Find an efficient algorithm to calculate $\sin(x) $ Suggest an efficient algorithm to determine the value of the
function $ \sin(x) $ for $ x \in [-4\pi, 4\pi] $.
You can use only Taylor series and $ +, -, *, /$.
I know, that $$\sin(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n + 1)!}$$
but I can't find an efficient algorithm.
Thank for your help.

Hint: The coefficients of the series are related to one another by the simple relation: 
$$
t_{n+1}=\frac{x^{2(n+1)+1}}{(2(n+1)+1)!}=\frac{x^2}{(2n+2)(2n+3)}\frac{x^{2n+1}}{(2n+1)!}=
\frac{x^2}{(2n+2)(2n+3)}t_n
$$
(Here $t_n$ represents the coefficient of $x^{2n+1}$, multiplied by $x^{2n+1}$.  So $\sin x=\sum_{n=1}^\infty t_n$.)  
This means that you don't need to work out each coefficient separately: once you've worked out $t_n$, you've done most of the work you need to do to work out $t_{n+1}$.  
The other thing you'll need to do is to work out how many terms of the series you'll need to ge a good enough approximation on $[-4\pi,4\pi]$.  
