Question

Find the exact value of $\tan \frac{\mathrm{\pi }}{3}$.

Answer 1

Compute the required value:

As we know, $\tan \theta =\frac{\sin \theta }{\cos \theta }$.

That is

$\tan \left(\frac{\mathrm{\pi }}{3}\right)=\frac{\sin \left({\displaystyle \frac{\mathrm{\pi }}{3}}\right)}{\cos \left({\displaystyle \frac{\mathrm{\pi }}{3}}\right)}$.

Since $\sin \left(\frac{\mathrm{\pi }}{3}\right)=\frac{\sqrt{3}}{2},\cos \left(\frac{\mathrm{\pi }}{3}\right)=\frac{1}{2}$, we can write that

$$\begin{gathered} \tan \left(\frac{\mathrm{\pi }}{3}\right)=\frac{{\displaystyle \frac{\sqrt{3}}{2}}}{{\displaystyle \frac{1}{2}}} \\ \tan \left(\frac{\mathrm{\pi }}{3}\right)=\frac{\sqrt{3}}{2}\times 2 \\ \tan \left(\frac{\mathrm{\pi }}{3}\right)=\sqrt{3} \end{gathered}$$.

Hence, the value of $\tan \frac{\mathrm{\pi }}{3}$ is $\sqrt{3}$.