In the middle of a PDE I'm trying to solve, I've gotten $$\sum_{n=1}^\infty T_n(0) \sin(nx) = \cos(3x)$$
Is this even possible? How can you expand a cosine (even) in terms of sines (odd)? Did I necessarily make a mistake (I don't see one)? If not, how could I solve for my $T_n$'s?
You find the coefficients in the usual way: $T_n(0) = \displaystyle \dfrac{2}{\pi} \int_0^\pi \sin(nx) \cos(3x)\; dx$.
The reason odd/even doesn't cause a problem is that for a sine series, you are implicitly extending the function from the interval $[0,\pi]$ to $[-\pi,\pi]$ so that it is odd, and then to the real line so it has period $2\pi$. So really what you are approximating with odd functions is not $\cos(3x)$ but $$ \cases{ \cos(3x) & for $0 < x < \pi$\cr -\cos(3x) & for $-\pi < x < 0$\cr}$$