Question

Find the exact value of the expression:

$\tan {195}^{°}$

Short answer

$\left(\frac{3-\sqrt{3}}{3+\sqrt{3}}\right)$

Step-by-step solution

Step 1. Given information 

We have been given an expression $\tan {195}^{°}$.

We have to find its exact value.

Step 2. Simplify the expression

We can also write the expression as:

$\tan {190}^{°}=\tan ({150}^{°}+45°)$

Step 3.  Apply the compound angle formula

$\tan ({150}^{°}+45°)$has the form $\tan (\alpha +\beta )$

Therefore, apply

$\tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-+\tan \alpha \tan \beta }$

$\begin{array}{r}\frac{\tan \left({150}^{\circ }\right)+\tan \left({45}^{\circ }\right)}{1-\tan \left({150}^{\circ }\right)\tan \left({45}^{\circ }\right)}=\frac{-\left(\frac{\sqrt{3}}{3}\right)+1}{1-\left(-\left(\frac{\sqrt{3}}{3}\right)\right)}\\ =\frac{\left(\frac{3}{3}\right)-\left(\frac{\sqrt{3}}{3}\right)}{\left(\frac{3}{3}\right)+\left(\frac{\sqrt{3}}{3}\right)}\\ \\ =\frac{\left(\frac{3-\sqrt{3}}{3}\right)}{\left(\frac{3+\sqrt{3}}{3}\right)}\\ =\left(\frac{3-\sqrt{3}}{3+\sqrt{3}}\right)\end{array}$