Earlier quoted context omitted.
Treating it as a function is the fiction (and the reason is probably that the average college freshman doesn't know what a distribution is). But it's "enough" like a function that unless you look too closely, no real problems arise from glossing over the function/distribution distinction.
Right. That was kind of my point. It is a "mathematical trick" in the sense that it's not a function, but a distribution.
Cos/sin are infinitely precise, and can't be reduced to algebra (as opposed to calculus/analysis). This in itself is an idealisation, a fiction.Sin/cos seem natural because they model the ideal unit circle, while the dirac delta is natural because it models an ideal impulse. The latter seems more abstract than the former because we all have been exposed to unit circles, but usually we are not exposed to unit impulses.
You can apply a step function to a mass-spring-damper system, then observer the response. Thereafter, you can make the step function narrower but taller and observer the response. As you continue this process, the response approaches something simple. Maybe this way of thinking about the dirac delta in obvious to everyone, but to me it was a major breakthrough because I couldn't imagine how something infinitely tall and infinitely narrow could model anything in the real world.
So my ultimate takeaway message is that cos/sin are to an n-gon what the dirac delta is to a step function, and that they are just as "real" as one another. Alternately, the dirac delta doesn't work for "symbolic" or algebraic reasons. It's an idealisation of reality, not an abstraction _away_ from reality for the sake of convenience (e.g. how we sometimes artificiallyl define 0^0 (zero to the zero) to be 1 in some cases or 0 in other cases)).