# Solve tan2x=sqrt3 for 0<x<360?

Jun 17, 2018

$x \in \left\{{15}^{\circ} , {105}^{\circ} , {195}^{\circ} , {285}^{\circ}\right\}$

#### Explanation:

Let $\theta = 2 x$
So that the given $\tan \left(2 x\right) = \sqrt{3}$
becomes $\tan \left(\theta\right) = \sqrt{3}$

This is one of the standard reference angles, namely ${30}^{\circ}$

Based on the CAST quadrant layout we know that this reference angle will apply to Quadrants 1 and 3;

that is to the angles ${30}^{\circ}$ and ${180}^{\circ} + {30}^{\circ} = {210}^{\circ}$
for $\theta \in \left[0 : {360}^{\circ}\right]$ i.e. for $x \in \left[{0}^{\circ} : {180}^{\circ}\right]$
and
to the angles ${360}^{\circ} + {30}^{\circ} = {390}^{\circ}$ and ${360}^{\circ} + {180}^{\circ} + {30}^{\circ} = {570}^{\circ}$
for $\theta \in \left({360}^{\circ} : {720}^{\circ}\right]$ i.e. for $x \in \left({180}^{\circ} : {360}^{\circ}\right]$

Therefore $\theta \in \left\{{30}^{\circ} , {210}^{\circ} , {390}^{\circ} , {570}^{\circ}\right\}$
and since $2 x = \theta$
$\textcolor{w h i t e}{\text{XXXXX}} x \in \left\{{15}^{\circ} , {105}^{\circ} , {195}^{\circ} , {285}^{\circ}\right\}$