Spherical to Cartesian transformation

$$f(r,\theta ,\phi )=\frac{C\sin\phi \cos\theta}{r^3}$$

Question

Spherical to Cartesian transformation

$$f(r,\theta ,\phi )=\frac{C\sin\phi \cos\theta}{r^3}$$

richyw · Answer

wikipedia says I must use 

$$x=r \, \sin\theta \, \cos\varphi$$$$y=r \, \sin\theta \, \sin\varphi$$$$z=r \, \cos\theta$$ but I don't really see how this is useful. I feel like it would be more useful to have $r, \theta$ and $\phi$ isolated and then plug those in. When I do I get $$f(x,y,z)=\sin\left( \arctan{\frac{y}{x}}\right )\frac{z}{(x^2+y^2+z^2)^2}$$which is quite messy.