power series

The theorem of Taylor predicts that a function that behaves sufficient neatly can be written as a power series of polynomials.
This series can be used to define a function (with curve y = f(x)). In that way trigonometric functions can be built, without using geometrical insights.

In the case of a finite number of polynomials the function is an algebraic function. In the infinite case, the function is transcendental.