Cone in spherical coordinates after rotation
$r$ is untouched by the rotation, we can work with $r=1$.
After rotation, the cone axis will go in some direction $(u, v, w)$ (a unit vector).
If suffices to express that the cone aperture is $\alpha$ by means of a scalar product:
$$u\cos\theta\sin\phi+v\sin\theta\sin\phi+w\cos\phi=\cos\alpha.$$
This gives you a $\theta,\phi$ relation.