Answer
Half-Cone
Work Step by Step
Conversion of rectangular to spherical coordinates is as follows:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and $\rho=\sqrt {x^2+y^2+z^2}$; $\cos \phi =\dfrac{z}{\rho}$; $\cos \theta=\dfrac{x}{\rho \sin \phi}$
Here, $\theta=\dfrac{\pi}{3}$
$\cos \phi =\cos \dfrac{\pi}{3} =\dfrac{1}{2}$; $\cos \theta=\dfrac{0}{2 \sin \dfrac{\pi}{6}} \implies \theta=0$
This implies that $\rho^2 \cos^2 \phi =\dfrac{1}{4}\rho^2$
or, $z^2=\dfrac{1}{4}(x^2+y^2+z^2$
Thus, we have $3z^2=x^2+y^2$
This equation represents the half cone.
