Extending f(x) as an Even Function: Obtain Cosine Fourier Series

[tomeatworld]

Homework Statement

f(x) = sin(x) for 0$$\leq$$x<$$\pi$$. Extend f(x) as an even function . Obtain a cosine Fourier series for f.

Homework Equations

$$a_{0}$$/2 + $$\sum$$ $$a_{n}$$cos(nx)

The Attempt at a Solution

So as far as I know, to extend sin(x) as an even function you have to make f(x)=-sin(x) for $$-\pi\leq$$x<0 and then just use that to integrate for a_n but this gives a series without an a₀ term which the question points to it having. What have I done wrong?

 

[elibj123]

Have you integrated correctly?

Did you remember that the integral over [-pi,pi] breaks to [-pi,0] where it's -sin(x) and [0,pi] where it's sin(x) (or alternatively, the integral over [-pi,pi] is double the integral over [0,pi])

Can you please show us your calculation?

 

[tomeatworld]

sure.
As you've said, I've just used:
$$\frac{2}{\pi}$$$$\int^{\pi}_{0}$$sin(x) dx
so integration gets
[-cos(x)$$]^{\pi}_{0}$$.
expanding gives [1 - 1] so I lose the $$a_{0}$$ term. Is that correct?

Oops, i see what I did. Forgot the - - so it should actually be 4/pi. Is that correct?

 

[vela]

Yeah, that's right.