EJMT Abstract

This paper introduces a family of infinite product formulae for π, developed through an original and foundational approach that blends pedagogy, history, and mathematical discovery. The methods are constructed from scratch, avoiding reliance on memorised angle-division identities or shortcuts. Instead, the derivations unfold step by step, making the complex structure of trigonometric and nested radical expressions accessible to students with only a basic mathematical background. This process not only facilitates genuine understanding but also enables learners to independently reconstruct the logic of the results—promoting a deeper conceptual grasp over rote techniques. Alongside these pedagogical advantages, the paper makes a distinct research contribution: reveals elegant and compact infinite product representations for π, derived through self-contained trigonometric constructions and supported by Dirichlet kernels. These products converge exponentially, with precision controlled by a single parameter; for instance, one specific choice yields 31 correct digits using just one term. Each trigonometric factor is expressible as a nested radical built from repeated halving and sign tracking, producing predictable patterns that enhance both computation and intuition. Framed within the historical evolution of π—from ancient approximations to modern infinite series—this work exemplifies how mathematical insight, when developed from first principles, can simultaneously advance learning and original inquiry.