TITLE: How to obtain the series of the common elementary functions without using derivatives?
QUESTION [5 upvotes]: First, I'm a freshman student of physics, not of mathematics, so please excuse my ignorance of mathematics :)
Well, I'm reading the book "Huygens and Barrow, Newton and Hooke" by Vladimir Arnold, and one excerpt (at the page 43) called my attention greatly: "he used Taylor's formula for calculating derivatives rather than using the derivatives for the expansion of functions" ("he" refers to Sir Isaac Newton).
My main question is: How to obtain the series of the common elementary functions (trigonometric, exponential...) without using derivatives? And, for example, could calculus be developed in a way that we would get $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ and also the one for cosine and then use this series to find the derivative of sine by seeing that it reduces to the series of the cosine?
P.S. I don't know English very well, I just hope the question asked is that I wanted to ask! Thanks in advance to everyone who answer.

REPLY [4 votes]: Defining $e^x = \lim_{n \to \infty}(1 + x/n)^n $, we can use the binomial expansion to get the familiar taylor series of $e^x$. Then one can use $e^{ix} = \cos(x) + i\sin(x)$ to get the taylor series for $\cos$ and $\sin$. 
This seems like a plausible path for Newton(/Euler) to use in deriving the taylor series for a large amount of important functions.