  We study the dynamics of iterated cosine maps $E\colon z \mapsto
ae^z+be^{-z},$ with $a,b \in \C\setminus \{0\}$. We show that the points which
converge to infinity under iteration are organized in the form of rays and, as
in the exponential family, every escaping point is either on one of these rays
or the landing point of a unique ray. Thus we get a complete classification of
the escaping points of the cosine family, confirming a conjecture of Eremenko
in this case. We also get a particularly strong version of the ``dimension
paradox'': the set of rays has Hausdorff dimension 1, while the set of points
these rays land at has not only Hausdorff dimension 2 but infinite planar
Lebesgue measure.
