Tangents of the multiples of π/η

• Autores: Julio Castiñeira Merino
• Localización: Boletín de la Sociedad Puig Adam de profesores de matemáticas, ISSN 1135-0261, Nº. 120 (Octubre), 2025, págs. 33-53
• Idioma: inglés
• Títulos paralelos:
  • Tangentes de los múltiplos de π/η
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• Resumen

  • This article shows the exact values of the tangents of the rational fractions of π by solving the equations derived from the multiple angle tangent formulae, using a procedure similar to that of calculating cyclotomic polynomials. We determine the values of η for which the equations are solvable by quadratic radicals and calculate the tangent values by simplifying the method used by Abel in his paper "Memoir on a particular case of algebraically solvable equations". Our method is related to Carlile's circles. Examples of other Abelian equations are also provided.