How do we really get the angle of a vector from the components? Usually when people discuss getting the polar form of a vector $v$, they present the following two formulas:
$$\text{Magnitude}(v) = \sqrt{x^2 + y^2}$$
$$\text{Angle}(v) = \arctan \left(\frac{y}{x} \right)$$
$$ \text{ Where } \space v = \begin{bmatrix} x \\ y \end{bmatrix}$$
I believe that this formula for the angle is only partially true. I think a better and more complete formula for the angle should be:
$$ \text{Angle}(v) = \begin{cases}
\arctan \left(\frac{y}{x} \right)  &; \space x \gt 0 \\
\pi +\arctan \left(\frac{y}{x} \right) &; \space x \lt 0 \\
{\begin{cases}
 \operatorname{sign}(y) \frac{\pi}{2}  &; y \neq 0 \\
\text{undefined} &; \space y = 0
\end{cases}} &; x = 0
\end{cases} $$
Is there some sort of way to simplify this or to better express this, or is this it?
 A: If you want an answer in form of a mathematical function definition, using only functions that were in common use in undergraduate instruction fifty years ago, I do not think you can do much better than the excellent answer by Rhys Hughes.
As noted in the answer, this is how mathematicians often define a function equivalent to yours in textbooks.
The only detail you might want to add is something to deal with the case $x = y = 0.$
Note that the formulas in that answer do not tell you how to find $\theta,$
but they do uniquely identify the output of the function for any possible input.
There is always some value of $\theta$ that will satisfy both equations when 
$x^2 + y^2 \neq 0,$ and there will never be more than one value of $\theta$ that satisfies both equations.
If you want a formula to compute the angle using only functions that were in common use in undergraduate instruction fifty years ago, I think the formula you wrote is close to the best you can get, though I would handle one or two cases a bit differently.
If you want a nice way to represent your function in other formulas, you can borrow the two-parameter arc tangent function that is defined in many software packages.
That is, define a function $\operatorname{atan2}(y, x)$
whose value is the angle of the vector $\begin{bmatrix} x \\ y \end{bmatrix}.$
You can define $\operatorname{atan2}(y, x)$ either in the style of the complex analysis textbooks described in the other answer, or you can define it in your style:
$$
\operatorname{atan2}(y, x) = \begin{cases}
\arctan\left(\frac yx\right)      &  x \gt 0 \\
\arctan\left(\frac yx\right) + \pi \quad &  x \lt 0, \ y \geq 0 \\
\arctan\left(\frac yx\right) - \pi \quad &  x \lt 0, \ y < 0 \\
\frac\pi2   & x = 0,\ y > 0 \\
-\frac\pi2  & x = 0,\ y < 0 \\
\text{undefined} & x = 0,\ y = 0.
\end{cases}
$$
When this is implemented in software I think the "undefined" case usually returns $0.$
If you have to write formulas involving the direction angles of several two-dimensional vectors in terms of their components, then you might find the notation
$\operatorname{atan2}(y, x)$ convenient.
A: This looks very similar to complex numbers, where $\text{Magnitude}(v)$ is displayed by $|v|$ and $\text{Angle}(v)$ is written $\text{arg}(v)$. We have that $\arg(v)$ is the unique angle $\in (-\pi, \pi]$ where:
$$\cos\theta =\frac{x}{|v|}=\frac{x}{\sqrt{x^2+y^2}}$$
and
$$\sin\theta=\frac{y}{|v|}=\frac{y}{\sqrt{x^2+y^2}}$$ 
