Exact evaluations and reciprocity theorems for finite trigonometric sums

TL;DR

This paper presents new closed-form evaluations of finite trigonometric sums using innovative methods involving roots of unity and contour integration, along with establishing reciprocity theorems for these sums.

Contribution

It introduces two novel methods for evaluating finite trigonometric sums and proves reciprocity theorems, advancing the theoretical understanding of these sums.

Findings

Closed-form evaluations of several classes of finite trigonometric sums.

Development of a new method involving sums of roots of unity.

Establishment of reciprocity theorems for specific trigonometric sums.

Abstract

We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of the authors. Reciprocity theorems for certain trigonometric sums are also established.