## richyw 3 years ago could someone remind me how to do this with spherical coordinates? I need to find the cartesian formula for this spherical surface $$\phi=\frac{\pi}{6}$$ I can easily see that this is a cone. no idea how to get the answer $$z^2=3x^2+3y^2$$ my textbook solutions don't even show how to do it once!

1. richyw

using these definitions

2. beginnersmind

I'm just guessing but why not try writing $x =r sin \theta cos\phi$ $y =r sin \theta sin\phi$ $z= rcos\phi$ Now look at x^2+y^2 and z^2 substituting $\phi=\pi/6$ Maybe...

3. richyw

I have tried that. I still get stuck.

4. beginnersmind

r from my equations is rho in your picture

5. beginnersmind

Ok, let me see if I can get anything useful.

6. beginnersmind

Ah, ok, I think I got it. Using the letters from your picture: Starting from $z^2 = 3x^2 +3 y^2$ we have $z^2 = 3\rho^2$ or $(z/\rho)^2 = 3$ where z/p is the arctangent of phi.

7. beginnersmind

I mean z^2 = 3r^2, sorry.

8. richyw

thanks but I can't figure out how to go backwards like the question requires!

9. beginnersmind

Sorry, didn't see your comment. You just need to reverse the argument. From the picture we see that ctg(phi) = z/sqrt[(x^2+y^2)]. Plugging in phi=pi/6 and squaring both sides gives 3=z^2/(x^2+y^2) or 3x^2+3y^2=z^2