Summary: We show that in the ring generated by the integers and the functions \(x, \sin x^{n}\) and \(\sin(x\cdot \sin x^{n})\) \((n=1,2,\ldots)\) defined on \(\mathbf{R}\) it is undecidable whether or not a function has a positive value or has a root. We also prove that the existential theory of the exponential field \(\mathbf{C} \) is undecidable.