Radiative transfer calculations have been performed for five cool stars: α Tau, β Gem, Procyon, ∊ Eri and the Sun, for the purpose of investigating the behaviour of the OI emission over a wide range of stellar types, and its dependence on coherent photon scattering. These stars span a range of spectral types from F5 IV-V to K5 III and surface gravities 1.25 < logg* < 4.75. Particular attention has been paid to the calculation of the flux in the resonance triplet around 1305 Å which is pumped by H Lyβ, including the effects of partial redistribution (PRD) and cross-redistribution of photons. These are the first calculations for the resonance triplet in giant stars using a full PRD treatment. Calculations of the predominantly collisionally excited intersystem doublet at 1355, 1358 Å are included, and it is found that the ratio of these fluxes shows the effects of opacity. The flux in the forbidden line at 1641 Å is calculated for the giant stars and the effects of coherent scattering on this line are investigated. The discrepancy between the calculated and observed fluxes in the OI lines is used to infer the inadequacy of single-component chromospheric models.