I have some vector function, call it $F_a\equiv F_a(r,\theta,\phi)$ where $\theta$ and $\phi$ are the polar and azimuthal angles of a spherical coordinate system respectively. If I convert this to Cartesian coordinates using: $$F_{a^\prime}=\frac{\partial x^a}{\partial x^{a^\prime}}F_a.$$ Once I do this, my goal is to find this vector along the z-axis only. Unfortunately, while transforming to Cartesian coordinates I use the equations: $$ \cos\phi=\frac{x}{\sqrt{x^2+y^2}}, \hspace{20pt} \sin\phi=\frac{y}{\sqrt{x^2+y^2}}.$$ Taking the limit as x and y goes to zero of the above equations yield different results, and it specifies a particular direction.
My question is, how do I define $\cos\phi$ and $\sin\phi$ on the z-axis? I know by definition that $\phi$ is undefined there, but how does this transfer to this transformation that I'm trying to do?
Please let me know if further information is needed.