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Magnetization plateaux are emerged from the collective quantum phenomena in quantum magnets and have been the focus of intense research over the last decades. After analyzing the ground state of the triangular lattice S = 1/2 antiferromagnet Ba₃CoSb₂O₉ at 0T, we investgate the dynamical properties of the 1/3 magnetization plateau phase under the magnetic field. By measuring the entire structure of the excitation spectrum and revealing excellent agreement with nonlinear spin wave theory, we demonstrate that magnons behave semiclassically in spite of the quantum origin of the underlying magnetic ordering. We also discuss implications of the present result to the elusive zero-field spin dynamics in the same material, which has been recently observed by different groups.