Usually I take this for granted, but lately I had an encounter with some infinitesimal calculus concepts from a computational point of view, Fourier transformations for the most part, and I can't wrap my head around what is the concept that makes possible to compute a transcendental function.
Typically you find a vague explanation about this that involves polynomials, and it's not really about computing analytical functions, it's about reducing or simplifying polynomial forms; also polynomials are already part of the simple algebra, there is nothing really analytic in something in a polynomial form, it's merely an expansion that tries to approximate the original behaviour, kinda like mimicking a certain behaviour.
So I guess that my question becomes, what are the principles that are used to transform a transcendental function into polynomials that are easy to compute ? In what point in time they were introduced/invented ?
There are other kind of expansions except the one based on polynomials ?
I'm not really sure I understand the question, but, here goes:
Given a (sufficiently smooth) function $f$, and a number $a$ in the domain of $f$, and a positive integer $n$, you can find a polynomial $p$ (which I should really write as $p_n(f;a)$) such that $p(a)=f(a)$ and $p'(a)=f'(a)$ and $p''(a)=f''(a)$ and ... and $p^{(n)}(a)=f^{(n)}(a)$ (that is, $p$ and $f$ agree at $x=a$, as do their first $n$ derivatives). Then it follows that $f(x)$ will be closely approximated by $p(x)$ for values of $x$ near $a$, and indeed "Taylor's Theorem with Remainder" gives you some idea of how good the approximation will be.
I don't know exactly when this was introduced, but it probably wasn't long after Newton & Leibnitz invented Calculus.
Before that, there were other methods for evaluating trigonometric functions, based on geometry; I'd refer you to texts on the history of math for details.
For more recent methods, you might be interested in CORDIC.