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