The role of bounce resonances in the cyclotron build-up of fast magneto-acoustic waves, by trapped alpha particles in a tokamak reactor is investigated theoretically. It is shown that bounce resonances can cause a further increase in the growth rate of fast magneto-acoustic waves by a magnitude of the order of 0.5 (ωBS/ωb)1/3, where ωBS is the mean cyclotron frequency of alpha particles on the magnetic surface, ωb is their bounce frequency and is the ratio of the minor and major radii of the tokamak.