One of the Dirichlet conditions, at least in the texts I've used, is: "$f$ must have a finite number of maxima and minima."
This is not true for the sine function as its derivative is the cosine which equals $0$ at an infinite number of points. However, isn't it rather strange that the sine function can't be expressed as a Fourier series when odd functions are described by a sum of sines?
Moreover, by orthogonality, taking $2π$ as the period, all of the cosine coefficients are $0$ and all of the $n \neq 1$ sine coefficients are also $0$, clearly demonstrating that there is a Fourier series for the sine function: namely, itself.
Is it simply that the Dirichlet conditions only apply to functions outside the basis of the Fourier vector space?