approximation of trigonometric functions in form $a + bx + cx^2 +...$ Is there any way to express the trigonometric functions as infinitely long polinomials? If so, how? If not why?
Obviously doing $x(x-\pi)(x-2\pi)(x-3\pi)...$ does not work as it doesn't match for values in between the zeroes.

Another answer has mentioned the infinite sum Taylor series for sine. Here is the infinite product Weierstrass factorization for sine:
$$  \sin(x) = x \left(1 - \left(\frac{x}{\pi}\right)^2 \right)
 \left(1 - \left(\frac{x}{2\pi}\right)^2 \right) 
 \left(1 - \left(\frac{x}{3\pi}\right)^2 \right) \cdots. $$
There is a similar factorization for cosine.
