I was finally able to figure out how to find the sine series for cos(x), but only for [0,2pi]. A question i have though is what is the interval of validity? is it only [0,pi]? Ie if I actually had to sketch the graph of the sum of the series, on all of R, would I have cosine or just a periodic extension of cosine from [0,2pi]?
Hey Konrad, welcome to PF. I am afraid I don't entirely understand your question. You say that you have managed to write cos(x) as a(n infinite) sum of sines on the interval [0, 2pi]. But both cos(x) and the sines you used are periodic with period 2pi, aren't they? So if the infinite sum converges to cos(x) on an interval with a length of at least one period, then it converges to cos(x) everywhere, doesn't it?
If you have expanded cos(x) in a sine series using [itex]p = 2\pi[/itex] in the formula [tex] b_n = \frac 2 p \int_0^p \cos(x) \sin{\frac{n\pi x}{p}}\,dx[/tex] what you are representing is the [itex]4\pi[/itex] periodic odd extension of cos(x). [edit - corrected typo: b_{n} not a_{n}]