Question

Fourier series..

Answer 1

I need to find the Fourier cosine series to\[ f(x)=\cos(3x)\sin^2(x)\]
Can someone help me?

Answer 2

I know the series are even and 2π-periodic...

Answer 3

The cosine series would be \(\displaystyle\sum_{n=1}^\infty a_n\cos(nx),\) where
\[a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}\cos(3x)\sin^2x\cos(nx)~dx\]
I'm guessing you need help with the actual integral?

I think writing \(\sin^2x=\frac{1}{2}(1-\cos(2x))\) will help, and from there make use of the sum/difference identities to further break up the integrand:
\[\cos(mx)\cos(nx)=\frac{1}{2}\bigg[\cos((m+n)x)+\cos((m-n)x)\bigg]\]
If all that works out nicely, you should be able to integrate term-by-term.

Answer 4

It does work out! It's tricky to type it out, since it gets cut off, it's so long... If you need more help, let me know.