How to calculate $\theta$ when we know $\tan \theta$. Hej
I'm having difficulties calculating the angle given the tangent.
Example:
In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this except for when I get to calculating the angle $\theta$.
I know that $\tan \theta = -\frac{1}{\sqrt{3}}$  but I do not know how to continue and compute the angle from that.

You shouldn't use the tangent for this kind of problems; compute
$$
|z|=\sqrt{z\bar{z}}=\sqrt{(\sqrt{3}-i)(\sqrt{3}+i)}=
\sqrt{3+1}=2
$$
Then you have $z=|z|u$, where
$$
u=\frac{\sqrt{3}}{2}-i\frac{1}{2}
$$
and you need an angle $\theta$ such that
$$
\cos\theta=\frac{\sqrt{3}}{2},\quad\sin\theta=-\frac{1}{2}.
$$
Since the sine is negative and the cosine is positive, you see that you can take
$$
\theta=-\frac{\pi}{6}
$$
(the pair of values is well known). If you need an angle in the interval $[0,2\pi)$, just take
$$
-\frac{\pi}{6}+2\pi=\frac{11\pi}{6}.
$$
