Answer
The spherical coordinates is $\left( {\rho ,\theta ,\phi } \right) = \left( {5,\frac{\pi }{6},0.644} \right)$.
Work Step by Step
We have in cylindrical coordinates: $\left( {r,\theta ,z} \right) = \left( {3,\frac{\pi }{6},4} \right)$.
Convert to spherical coordinates:
1. the radial coordinate is
$\rho = \sqrt {{x^2} + {y^2} + {z^2}} = \sqrt {{r^2} + {z^2}} = \sqrt {{3^2} + {4^2}} = 5$
2. the angular coordinate $\theta = \frac{\pi }{6}$
3. the angular coordinate $\phi$ satisfies
$\cos \phi = \frac{z}{\rho } = \frac{4}{5}$, ${\ \ }$ $\phi = 0.644$
Therefore, the spherical coordinates is
$\left( {\rho ,\theta ,\phi } \right) = \left( {5,\frac{\pi }{6},0.644} \right)$.
