Find an equation in spherical coordinates for the surface represented by the rectangular equation

Question

The rectangular equation is $$x^2+y^2-8z^2=0$$ $$x^2+y^2=8z^2$$ Know in the relationship between rectangular and spherical coords. we can manipulate our given to fit the form: $$x^2+y^2+z^2=9z^2$$ $$\rho=x^2+y^2+z^2, \space z=\rho\cos(\phi)$$ $$\rho^2=9\rho^2\cos^2(\phi)$$ $$1=9\cos(\phi)$$ $$\frac{1}{3}=\cos(\phi)$$ $$\arccos(\frac{1}{3})=1.23 \space rads$$ And so the equation in spherical coords. is $\phi=1.23$ I know my math is correct but I have the wrong answer so I'm not sure where I went wrong.

Answer 1

Your calculation is almost right indeed the surface is a cone with equation in spherical coordinates $$\phi=\arctan (2\sqrt 2)=\arccos \left(\frac 13\right)$$ $$\phi=\pi-\arctan (2\sqrt 2)=\pi-\arccos \left(\frac 13\right)$$

assuming $\phi \in[0,\pi]$.

Following your steps, from here

$$1=9\cos^2(\phi)$$

we obtain indeed

$$\cos (\phi)=\pm \frac13 \implies \phi=\arccos \left(\frac 13\right),\,\phi=\pi-\arccos \left(\frac 13\right)$$