In this paper we give a characterization of the Gromov hyperbolicity
of trains (a large class of Denjoy domains which contains
the flute surfaces) in terms of the behavior of a real function.
This function describes somehow the distances between some
remarkable geodesics in the train. This theorem has several
consequences; in particular, it allows to deduce a result
about stability of hyperbolicity, even though the original
surface and the modified one are not quasi-isometric. In order
to obtain these results we also prove some trigonometric lemmas
that are interesting by themselves, since they provide very
simple estimates on some hyperbolic distances.