The infinite series of cosine and sine can be approximated by a partial sum of the first $n+1$: $$\begin{array}{rcl}\cos(x)&=&\sum_{k=0}^\infty (-1)^k\frac{x^{2k}}{(2k)!}=\sum_{k=0}^n(-1)^k\frac{x^{2k}}{(2k)!}1+r_{2n+2}(x),\\ \sin(x)&=&\sum_{k=0}^\infty (-1)^k\frac{x^{2k+1}}{(2k+1)!}=\sum_{k=0}^n(-1)^k\frac{x^{2k+1}}{(2k+1)!}+r_{2n+3}(x),\end{array}$$
with some error remainder terms not greater than
$$\begin{array}{rcl}|r_{2n+2}(x)|\le\frac{|x|^{2n+2}}{(2n+2)!}&\text{ for }&|x|\le 2n+3,\\ |r_{2n+3}(x)|\le\frac{|x|^{2n+3}}{(2n+3)!}&\text{ for }&|x|\le 2n+4.\end{array}$$
Proofs: 1
