Fourier sine series

Question

Compute the Fourier sine series of $f(t)=t$ over the interval $[1,3]$.

The question I have is that over $[-L,L]$, the cosine series is $0$ but does this still apply over the interval $[1,3]$? So would I only have to compute the sine series?

Answer 1

Hint:

see the plot to conclude the answer

Answer 2

$f(t) = t$ is odd ($f(t) = -f(-t)$) on [-L,L] and hence the cosine series is 0 (cosine being even). This is not the case for [1,3].

Easiest way of soling the problem is translating your t to t' such that $t' \in [-L,L]$. L is 1 here so we translate by saying $t = t' + 2$. Thus the problem now becomes:

Find the Fourier series of $f(t') = t' + 2$ on [-L,L].