Functions satisfying $\int_0^1\cos(\pi x)f(x) = \int_0^1\sin(\pi x )f(x) = 0$

Question

Are there any continuous, nonzero functions that satisfy: $$ \int_0^1\cos(\pi y)f(y)\mathrm{d}y = \int_0^1\sin(\pi y )f(y) \mathrm{d}y = 0 $$ ?

For reference: I am trying to find the kernel of the integral kernel given by $\sin(\pi(x-y))$ where integration is done with respect to $y$. Can we completely characterize this space somehow (I.e some closed form representation?)

Answer 1

$f(y)=\sin (3\pi y)$ is one such function.