Provided are audio encoding and decoding apparatuses capable of recovering a high-quality audio signal at a low bit rate. The audio encoding method includes: detecting at least one sinusoidal wave from an input audio signal; calculating components of additional basis vectors based on residual audio signals and the additional basis vectors of the sinusoidal wave; determining transmission of components of the additional basis vectors; and at least one of (a) encoding frequencies and (b) phases and amplitudes of the sinusoidal waves when the transmission of the components of the additional basis vectors are determined, wherein the residual audio signals are obtained by excluding the detected sinusoidal waves from the input audio signal.