How do I find a fourier series for the function of period $2\pi$ satisfying $$f(t)= \begin{cases}\sin t &0 \le t<\pi\\0 &\pi\le t<2\pi\end{cases} $$
Do I find $b_n$ as usual (because it's an odd function) and then give the Dirichlet conditions? I'm a bit thrown by the zero.
The function is not odd. Since its period is given as $2\pi$, the function is also $0$ from $-\pi$ to $0$. So you'll have to grind out the integrals
$$a_n =\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos \frac{n\pi x}{\pi} \; dx =\frac{1}{\pi} \int_{0}^{\pi} \sin(x) \cos nx\; dx$$
and so on.