We show that the asymptotic expansion of the sequence $x_n = \sin x_{n-1}$ with $x_0 = x(x\in]0,\pi[)$, as $n$ goes to $+\infty$, uses a family of polynomials (with rational coefficients) which are linked by relations of recurrency. The study applies to a large class of sequences. We finish by a sharp study of the sinus function.