Trigonometric Ratios Can Prove the Pythagorean Theorem

TL;DR

This paper presents three noncircular trigonometric proofs of the Pythagorean theorem, emphasizing the role of identities like the double-angle formula to clarify foundational aspects of trigonometry.

Contribution

It introduces new proofs based on isosceles triangles and angle-bisector theorem, extending previous work and highlighting the significance of trigonometric identities.

Findings

Proofs avoid circular reasoning

Unified approach using angle-bisector theorem

Clarifies role of double-angle formula

Abstract

Recent interest in noncircular trigonometric proofs has underscored the need for alternative methodologies. Jackson and Johnson's 2024 study addresses a longstanding gap in the foundations of trigonometric proofs. Inspired by the work of Jackson and Johnson [JJ24], we present three noncircular proofs of the Pythagorean theorem based on trigonometric identities. First, we establish the Pythagorean theorem via an isosceles triangle construction and the tangent double-angle formula. Second, we present an alternative proof utilizing an isosceles-triangle and the angle-bisector theorem. Third, we derive a novel trigonometric relation from the angle-bisector theorem, thereby unifying and extending the two preceding approaches. These approaches collectively demonstrate that the principal contribution of Jackson and Johnson lies in their strategic use of the double-angle formula. These proofs clarify the role of trigonometric identities independent of infinite series.