the thin obstacle problem is a classical free boundary problem arising from the study of an elastic membrane resting on a lower-dimensional obstacle. concerning the behavior of the solution near a contact point between the membrane and the obstacle, many important questions remain open. in this talk, we discuss a unified method that leads to a rate of convergence to `tangent cones' at contact points with integer frequencies. \\ \\ this talk is based on a recent joint work with ovidiu savin.