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...