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.

You already received good answers to the question.
In this area, we can do other things which are amazing. If you look here, being myself fascinated by the approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$  which wasproposed, more than $1400$ years ago by Mahabhaskariya of Bhaskara I (a seventh-century Indian mathematician), I wrote
$$\sin(x)=\sum_{k=1}^n a_k [(\pi-x)x]^k$$ which respects the value of $0$ at $x=0$ and $x=\pi$. The first coefficients are
$$\left(
\begin{array}{cc}
 k & a_k \\
 1 & \frac{1}{\pi } \\
 2 & \frac{1}{\pi ^3} \\
 3 & \frac{12-\pi ^2}{6 \pi ^5} \\
 4 & \frac{10-\pi ^2}{2 \pi ^7} \\
 5 & \frac{1680-180 \pi ^2+\pi ^4}{120 \pi ^9} \\
 6 & \frac{1008-112 \pi ^2+\pi ^4}{24 \pi ^{11}} \\
 7 & \frac{665280-75600 \pi ^2+840 \pi ^4-\pi ^6}{5040 \pi ^{13}} \\
 8 & \frac{308880-35640 \pi ^2+450 \pi ^4-\pi ^6}{720 \pi ^{15}}
\end{array}
\right)$$ The maximum error is at $x=\frac \pi 2$; for example, using $n=8$, the error is just $4.99 \times 10^{-13}$.
