[Math] General term of Taylor Series of $\sin x$ centered at $\pi/4$

Question

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$?

It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$

What power is $(-1)$ supposed to be raised to?

Answer 1

Write $f(x)=\sin x$. Then your Taylor series at $\pi/4$ is $$ \sum_{n\geq 0}\frac{f^{(n)}(\pi/4)}{n!}\left(x-\frac{\pi}{4}\right)^n $$

Compute the first derivatives at $\pi/4$ and see the pattern. This is periodic. You can guess the sign.

Edit: So the sequence $f^{(n)}(\pi/4)$ is, from $n=0$: $$ +\frac{\sqrt{2}}{2}, +\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},+\frac{\sqrt{2}}{2},+\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},\ldots $$ This can be given by: $$ f^{(n)}(\pi/4)=(-1)^{\lfloor n/2\rfloor}\cdot\frac{\sqrt{2}}{2}. $$