6}\right)}{1+\tan \left(\pi \right)\tan \left(\frac{\pi }{6}\right)}\left[\because \tan \left(a+b\right)=\frac{\tan a+\tan b}{1+\tan a\tan b}\right]\\ & =& -\tan \left(\frac{\pi }{6}\right)\left[\because \tan \left(\pi \right)=0\right]\\ & =& -\frac{1}{\sqrt{3}}\left[\because \tan \left(\frac{\pi }{6}\right)=-\frac{1}{\sqrt{3}}\right]\\ & =& -\frac{\sqrt{3}}{3}\end{array}$

Therefore, the value of $\tan \left(\frac{5\mathrm{\pi }}{6}\right)$is $-\frac{\sqrt{3}}{3}$.