Answer
The cylindrical coordinates is $\left( {r,\theta ,z} \right) = \left( {\frac{3}{2}\sqrt 3 ,\frac{\pi }{6},\frac{3}{2}} \right)$.
Work Step by Step
We have in spherical coordinates $\left( {\rho ,\theta ,\phi } \right) = \left( {3,\frac{\pi }{6},\frac{\pi }{3}} \right)$.
The relations between cylindrical and spherical coordinates can be found using rectangular coordinates $x$, $y$, $z$ as in the following:
$x = r\cos \theta = \rho \sin \phi \cos \theta $
$y = r\sin \theta = \rho \sin \phi \sin \theta $
$z = \rho \cos \phi $
So, $r = \rho \sin \phi $ and $z = \rho \cos \phi $. Whereas $\theta$ is the same in both cylindrical and spherical coordinates.
Convert to cylindrical coordinates:
1. the radial coordinate is
$r = \rho \sin \phi = 3\sin \frac{\pi }{3} = \frac{3}{2}\sqrt 3 $
2. the angular coordinate is the same, $\theta = \frac{\pi }{6}$
3. the $z$-coordinate satisfies
$z = \rho \cos \phi = 3\cos \frac{\pi }{3} = \frac{3}{2}$
Therefore, the cylindrical coordinates is
$\left( {r,\theta ,z} \right) = \left( {\frac{3}{2}\sqrt 3 ,\frac{\pi }{6},\frac{3}{2}} \right)$.
