Are angles ever multiplied? I recently explained multiplication of (non-zero) complex numbers to my Mathematics Fundamentals students, the usual bit about “multiply their lengths, and add their angles”. Of course, there is always at least one student who wants to know why the angles are added instead of multiplied – after all, this is multiplication, right? – and so I go through a non-trivial example to establish that it is at least highly plausible that addition, not multiplication, of angles is what occurs. This quiets their objection, but it always gets me to wondering: Indeed, are angles EVER multiplied anywhere in Mathematics? I don’t recall ever having seen such a thing. And since this wonderful wonderland of MSE now exists, I will pass this question on to the community.
Of course, there are exercises of the form “If cos(f(x)) = sin(g(x)), solve for x.”, where this could happen, but of course what I am asking is whether this happens as part of a significant theorem, or as part of solving a problem of real physical interest. The non-multiplication of angles seems all the more counter-intuitive because they are actually dimensionless, in spite of the use of dimension-sounding phraseology such as “degrees” and “radians”. So, angles are just numbers, and surely numbers can be multiplied together, right?
(Of course, not being dimensionless is no barrier to getting multiplied by objects of the same/different type. After all, we deal with the square of seconds in regard to acceleration, and with the square of grams in regard to the (statistical) variance of weights, and so on. I’m just saying that it would seem all the more plausible that they would occasionally get multiplied together if they are dimensionless.)
I’m going to go out on a limb and conjecture that there is no such case. The basis of my conjecture is simply that I have never seen it happen, and also the fact that angles do not exist for vectors of zero length. I know this is quite tenuous, but what I’m guessing is that only quantities that behave well for 0 are eligible, so to speak, to be multiplied together.
 A: Perhaps this is a bit too obvious to be a useful answer, but I will post it anyway:
Angles are only defined up to congruence $\!\!\mod2\pi$. Addition preserves this symmetry:
$$
  (\vartheta+2\pi k) + (\varphi+2\pi j) = \varphi + \vartheta + 2\pi(k+j) =: \xi + 2\pi l, \qquad k,j,l\in\mathbb{Z}
$$
while multiplication doesn't. Therefore, the multiplication of two angles cannot be well-defined, unless you come up with some additional constraint on the range of the angles (in physics, you would call it a gauge). But this would not represent the mathematical/physical meaning of angles.
A: Maybe look in spherical trig for more.  Area of a spherical triangle, and such things.
A: For what it's worth, there is such a thing as a "square degree" and a "square radian", better known as a steradian. As GEdgar notes, these come up in spherical trig. 
A: While it's normal in mathematics to eliminate the units of Radians, it will be useful to remember for this example that the radian is a meter per meter (m/m).
Radians are used to represent the planar (plane) angle. But there are other kinds of angles worth considering. One, worth considering enough, has it's own associated unit. That angle is the solid angle, and the unit is the steradian, with dimensions (m²/m²). While of course one could divide an area by area to arrive at this value, it's also plausible, and sometimes necessary, to have to multiply two angles.
Multiplying angles, like any other dimension, results in a new dimension. As such, it's very different from multiplying universal integers, or other universal numerics.
https://en.wikipedia.org/wiki/Steradian
https://en.wikipedia.org/wiki/Solid_angle
And others, but I can't add the links yet. :/
