# Solve the equation?

## Solve $\tan x - \sqrt{3} = 0$ , $x$$\in$$\left(- \pi , - \frac{\pi}{2}\right) \cup \left(- \frac{\pi}{2} , \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2} , \pi\right)$

Apr 9, 2018

$x = \frac{\pi}{3}$ or $x = - \frac{2 \pi}{3}$

#### Explanation:

$\tan \left(x\right) - \sqrt{3} = 0$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow \tan \left(x\right) = \sqrt{3}$

In Quadrant I, this is one of the standard triangles: Using the CAST notation for the Quadrants, a reference angle in Quadrant III will have the same $\tan \left(x\right)$ value i.e. $\left(- \pi + \frac{\pi}{3}\right)$ will have the same value. Apr 9, 2018

$x = \frac{\pi}{3} + k \pi$

#### Explanation:

$\tan x = \sqrt{3}$
Trig table and unit circle give 2 solutions:
$x = \frac{\pi}{3}$ and $x = \frac{\pi}{3} + \pi = \frac{4 \pi}{3}$
$x = \frac{\pi}{3} + k \pi$
Inside the interval $\left(- \pi , - \frac{\pi}{2}\right)$, the answer is $\frac{4 \pi}{3}$
Inside the interval $\left(- \frac{\pi}{2} , \frac{\pi}{2}\right)$, the answer is $\left(\frac{\pi}{3}\right)$
No answer in the interval $\left(\frac{\pi}{2} , \pi\right)$