Calculating $\int_0^{2\pi} \sin(nt)\cos(mt)dt$

Question

Is there a way to calculate the 2nd integral quickly, if I already know what the first one is?

$\int_0^{2\pi} \exp(int)\exp(-imt)dt$

$\int_0^{2\pi} \sin(nt)\cos(mt)dt$

Answer 1

for 2) use that $$\sin(x)\cos(y)=\frac{1}{2}\left(\sin(x-y)+\sin(x+y)\right)$$

Answer 2

The trig identity $\sin a \sin b = (1/2)(\cos (a-b)-\cos(a+b)$ is the normal way to do these. If you need to use the first integral, then replace the $\sin a$'s by $\frac{\exp(ia) - \exp(-ia)}{2i}$ and multiply things out.