A Generalisation of Niven's Theorem for Trigonometric Functions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866912527900737536 |
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| author | Keilthy, Adam Ruairí, Ailbhe Ní |
| author_facet | Keilthy, Adam Ruairí, Ailbhe Ní |
| contents | Niven's Theorem asserts that $\{\cos(rπ)|r\in \mathbb{Q}\}\cap\mathbb{Q} = \{0, \pm 1, \pm\frac{1}{2}\}$. This paper uses elementary methods to classify all elements in the sets $\{\cos^n(rπ)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$ and $\{\sin^n(rπ)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$. Using some algebraic number theory, we extend this to a classification of all elements in $\{\tan^n(rπ)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$. Finally, we present a short Galois theoretic argument to provide a more conceptual understanding of the results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Generalisation of Niven's Theorem for Trigonometric Functions Keilthy, Adam Ruairí, Ailbhe Ní Number Theory 11R04 Niven's Theorem asserts that $\{\cos(rπ)|r\in \mathbb{Q}\}\cap\mathbb{Q} = \{0, \pm 1, \pm\frac{1}{2}\}$. This paper uses elementary methods to classify all elements in the sets $\{\cos^n(rπ)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$ and $\{\sin^n(rπ)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$. Using some algebraic number theory, we extend this to a classification of all elements in $\{\tan^n(rπ)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$. Finally, we present a short Galois theoretic argument to provide a more conceptual understanding of the results. |
| title | A Generalisation of Niven's Theorem for Trigonometric Functions |
| topic | Number Theory 11R04 |
| url | https://arxiv.org/abs/2508.06415 |