How to solve $\mathrm{cot}(2x)=\frac {-7}{6}?$

Question

This has been bothering me for a while now.

$\mathrm{cot}(2x)=\frac{-7}{6}$

How do I solve for an angle x in a double angle situation such as this? I honestly don't even know where to start.

Trying anything at one point, I thought I could rearrange to get $\tan(2x)$ then apply $\mathrm{arctan}$ and divide by $2$ but that doesn't work.

The angle $x$ is supposed to be $69.7$ degrees from the solutions I'm working with.

Any help appreciated.

Answer 1

$$\tan(2x)=-\frac{6}{7}$$ If you just take the arctan, you get $-40.6^\circ$. Note that the tangent function has a periodicity of $180^\circ$, so you can use $2x=180^\circ-40.6^\circ=139.4^\circ$, which yields $x=69.7^\circ$

Answer 2

$$ \cot (2x) = -7/6$$

$$ \tan (2x) = -6/7$$

$$ 2x= \tan ^{-1} (-6/7)$$

$$x= (1/2) \tan ^{-1} (-6/7)\approx -20.30 $$degrees

For a positive solution you may add 90 degrees to get

$$ x= (1/2) \tan ^{-1} (-6/7) + 90 \approx 69.69 $$degrees