Arctan of $\frac{1}{0}$ I applied the formula $\rho=\sqrt{z^2+r^2},\ \  \theta=\theta , \ \ \phi=\arctan \left (\frac{r}{z}\right )$ for the point in cylindrical coordinates $\left (1, -\frac{\pi}{6}, 0\right )$.
To calculate $\phi$ do we write it as followed?? $$\phi=\arctan \left (\frac{1}{0}\right )$$
Or is there an other way to write it?? 
Do we say: "Since it $\frac{1}{0}$ is not defined and we know that $\tan $ is not defined at $\frac{\pi}{2}$, we conclude that $\phi=\frac{\pi}{2}$." ?? 
Is the formulation correct?? Could I improve something??
 A: Mathematically, $\arctan(1/0)$ isn't well-defined; it would be more accurate to write $\arctan(1/0^+)$ and $\arctan(1/0^-)$, as appropriate. This indicates that we consider the limits of $\arctan(1/x)$ as $x$ tends to zero. The one-sided limits are different: $\pi/2$ vs $-\pi/2$. 
A bigger issue with the formula $\phi=\arctan \left (\frac{r}{z}\right) $ is that for negative   $z$ it gives negative values of $\phi$. But this is not what you want: $\phi$ normally ranges from $0$ to $\pi$ in spherical coordinates.  
A similar issue already comes in two-dimensional geometry, where the polar angle $\theta$ satisfies $\theta = \arctan(y/x)$... except for all the cases when it doesn't. 
In programming this is solved by using the two-variable function atan2 which takes $x$ and $y$ values, and returns polar angle. In mathematics,  we have to be aware that "take arctangent of ratio of coordinates" is not a universal recipe, and use other trigonometric   functions when appropriate. 
In particular, knowing that $z=\rho\cos\phi$ in spherical coordinates tells that $z=0$ if and only if $\phi=\pi/2$. (With the exception of the origin $(0,0,0)$, where the angles are not defined.)
