How to evaluate $\tan 195^{\circ}$ without using the calculator, and how to give the answer in the form $a+b \sqrt{3}$, where $a$ and $b$ are integers?
Evaluate $\tan 195^{\circ}$ without using the calculator
trigonometry
trigonometry
How to evaluate $\tan 195^{\circ}$ without using the calculator, and how to give the answer in the form $a+b \sqrt{3}$, where $a$ and $b$ are integers?
Best Answer
Using the identity $\tan \left(180^\circ+\theta\right)=\tan\theta$ and $\tan \left(\alpha-\beta\right)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$,
$\quad\tan195^\circ \\=\tan 15^\circ\\=\tan \left(60^\circ-45^\circ\right)\\=\dfrac{\tan60^\circ-\tan45^\circ}{1+\tan60^\circ\tan45^\circ}\\=\dfrac{\sqrt{3}-1}{\sqrt{3}+1}\\=\dfrac{\left(\sqrt{3}-1\right)\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}\\=\dfrac{4-2\sqrt{3}}{2}\\=\boxed{2-\sqrt{3}}$