Why $\theta = \arctan(\frac{y}{x})$ in the polar coordinate system?

Question

Many authors write $\theta = \arctan(\frac{y}{x})$ in the polar coordinate system.

If $(x, y) = (1, 1)$, then $\theta = \frac{\pi}{4} = \arctan(\frac{1}{1})$.
In this case, $\theta = \arctan(\frac{y}{x})$ holds.
If $(x, y) = (-1, -1)$, then $\theta = \frac{5 \pi}{4} \ne \frac{\pi}{4} = \arctan(\frac{-1}{-1})$.

Why do many authors write $\theta = \arctan(\frac{y}{x})$?

I cannot understand that.

Answer 1

The notation is correct for $\theta $ betwen $-\pi/2$ and $\pi/2$

The better notation would be $$\tan \theta = \frac {y}{x}$$ and even better is

$ x=r\cos \theta$ and $y=r\sin \theta$

Note that the polar coordinate is not unique so we have to know which quadrant we are talking about to get to a correct value for $r$ and $\theta$