ART

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.

Trigonometry

Higher-dimensions

Trigonometric functions

Other

Polar/Trigonometric forms of hypercomplex numbers[11][12]

Polygonometry - trigonometric identities for multiple distinct angles[13]

See also

The Pythagorean theorem in non-Euclidean geometry

References

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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, CiteSeerX 10.1.1.160.1580, MR 1468236
Leuzinger, Enrico (1992), "On the trigonometry of symmetric spaces", Commentarii Mathematici Helvetici, 67 (2): 252–286, doi:10.1007/BF02566499, MR 1161284
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
Richardson, G. (1902-03-01). "The Trigonometry of the Tetrahedron" (PDF). The Mathematical Gazette. 2 (32): 149–158. doi:10.2307/3603090. JSTOR 3603090.
West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003), Physics of fractal operators, Institute for Nonlinear Science, New York: Springer-Verlag, p. 101, doi:10.1007/978-0-387-21746-8, ISBN 0-387-95554-2, MR 1988873
Harkin, Anthony A.; Harkin, Joseph B. (2004), "Geometry of generalized complex numbers", Mathematics Magazine, 77 (2): 118–129, doi:10.1080/0025570X.2004.11953236, JSTOR 3219099, MR 1573734
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, archived from the original (PDF) on 2011-07-22
Antippa, Adel F. (2003), "The combinatorial structure of trigonometry" (PDF), International Journal of Mathematics and Mathematical Sciences, 2003 (8): 475–500, doi:10.1155/S0161171203106230, MR 1967890


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