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So I was watching this video on spherical coordinates via a rotation matrix:

and in the end, he gets:

x = \rho * sin(\theta) * sin(\phi)

y = \rho* cos(\theta) * sin(\phi)

z = \rho cos(\theta)

Clearly, this doesn't look like the equations we normally get for spherical coordinates, which are:

x = \rho* cos(\theta) * sin(\phi)

y = \rho * sin(\theta) * sin(\phi)

z = \rho * cos(\theta)

What I don't understand is he says that they are not the same as some of the spherical coordinates as we've seen, but selecting measures is an arbitrary choice. Additionally, most resources I find that get the answer I'm more used to.

However, it seems to me that if someone was using these equations, they would get completely the wrong stuff. So is there a way to get our usual equations using this method? Or what's going on here that makes this ok? I don't see how we should get a completely different result. I really like this method because it makes a lot of sense to me, but I don't like how it gets what I consider a completely wrong result.

From what I can tell, the issue seems to come from the rotation table/matrix from the a to n coordinate frame which is:

R = [cos(\theta), sin(\theta), 0

sin(\theta), cos(\theta), 0

0, 0, 1]

However, I can't seem to think of a way that this is inherently wrong given the geometry, nor how we would get the usual equations from this.

Can someone please help/elaborate/explain this situation to me?