= Solution
Use $s$ for <cylindrical radius>, to distinguish it from spherical radius $r$. In <cylindrical coordinates> $(s,\phi,z)$, the <azimuthal angle> $\phi$ is measured from positive $x$ in the horizontal plane:
$$
\boxed{x=s\cos\phi,\qquad y=s\sin\phi,\qquad z=z,\qquad s\geq0.}
$$
Here $0\leq\phi<2\pi$ and $z\in\mathbb R$.
In <spherical polar coordinates> $(r,\theta,\phi)$, take $\theta$ as the polar <angle> from positive $z$ and use the same <azimuthal angle> $\phi$:
$$
\boxed{x=r\sin\theta\cos\phi,\qquad y=r\sin\theta\sin\phi,\qquad z=r\cos\theta.}
$$
The ranges are $r\geq0$, $0\leq\theta\leq\pi$ and $0\leq\phi<2\pi$. The radial direction has horizontal projection of length $r\sin\theta$, which explains the extra factor in $x,y$. The azimuth is undefined on the polar axis, and both angles are nonunique at the origin; these are coordinate degeneracies rather than geometric singularities.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2019-polar-coordinate-systems.png]
{title=Cylindrical and spherical polar coordinates; the spherical polar angle is measured from positive z.}
{description=The same point and its projection onto the horizontal plane are shown in both panels. The cylindrical radius is the projection length; the spherical radius is the distance from the origin.}
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