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!

Question

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!

richyw · Answer

using these definitions

beginnersmind · Answer

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...

richyw · Answer

I have tried that. I still get stuck.

beginnersmind · Answer

r from my equations is rho in your picture

beginnersmind · Answer

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

beginnersmind · Answer

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.

beginnersmind · Answer

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

richyw · Answer

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

beginnersmind · Answer

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