Rectangular to polar conversion angle error

Question

I am trying to determine the polar form of the following rectangular vector:

-105 + 140j

The polar form is $\sqrt(-105)^2+140^2$ = 175 and the angle is determined by $\arctan(\frac{140}{-105}) = -53.13$

The answer provided gives an angle of $\theta=126.87$, which is 180 degrees later. Also this website gives the 126.87 degrees.

Why is that and which is correct?

Answer 1

Here is a visualization:

(Large Version)

So $-53.13$ is not a correct answer, it would run clockwise, assigning a point in the $4$-th quadrant.

It is a problem with how to pick $\tan^{-1}$.

(Large Version)

The $\color{green}{\text{dark green curve}}$ is the graph of $y = \tan(x)$. If one inverts it, by mirroring it along $y = x$, you would get a multivalued relation, but a function must have at most one value per argument.

So one limits the inverse to a strip parallel to the $x$-axis of width $\pi$.

The $\color{red}{\text{red curve}}$ is the graph of the $\arctan$ function your calculator offered, and is offered by much of the available mathematical software.

The $\color{purple}{\text{purple curve}}$ is the graph of the inverse function you would need, which is $y = \arctan(x) + \pi$

Note that there is nothing wrong with your calculator, instead of offering many versions of $\tan^{-1}$ the user is expected to do the shift himself.