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This paper introduces a complete family of trigonometric transforms over prime finite fields (FFTTs). Eight cosine transforms and eight sine transforms are defined, denoted FFCT and FFST, respectively. The property of symmetric convolution for these new tools is introduced. It is shown that the use of this operation, together with the development of fast algorithms, allows efficient computation of linear convolutions via FFTTs. A discussion concerning application of these transforms in the field of digital image processing is presented.