$\sin$ and $\cos$ are the basis of what space?
It's the Hilbert space of $L^2$ functions on the circle. More explicitly, it's the space of Lebesgue measurable functions $f : \mathbb{R} \to \mathbb{R}$ which are periodic with period $2\pi$ and such that the integral
$$\int_0^{2\pi} |f(x)|^2\,dx$$
converges, modulo the equivalence relation where $f \sim g$ if $\int_0^{2\pi} |f(x) - g(x)|^2\,dx = 0$.