taylor series and trigonometry, questions

I have some trouble with taylor series,but it all comes down to basic trig questions,for example

1)for the taylor series of f(x)=sinx,the textbook states that f^{(}^{n)}(x)=sin(x+nπ/2)

Is it because for every n+1 it just goes form cosx to -sinx and from -sinx to -cosx and so on?

2)Find the first three terms of the Taylor series for the function f(x)=sin(-x^{2}+πx+π/2).When I try to solve this one I get cos(-1+3π/2) among others,for example f'(1)=cos(3π/2 -1)(π-2).How do I approach that?

EDIT: Is it f'(1)=(cos(π/2-1))(π-2)=(sin(-1))(π-2) ?

(my mistake,this should've been in pre calculus forum I guess)

Re: taylor series and trigonometry, questions

1) Recall the identities

$\displaystyle \sin{(x + \pi/2)} = \cos{x} \ , \ \cos{(x + \pi/2)} = -\sin{x} $

So for $\displaystyle f(x) = \sin{x} $

$\displaystyle f'(x) = \cos{x} = \sin{(x + \pi/2)} $

$\displaystyle f''(x) = -\sin{x} = \sin{(x + \pi)} $

$\displaystyle f'''(x) = -\cos{x} = \sin{(x + 3 \pi/2)} $

and so on...

Re: taylor series and trigonometry, questions

2) What value of x are you expanding around? x = 1?

Simplifying your function using the above identities makes the problem a bit easier.

$\displaystyle f(x) = \sin{(-x^2 + \pi x + \pi/2)} = \cos{(-x^2 + \pi x)} = \cos{(x^2 - \pi x)} $

The first three terms of the expansion about x = a are

$\displaystyle f(x) = f(a) + (x-a)f'(a) + \frac{1}{2!} (x-a)^2 f''(a) + ... $

Compute the required derivatives of f(x) and plug into the expression for the expansion. If the problem requires that you express your answer as polynomials of x, then do expansions on the trig functions in your answer, and organize it to give the three leading terms.