# How do you find the exact value of tanx-3cotx=0 in the interval 0<=x<360^@?

Nov 17, 2016

$x \setminus \in \left\{{60}^{\circ} , {120}^{\circ} , {240}^{\circ} , {300}^{\circ}\right\}$

#### Explanation:

It goes as follows:
$\tan x - 3 \cot x = 0$
$\tan x = 3 \cot x$
$\tan x = 3 \setminus \cdot \frac{1}{\tan} x$
${\tan}^{2} x = 3$
$\tan x = \sqrt{3}$ or $\tan x = - \sqrt{3}$
$x = {60}^{\circ} + k \cdot {180}^{\circ}$ or $x = {120}^{\circ} + k \cdot {180}^{\circ}$ where $k$ is an integer

Now we can simply list all solutions that fall between ${0}^{\circ}$ and ${360}^{\circ}$:
$x \setminus \in \left\{{60}^{\circ} , {120}^{\circ} , {240}^{\circ} , {300}^{\circ}\right\}$

Note: at the very begining we should also include the domain of this equation; we want the $\tan x$ and $\cot x$ to exist so:
$x \setminus \in \setminus m a t h \boldsymbol{R} \setminus \setminus \left\{k \cdot {90}^{\circ} | k \text{ is integer}\right\}$
Fortunately enough, the solutions that we find fall into the domain.