Finite trigonometric sums arising from Ramanujan's theta functions

TL;DR

This paper evaluates two classes of finite sine-based trigonometric sums in closed form using Ramanujan's theta functions and modular equations, providing new insights into their structure and relationships.

Contribution

It introduces original proofs for finite trigonometric sums derived from Ramanujan's theta functions and modular equations, expanding the understanding of these sums.

Findings

Closed-form evaluations of two classes of sine sums

Connections established between trigonometric sums and Ramanujan's theta functions

New proofs based on modular equations

Abstract

Two classes of finite trigonometric sums, each involving only $\sin$'s, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations.