I don't think this is how it was historically developed, but the quickest way to see how the series expansions fit in is by expanding out the left side of .
I was wondering... Surely, the trigonometrical functions began as mere observations about the proportions between the sides of right angled triangles. Probably people were just drawing triangles in the sand, measured the sides and dividing, noticing that similar angles gave similar quotients.
Now, how did we get to were we are today? I know you can approximate a trig. func. by its Taylor expansion, but in order to do this you first have to know the value of the function in a certain point and the value of it's derivatives.... How do know this?
I hope my question isn't to vague...
There are very old documents (tablets, etc) with incremental values that much later came to be known as values of trigonometric functions as we know them today. Many of these appear to have been useful for the engineers of the day - building things and dividing property and the like. We still do things like that.
an (educated) guess on my part, is that trigonometry arose as a way of measuring "round things", in particular, circular arcs. these, in turn, are intimately bound up with the notion of the angle between two rays. in the old days, the line joining the endpoints of a circular arc, was called a chord, and these (chords) were seen as the basic things of study for some time.
(if something is moving, or built along a circular path, chords answer the question: how far is it from here to there?). circular things are easy to construct, you only need one point of reference (the center) and one unit of measurement (the radius). when circles and lines get together, all sorts of interesting things happen, and eventually people gave names to certain sides of the triangles you could make this way.
the sine function (or "half-chord") didn't come into prominence until some time later. and the cosine even later than that, for some time the "second trig function" was the "turned sine" or versine function (versed sine). the deeper connections between orthogonality ("right-angledness") and the circle came somewhat later. the pythagorean theorem, is a very profound thing, popping up everywhere from metric spaces, to complex analysis, to number theory, and many other branches of mathematics. it says something intrinsic about the relationship of number to distance, which cuts much deeper than the simple patterns noticed in certain kinds of triangles.
the earliest tabulated values for sines came at great difficulty, a mathematician could easily spend the better part of his life making a reliable table. remember, in the early days of civilization, the only reliable clock we had was the sky. early astronomy was crucial to our reckoning of time. and trigonometry did not develop as a full-blown field, but evolved over time, in fits and starts.
of course, form our current vantage point, we can form power series, and use the properties of the real numbers to show that they converge (under certain conditions). this allows us to calculate trigonometric functions with relative ease, and great precision. it is not hard to show (given a power series definition of sine and cosine) that they both satisfy the differential equation:
f + f" = 0
(just differentiate term-by-term).
it is also possible to show that any solution of that equation has the form asin(x) + bcos(x), and that specifying f(0) and f'(0) allows us to calculate f(x) for ALL x.
so to calculate the Taylor series for sine and cosine, we really only need two facts: sin(0) = 0, and cos(0) = 1, and that they both satisfy the equation f + f'' = 0.
it IS remarkable that sine and cosine both satisfy that differential equation. it's...unexpected. and it is even more remarkable that when you combine sine and cosine in the right way, you get an exponential function. such remarkable facts often lend a sense of the mystical to mathematics.