Trigonometric Ratios Can Prove the Pythagorean Theorem
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910992959537152 |
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| author | Kise, Shoya Uehara, Takesa Shinzato, Takashi |
| author_facet | Kise, Shoya Uehara, Takesa Shinzato, Takashi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trigonometric Ratios Can Prove the Pythagorean Theorem Kise, Shoya Uehara, Takesa Shinzato, Takashi History and Overview Algebraic Geometry 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. |
| title | Trigonometric Ratios Can Prove the Pythagorean Theorem |
| topic | History and Overview Algebraic Geometry |
| url | https://arxiv.org/abs/2506.06304 |