Is the definition of multiplication extended to include the product of actual angles? For instance, is there any sensible relation between or is there even a notion of things like (for an angle like a radian $R$) $R^3,R^2,R,1,$ and $\frac{1}{R}$ (or $R^{-1}$ like an inverse radian).
I'm talking about angles themselves, NOT angle measurements in the form of numbers like $\frac{\pi}{3}$. I searched elsewhere, but it wasn't conclusive (e.g. I'm not sure why a steradian is a square radian and etc.).
Angles are a useful concept in geometry. You can draw a triangle or a rectangle or some other shape. And where two lines (or planes) intersect there is an angle. It is well defined, it can be measured (for example in units of degrees or radians) and has an easily understood intuitive meaning: for example sharp angles are smaller than obtuse angles.
In pure mathematics the concepts of geometry have been formalized in terms of trigonometry. This is essentially the study of the sine, cosine and tangent function. For these functions one has constructed the inverse functions. For example if we start with a function $y = sin(x)$, we may invert the relationship and thus conclude that $x$ is the inverse sine of $y$. Going back to good old geometry, we can interpret $x$ as a particular angle (in radians) in some triangle.
On the other hand, from a pure mathematical viewpoint, $x$ is also simply just another real variable. It can be manipulated in any way we want ! So we can square it, or take the third power, or the reciprocal. We can perform any mathematical operation we want. Of course it then becomes very difficult (impossible?) to associate it with the concept of "angle" as something geometric.