Answer
Half-cone
Work Step by Step
Convert rectangular coordinates to spherical coordinates.
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and $\rho=\sqrt {x^2+y^2+z^2}$
Here, $\theta=\dfrac{\pi}{3}$
$ \phi =\cos (\dfrac{\pi}{3}) =\dfrac{1}{2}$
and $\theta=\cos ^{-1}(\dfrac{0}{2 \sin \dfrac{\pi}{6}})=0$
Re-arrange as $\rho^2 \cos^2 \phi =\dfrac{1}{4}\rho^2$
and $z^2=\dfrac{1}{4}(x^2+y^2+z^2)$
or, $3z^2=x^2+y^2$
Thus, this shows an equation for the half cone.
