Question

Find (a) the Fourier cosine series, (b) the Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. f(x)=sin x(0<x<π)

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(a)

To find the Fourier cosine series, we suppose that the given function is even. That is, we expand the graph of the function so that it would be symmetric with respect to the yy axis.

We assume the function is periodic with period p=2πp=2\pi. This gives us

L=p2=πL=\frac{p}{2}=\pi

Since we are looking for the cosine series, we suppose that the Fourier coefficients bn=0b_n=0 and we need to find coefficients a0a_0, ana_n for n=1,2,n=1,2,\ldots

a0=1π0πf(x)dx=1π0πsinxdx=cosxπ0π=2π\begin{align*} a_0&=\frac{1}{\pi}\int_{0}^{\pi}f(x)\,dx\\[7pt] &=\frac{1}{\pi}\int_{0}^{\pi} \sin x \,dx \\[7pt] &=-\frac{\cos x}{\pi}\Bigg|_0^\pi\\[7pt] &=\frac{2}{\pi } \end{align*}

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