<s>[INST] 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.
 [/INST] 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 $4.99 \times 10^{-13}$.
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