I would like to find the fourier transform of $\sin(ax)$ which is non zero only on $-c\le x \le c$.
Calculating, I get an answer with sines and cosines but I'm not sure if this is correct... Should I be using the delta function in my answer? Or is that if the interval is infinite?
You can represent your function as $$ f(x)=[\theta(x+c)-\theta(x-c)]\sin(ax) $$ being $\theta(x)$ the Heaviside step function. Then, the Fourier integral becomes rather easy as $$ {\cal F}(f)(k)=\int_{-\infty}^\infty dx e^{ikx}f(x)=\int_{-c}^c dx e^{ikx}\sin(ax). $$ The latter integral is very well-known and rather easy to evaluate.