  Current searches for compact binary mergers by ground-based
gravitational-wave detectors assume for simplicity the two bodies are not
spinning. If the binary contains compact objects with significant spin, then
this can reduce the sensitivity of these searches, particularly for black
hole--neutron star binaries. In this paper we investigate the effect of
neglecting precession on the sensitivity of searches for spinning binaries
using non-spinning waveform models. We demonstrate that in the sensitive band
of Advanced LIGO, the angle between the binary's orbital angular momentum and
its total angular momentum is approximately constant. Under this \emph{constant
precession cone} approximation, we show that the gravitational-wave phasing is
modulated in two ways: a secular increase of the gravitational-wave phase due
to precession and an oscillation around this secular increase. We show that
this secular evolution occurs in precisely three ways, corresponding to
physically different apparent evolutions of the binary's precession about the
line of sight. We estimate the best possible fitting factor between \emph{any}
non-precessing template model and a single precessing signal, in the limit of a
constant precession cone. Our closed form estimate of the fitting-factor
depends only the geometry of the in-band precession cone; it does not depend
explicitly on binary parameters, detector response, or details of either signal
model. The precessing black hole--neutron star waveforms least accurately
matched by nonspinning waveforms correspond to viewing geometries where the
precession cone sweeps the orbital plane repeatedly across the line of sight,
in an unfavorable polarization alignment.
