# How do you find the exact value of tan(pi/3)?

Mar 18, 2018

The value of $\tan \left(\frac{\pi}{3}\right)$ is $\sqrt{3}$.

#### Explanation:

We can use this fundamental trigonometric identity:

$\tan \theta = \sin \frac{\theta}{\cos} \theta$

Here's a reference triangle with our $\angle \theta$:

Since we know $\sin \left(\frac{\pi}{3}\right)$ is $\frac{\sqrt{3}}{2}$ and $\cos \left(\frac{\pi}{3}\right)$ is $\frac{1}{2}$, we can use the previously stated identity to figure out the value of $\tan \left(\frac{\pi}{3}\right)$:

$\tan \left(\frac{\pi}{3}\right) = \frac{\quad \sin \left(\frac{\pi}{3}\right) \quad}{\cos} \left(\frac{\pi}{3}\right)$

$\textcolor{w h i t e}{\tan \left(\frac{\pi}{3}\right)} = \frac{\quad \frac{\sqrt{3}}{2} \quad}{\frac{1}{2}}$

$\textcolor{w h i t e}{\tan \left(\frac{\pi}{3}\right)} = \frac{\sqrt{3}}{2} \cdot \frac{2}{1}$

$\textcolor{w h i t e}{\tan \left(\frac{\pi}{3}\right)} = \frac{\sqrt{3}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}} \cdot \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}}{1}$

$\textcolor{w h i t e}{\tan \left(\frac{\pi}{3}\right)} = \frac{\sqrt{3}}{1} \cdot \frac{1}{1}$

$\textcolor{w h i t e}{\tan \left(\frac{\pi}{3}\right)} = \frac{\sqrt{3}}{1} \cdot 1$

$\textcolor{w h i t e}{\tan \left(\frac{\pi}{3}\right)} = \frac{\sqrt{3}}{1}$

$\textcolor{w h i t e}{\tan \left(\frac{\pi}{3}\right)} = \sqrt{3}$

That's the value of $\tan \left(\frac{\pi}{3}\right)$. Hope this helped!