I have this function which I should write the Fourier series for: $f(t)=|\sin t|$
I now that the period is $\pi$ and that it is an even function. Because it is even, I only need to calculate the cos coefficient which is:
$$a_n = \frac{2}{T}\int_0^T f(t)\cos(n\omega t)dt = \frac{2}{\pi}\int_0^\pi |\sin t| \cos(n\omega t)dt$$
My problem here is that I do not know how to handle the abs sign affecting sin. If it has just been $\sin t \cos (n\omega t)$ then I could use the formula $$ \sin\alpha\cos\beta=\frac{1}{2}(\sin(\alpha -\beta)+\sin(\alpha+\beta))$$ but I don't know if I could do that now... How should I solve this?
Notice that if $t\in[0,\pi]$, then $\sin t$ is non-negative. Consequently, we can directly use the formula $$\sin\alpha\cos\beta=\frac{1}{2}(\sin(\alpha -\beta)+\sin(\alpha+\beta)).$$