Question

How do you find the taylor series for `sinx` in powers of `x-pi/4`?

Answer 1

According to the definition of Taylor series it should be

`\sum_{n\geq 0}\frac{f^{(n)}(\pi/4)}{n!}*(x-\frac{\pi}{4})^n`

Now you have to compute the first derivatives at `x=pi/4` for
`f(x)=sinx`

You will see a pattern to arise which is periodic.

Finally (if my calculations are right) you should find tht the Taylor pattern is

`f^{(n)}(\pi/4)=(-1)^{\lfloor n/2\rfloor}\cdot\frac{\sqrt{2}}{2}`

The "strange" bracket sign you see in the formula above is the integer part or floor function

Here `n` is an integer of course.

Footnote

See the definition for Taylor series and integer part function

Taylor Series

Integer Part function