How do we deal with ArcTan (or other inverse functions) of undefined values?

Question

When converting coordinates from rectangular to cylindrical, to spherical, etc. we will eventually come across having to use $ArcTan(y/x)$, and/or $ArcCos(z/\rho)$ to derive $\theta$ and/or $\phi$. However, sometimes the $x$ coordinate used in $ArcTan(y/x)$ may very well be $0$, which makes the fraction undefined. How do we retrieve the angle we need in this situation?

Answer 1

if $x = \rho\cos\theta\sin\phi\\y = \rho\sin\theta\sin\phi\\z = \rho\cos\phi$

then $\frac {y}{x} = \tan \theta$ and $\arctan \frac{y}{x} = \arctan (\tan\theta) = \theta$ if $\theta \in (-\frac{\pi}{2},\frac {\pi}{2})$

If $\theta$ is not in the correct interval you will need to add (or subtract) some multiple of $\pi$

If $x=0$ then $\arctan \frac{y}{x}$ is undefined, but you may be able to find a limit as x approaches 0.

Answer 2

When converting from rectangular to cylindrical, the statement that $\theta=\arctan \frac yx$ is a bit sloppy. You need to start with if $x=0, \theta = \pm \frac \pi 2$ depending on the sign of $y$ because the arctangent never returns $\pm \frac \pi 2$. Then you can use the arctangent formula, but you may have to add $\pi$ depending on the quadrant. Computer languages have an atan2 function that takes care of all that for you. The conversion to spherical is similar.