Generalized trigonometry

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Ordinary trigonometry studies triangles in the Euclidean plane R2. There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers: right-angled triangle definitions, unit-circle definitions, series definitions, definitions via differential equations, definitions using functional equations. Generalizations of trigonometric functions are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of geometry or space. A triangle is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angles and polytopes such as tetrahedrons and n-simplices.


  • Fuzzy qualitative trigonometry[4]
  • Operator trigonometry[5]
  • Lattice trigonometry[6]


Trigonometric functions[edit]


See also[edit]


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  2. ^ Thompson, K.; Dray, T. (2000), "Taxicab angles and trigonometry" (PDF), Pi Mu Epsilon Journal, 11 (2): 87–96, arXiv:1101.2917free to read 
  3. ^ Herranz, Francisco J.; Ortega, Ramón; Santander, Mariano (2000), "Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry", Journal of Physics A, 33 (24): 4525–4551, arXiv:math-ph/9910041free to read, doi:10.1088/0305-4470/33/24/309, MR 1768742 
  4. ^ Liu, Honghai; Coghill, George M. (2005), "Fuzzy Qualitative Trigonometry", 2005 IEEE International Conference on Systems, Man and Cybernetics (PDF), 2, pp. 1291–1296 
  5. ^ Gustafson, K. E. (1999), "A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin", Вычислительные технологии, 4 (3): 73–83 
  6. ^ Karpenkov, Oleg (2008), "Elementary notions of lattice trigonometry", Mathematica Scandinavica, 102 (2): 161–205, arXiv:math/0604129free to read, MR 2437186 
  7. ^ Aslaksen, Helmer; Huynh, Hsueh-Ling (1997), "Laws of trigonometry in symmetric spaces", Geometry from the Pacific Rim (Singapore, 1994), Berlin: de Gruyter, pp. 23–36, MR 1468236, CiteSeerX: 
  8. ^ Leuzinger, Enrico (1992), "On the trigonometry of symmetric spaces", Commentarii Mathematici Helvetici, 67 (2): 252–286, doi:10.1007/BF02566499, MR 1161284 
  9. ^ Masala, G. (1999), "Regular triangles and isoclinic triangles in the Grassmann manifolds G2(RN)", Rendiconti del Seminario Matematico Università e Politecnico di Torino., 57 (2): 91–104, MR 1974445 
  10. ^ Richardson, G. (1902-03-01). "The Trigonometry of the Tetrahedron". The Mathematical Gazette. 2 (32): 149–158. doi:10.2307/3603090. 
  11. ^ West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003), Physics of fractal operators, Institute for Nonlinear Science, New York: Springer-Verlag, p. 101, ISBN 0-387-95554-2, MR 1988873 
  12. ^ Harkin, Anthony A.; Harkin, Joseph B. (2004), "Geometry of generalized complex numbers", Mathematics Magazine, 77 (2): 118–129, JSTOR 3219099, MR 1573734 
  13. ^ Yamaleev, Robert M. (2005), "Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics" (PDF), Advances in Applied Clifford Algebras, 15 (1): 123–150, doi:10.1007/s00006-005-0007-y, MR 2236628 
  14. ^ Antippa, Adel F. (2003), "The combinatorial structure of trigonometry" (PDF), International Journal of Mathematics and Mathematical Sciences (8): 475–500, doi:10.1155/S0161171203106230, MR 1967890