Understanding polar coordinates

Question

Express $z= -\sqrt{3} - j $

According to the rectangular form:

$a= -\sqrt{3} $, $b= -1$

In polar form ... $z = r\angle \theta$

$r = \sqrt{(-\sqrt{3}^2 + (-1)^2} = 2 $

$ \theta = 210 $ degrees

Therefore ...

$z = 2 \angle 210$

why is this not the final answer but

$z = 2 \angle -150$

I have been told that it’s because of the principal value that $\theta$ will be bigger than $-180$ degrees but not bigger or equals to $180$ degrees .

Why is this so ?

I have searched online and found out that it is because there may be many values for $\theta$ that satisfy the given conditions.

What does this mean ? And why (again) $\theta$ will be bigger than $-180$ degrees but not bigger or equals to $180$ degrees ? Why these 2 numbers ? Thanks

Answer 1

Well, for an angle we know that:

$$210^\circ=210^\circ-360^\circ=-150^\circ\tag1$$

Answer 2

This is because all periodic functions with period $2\pi$ of $\theta$ such as $sin\theta ,cos\theta,tan\theta$ repeat at any interval by length $2\pi$. For example for any complex number $z=re^{i\theta}$ we have:

$$ln \ z=ln \ r +i\theta \qquad \forall \theta\in\Bbb R$$

as logarithm of $z$ (Containing all of its branches) and

$$Ln \ z=ln \ r +i\theta \qquad -\pi\le\theta<\pi$$

as the main branch of logarithm function in $z$. This is just a definition. Of course you should mind it when integrating complex functions on a Jordan curve (those of closed perimeter).