A Few Finite Trigonometric Sums

TL;DR

This paper explores finite trigonometric sums, especially those involving products of functions, deriving new sums using residue methods and highlighting their interesting properties and simple values.

Contribution

It introduces new finite trigonometric sums involving products of functions, derived through residue calculus, expanding existing mathematical literature.

Findings

Derived new finite trigonometric sums using residue method

Identified sums with surprisingly simple values

Extended known results on trigonometric sums

Abstract

Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums with products of trigonometric functions can get complicated and may not have a simple expressions in a number of cases. Some of these sums have interesting properties and can have amazingly simple value. However, only some of them are available in literature. We obtain a number of such sums using method of residues.