Find the Fourier series of the function which is $8$ on $\left[0, \pi/4\right]$ and zero everywhere else.
We start by noting that this is an even function if we expand the range to [-π/4, π/4], so it has a cosine series. We get:
$a_0$ = $\frac{1}{\sqrt{2}}$
$a_n =\frac{1}{2}\left[\frac{1}{\pi} \int_{-\pi/4}^{\pi/4} 8\cos(nx)\right] = \frac{1}{2}\left[\frac{8}{\pi n}\sin(nx)\vert_{-\pi/4}^{\pi/4}\right] = \frac{1}{2} \left[\frac{8}{\pi n}\sin(\frac{\pi}{4}n) + \frac{8}{\pi n}\sin\left(\frac{\pi}{4}n\right) \right].$
I'm confused as to how to move on from here. I think I have the basic gist, but can someone really walk me through it?