Answer
The cylindrical coordinates: $\left( {r,\theta ,z} \right) = \left( {2\sqrt 2 ,0,2\sqrt 2 } \right)$.
Work Step by Step
We have in spherical coordinates $\left( {\rho ,\theta ,\phi } \right) = \left( {4,0,\frac{\pi }{4}} \right)$.
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.
In cylindrical coordinates:
1. the radial coordinate is $r = \rho \sin \phi = 4\sin \frac{\pi }{4} = 2\sqrt 2 $
2. the angular coordinate $\theta=0$
3. the $z$-coordinate satisfies
$z = \rho \cos \phi = 4\cos \frac{\pi }{4} = 2\sqrt 2 $
Therefore, the cylindrical coordinates is $\left( {r,\theta ,z} \right) = \left( {2\sqrt 2 ,0,2\sqrt 2 } \right)$.
