Answer
Half-cone
Work Step by Step
The conversion of rectangular coordinates to spherical coordinates is given as:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
Here, $\rho=\sqrt {x^2+y^2+z^2}$; $\phi =\cos^{-1} [\dfrac{z}{\rho}]; \theta=\cos^{-1}[\dfrac{x}{\rho \sin \phi}]$
Here, we have $\theta=\dfrac{\pi}{3}$
$ \phi =\cos (\dfrac{\pi}{3})=\dfrac{1}{2}$; $\theta=\cos^{-1}[\dfrac{x}{\rho \sin \phi}]=\cos^{-1}\dfrac{0}{2 \sin \dfrac{\pi}{6}}=0$
Now, we have $\rho^2 \cos^2 \phi =\dfrac{1}{4}\rho^2$
This gives: $z^2=\dfrac{1}{4}(x^2+y^2+z^2)$
and $3z^2=x^2+y^2$
Thus, this is the equation for the half cone.
