generally when you learn about calculating anything involving a sphere with integrals, you first learn about
cylindrical coordinates (r, θ, z) and
spherical coordinates (ρ, θ, φ). these are just different ways of drawing up lines in 3D space that are like the
cartesian coordinates (x, y, z) you already know.
the reason why you learn about these coordinate systems? because, among many other things, it makes calculating the volume of a sphere with integrals way nicer. we're talking about a volume, which is three-dimensional, so typically there's some way to write that as a
triple integral. and in fact, you can generally write any calculation for two-dimensional area as a
double integral, so that makes sense.
so here's what the equation ends up looking like when you want to calculate the area of a sphere with
cartesian coordinates:
this is a huge pain.
cylindrical coordinates, which would substitute slightly different terms into the x and y variables, will actually simplify this triple integral (although when you get into a calculus 3 class, these substitutions will make a lot more sense):
and
spherical coordinates, which were basically born to do this, make it look like this:
and there's your sine.
(pictures shamelessly stolen from this pdf)these equations can be expanded and compressed a thousand different ways to
look different but actually still mean the same thing. in fact, all three of these equations can be dumbed down to V = (4/3)πr
3 if you know what you're simplifying.
but your intuition is definitely right: in the most elegant way, a sine is involved in calculating the volume of a sphere.
