Representing real function as integral over trigonometric functions

Question

Since one can clearly express any function $g(x)$ as $$\int_0^{\infty} A(k)\cos(kx)dk+\int_0^{\infty} B(k)\sin(kx)dk,$$ how would $G(k)$ relate to $A(k)$ and $B(k)$? In other words, what would how would we express $G(k)$ in terms of $A(k)$ and $G(k)$ in terms of $B(k)$ where $G(k)$ is the Fourier transform of $g(x)$?