My question is regarding the geometric understanding of things such as $\sin (A+B+C+D)$ where one moves out of the typical 'trigonometry' setting because this is dealing with more than $3$ angles. I cannot grasp what this would look like if we were dealing with this in classic Euclidean geometry. So what exactly is the sine of the sum of multiple angles, geometrically speaking? What does it look like? Can this be studied further or has it been already? Does this have reverberations deeper in mathematics that can be exploited to ease other problems?
A bit of a sidenote: I have written out several expansions of these formulas so I do not need the general expansion formulas for this sort of stuff.
As a concrete example, here’s a visual explanation of where the double angle formula for sine comes from:
If $A$ and $B$ are angles, then $A+B$ is how much plane angle is encompassed by the both of them—for example, if they’re adjacent. You’re right to be confused by adding two angles, as you can’t really add things that aren’t numbers. That’s why geometers write $m\angle A$ to indicate the number associated with the angle as opposed to the angle itself. So, really we should say that $A$ and $B$ are the measures of two plane angles.
To familiarize yourself with this concept, you could look at altitudes of triangles or divide an isosceles triangle into two right triangles. In both of these instances, one of the corners of the triangle has a third segment intersecting it, dividing that corner into ‘two’ angles.
Below is another example that I just drew up. It shows how you might interpret the corner of a triangle as one angle or as the sum of the measures of two angles.
If you’re a geometry student, you’ve probably seen some variation of this problem. You would have been given $x$ and $y$ as well as any two of $α$, $β$, $γ$, $δ$ and $\varepsilon$ to solve for $z$, which you could do by using the definition of sine or the law of cosines.