Suppose that instead of wanting to express the roots of a polynomial equation with arithmetic operations and radicals we instead wanted expressed it with arithmetic operations and $\sin(x)$ ?
What (if anything) does Galois theory say about this situation?
If you mean, when can the roots be expressed using arithmetical operations and sines of rational multiples of $\pi$: because $e^{ix}=\cos(x)+i\sin(x)$ the question then boils down when can the roots be expressed in terms of roots of unity? By Kronecker-Weber, this is possible whenever the Galois extension is abelian.