So I'm given the Taylor Series expansion of the sine function and I've been asked to prove it (Done) and then construct the following by my lecturer:

Explain why the Taylor series containing $N$ terms is: $$\sin x = \sum_{k=0}^{N-1} \frac{(-1)^k}{(2k+1)!}x^{2k+1}+r_{2N-1}(x)$$ with a remainder $r_{2N-1}(x)$ that satisfies: $$|r_{2N-1}(x)| \leqslant \frac{|x|^{2N}}{(2N)!}$$

How many terms of the series do you need to include if you want to compute $\sin x$ with an error of at most $10^{-3}$ for all $x \in [-\pi/2, \pi/2]$?

Compute $\sin(\pi/2)$ from the Taylor Series. How large is the actual error?'

I'm fairly certain I can do the third part, but the first and second have completely thrown me. Can anybody help, or at least point me in the right direction?

Cheers.