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In this work, we present a new method for quantization of sinusoidal amplitudes and phases, and apply the method to sinusoidal coding of speech and audio signals. The method is based on unrestricted polar quantization, where phase quantization accuracy depends on amplitude. Amplitude and phase quantizers are derived under an entropy (average rate) constraint using high-rate assumptions. First, we derive optimal quantizers for one sinusoid and a mean-squared error distortion measure. We provide a detailed analysis of entropy-constrained unrestricted polar quantization, showing its high performance and practicality even at low rates. Second, we find optimal quantizers for a set of sinusoids that model a short segment of an audio signal. The optimization is performed using a weighted error measure that can account for the masking effect in the human auditory system. We find the optimal rate distribution between sinusoids, as well as the corresponding optimal amplitude and phase quantizers, based on the perceptual importance of sinusoids defined by masking. The new method is used in an audio-coding application and is shown to significantly outperform a conventional sinusoidal quantization method where phase quantization accuracy is identical for all sinusoids.