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