Trigonometry in finite fields was first introduced in [ISIT 1998 (1998; doi:10.1109/ISIT.1998.708898)] and further developed in [J. B. Lima et al., in: Applied algebra and number theory. Essays in honor of Harald Niederreiter on the occasion of his 70th birthday. Cambridge: Cambridge University Press, 255–279 (2014; Zbl 1346.11064)] giving many properties similar to the trigonometric functions over the reals. Those explorations used a degree-2 extension field of a base field. In the paper under review, the authors generalize the definitions of trigonometric functions and their related Chebyshev polynomials to arbitrary degrees.
In the second reference above, Lima, Panario, and Campello de Souza developed trigonometry in finite fields \(\mathbb{F}_q\) of odd characteristic. It was shown how these functions have many properties shared with the trigonometric functions on the real numbers, such as an analogue of the unit circle law, an angle sum formula, symmetries, and something like the Euler equation. In this paper, the authors show that these definitions and properties can be significantly generalized.