# Thread: Fourier sine series...

1. ## Fourier sine series...

Let f(x) = cos x on 0<= x <= pi. Calculate the Fourier sine series representation of f(x).

Can anybody possibly start me off on this question or give me any help at all as I don't understand my lecture notes on this topic at all

2. Originally Posted by hunkydory19
Let f(x) = cos x on 0<= x <= pi. Calculate the Fourier sine series representation of f(x).

Can anybody possibly start me off on this question or give me any help at all as I don't understand my lecture notes on this topic at all

What do you mean by fourier sine? is it the sine part of the Fourier Series?
if does, this should be a good hint:

If $\displaystyle f(x)$ is an even function, then $\displaystyle b_n = 0$, i.e $\displaystyle f(x) \approx \frac{a_0}{2} + \sum a_n \, \cos nx$

If $\displaystyle f(x)$ is an odd function, then $\displaystyle a_n = 0$, i.e $\displaystyle f(x) \approx \frac{a_0}{2} + \sum b_n \, \sin nx$

3. Thanks kalagota,

So $\displaystyle a_n = 0$

and $\displaystyle b_n = \frac{2}{\pi}\int^\pi_0 cos x sin nx \, \mathrm{d}x$

But how do integrate this, I tried doing it by parts but realised that it will just go on forever!

4. You can turn a product of sine and cosine into a sum, do you know that formula? Applyin' that the rest follows.

5. You need to know the orthogonal properties of sine and cosine.
(If L>0 we have):

$\displaystyle \int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m$.

$\displaystyle \int_{-L}^L \cos \frac{\pi n x}{L} \cos \frac{\pi m x}{L} ~ dx= \mbox{ same as above}$.

$\displaystyle \int_{-L}^L \sin \frac{\pi n x}{L}\cos \frac{\pi m x}{L} ~ dx = 0$.

Here $\displaystyle L=\pi$.

6. Originally Posted by ThePerfectHacker
You need to know the orthogonal properties of sine and cosine.
(If L>0 we have):

$\displaystyle \int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m$.

$\displaystyle \int_{-L}^L \cos \frac{\pi n x}{L} \cos \frac{\pi m x}{L} ~ dx= \mbox{ same as above}$.

$\displaystyle \int_{-L}^L \sin \frac{\pi n x}{L}\cos \frac{\pi m x}{L} ~ dx = 0$.

Here $\displaystyle L=\pi$.
indeed! and the first two are 0 if $\displaystyle m \ne n$

you know i didn't know about these rules until after my differential equations class was over! i always had to do it the hard way (the product to sum formulas)