Is there a solution to the equation $tan({\phi})=\frac{0}{0}$ I've been reading about conversion from Cartesian ($x,y,z$) to Spherical (r, $\theta$, $\phi$) coordinates. The formula to find the value of ${\phi}$ is given as:
$\tan({\phi})=\frac{y}{x}$
My problems begin when I have a point in Cartesian coordinates like $(0,0,1)$.
To get $\phi$ I should find the value of 
$\tan^{-1}(\frac{0}{0})$ 
and for me it does not make sense.
Intuitively and graphically, I can see $\phi$ can take any value, because $\theta=0$, but I cannot find a way to put the idea on paper.
Thanks.
 A: This equation breaks down for points on the $z$-axis, for a reason. 
The three spherical coordinates, like you say, are $(r,\theta, \phi)$, or the radial distance, the polar angle, and the azimuthal angle respectively.
Your point, $(0,0,1)$ is directly above the origin on the 2-D Cartesian plane. It is a distance $r=1$ from the origin. 
As for $\theta$ it is obviously $90º$ above the horizon, pointing directly up. 
The reason the tangent formula fails is because it doesn't matter what $\phi$ is, though convention would have you leave it at $0$ or a similar number. 
No matter how you rotate the point around the $z-axis$, it won't change position. If it were off the $z$-axis, then it would have either an $x$-coordinate, a $y$-coordinate, or even better, both. 
Notice that if $x=0$, then $\phi$ is similarly undefined. This is because, if $x=0$, then it is geometrically evident that it is an angle of $\frac{π}{2}$ about the $z$-axis, and $\tan$ has an asymptote at that angle value. 
Imagine, if you're looking down at the 2-D cartesian plane. The tangent formula is valid when $x≠0$. When $x=0$ just use common sense about what the angle must be (that is, $90º$ away from the positive $x$-axis, in some sense). 
