In this paper, we propose a method for obtaining eigenvectors of discrete cosine and sine transforms of types I and IV. The approach is based on constructing an initial eigenvector of one of such trigonometric transforms and a generating matrix. Multiplying powers of the matrix by the initial eigenvector, new eigenvectors are obtained. It is shown how the generated eigenvectors can be used in the fractionalization of the respective transform. Finally, we illustrate the applicability of the developed theory in the scenario of image encryption.