Question

Find the cosire series For f (x) = sin x in (og x)​

Answer 1

Answer:

The question I have been given states;

Consider the function f:(0,π)→R defined by x⟼sinx

Show that the Fourier cosine series (i.e. the Fourier series of the even extension of f) is given by

sinx∼2π−∑n=2∞2(1+(−1)n)π(n2−1)cosnx

Now I know that f(x)∼a02+∑n∈Nancosnx

So far I have gotten a0=4π and I know the equation I must solve for an is

an=2π∫π0sinxcosnxdx

My next step is to use integration by parts to get

=2π((−1)n+1−1)−n∫π0cosxsinnx

However, I am stuck from here.