Question

What are the cylindrical coordinates of the point whose spherical coordinates are (4, 2, 1π/6) ?

Answer 1

Assuming the usual spherical coordinate system, `(r,theta,phi)=(4,2,pi/6)` equates to `(R,psi,Z)=(2,2,2sqrt(3))`.

Explanation:

There are several different conventions as to the order of the angles in spherical coordinates. See http://mathworld.wolfram.com/SphericalCoordinates.html for more info.

I will assume that the order of coordinates wanted here is radial, azimuthal, polar, which matches that link, also has the advantage of being a right-handed coordinate system, and is the system most often seen.

Relation of spherical to Cartesian coordinates:

Spherical radius: `r=sqrt(x^2+y^2+z^2)`
Spherical azimuth: `theta=arctan(y/x)`
Spherical polar angle: `phi=arccos(z/sqrt(x^2+y^2+z^2))`

Cylindrical coordinates are less confused, and the standard definition is shown here: http://mathworld.wolfram.com/SphericalCoordinates.html.

Cylindrical radius: `R=sqrt(x^2+y^2)`
Cylindrical azimuth: `psi=arctan(y/x)`
Height: `Z=z`

From these, we may relate the cylindrical coordinates to the spherical ones:

`R=rsinphi` (via a little trig simplification using the below `Z`)
`psi=theta`; the two azimuths are identical
`Z=rcosphi`

So in this case, `(r,theta,phi)=(4,2,pi/6)` equates to `(R,psi,Z)=(2,2,2sqrt(3))`. Assuming I chose the same spherical system as you did - the details are easily enough altered if some other spherical system is wanted.