Approximating $\sin 100$

Question

If Taylor polynomial for $\sin(x)$ is $\sum_{n=0}^{+\infty} \frac{(-1)^{n}}{(2n+1)!}x^{2n+1} $. What do I have to do to find what degree of Taylor polynomial I have to use so the error is not greater than $10^{-4}$ in approximation of $\sin(100)$?

Answer 1

It is a(n eventually) converging alternating series, so the alternating series theorem applies. Find the first term after it starts decreasing that is less than $10^{-4}$ in magnitude and you are done.

Probably you are expected to look up the error term for the Taylor series. Note that all the derivatives of $\sin x$ are less than $1$ in magnitude, so you can ignore that.

As the comments point out, you will get there with many fewer terms if you are allowed to center the Taylor series at $32\pi$ instead of $0$.