Chebyshev polynomials have as roots nth roots of unity, so of course these are going to show up. It's one way to define them.
https://en.wikipedia.org/wiki/Chebyshev_nodes
The nth roots of unity are incredibly well studied, and some of that stemmed in the 1700-1800s on trying to factor things. The entire field of analytic numbers theory has taken these ideas to incredible (think decades of study and research to be state of the art) depths.
More explicitly: Chebyshev polynomials are what you get when you take trigonometric polynomials for a periodic interval or Laurent polynomials on the unit complex circle and project onto a diameter of the circle.