Generalized trigonometry

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Study of triangles in other spaces than the Euclidean plane

title: "Generalized trigonometry" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["trigonometry"] description: "Study of triangles in other spaces than the Euclidean plane" topic_path: "general/trigonometry" source: "https://en.wikipedia.org/wiki/Generalized_trigonometry" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Study of triangles in other spaces than the Euclidean plane ::

Ordinary trigonometry studies triangles in the Euclidean plane . There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions, definitions via differential equations, and 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

  • In spherical trigonometry, triangles on the surface of a sphere are studied. The spherical triangle identities are written in terms of the ordinary trigonometric functions but differ from the plane triangle identities.
  • Hyperbolic trigonometry:
    1. Study of hyperbolic triangles in hyperbolic geometry with hyperbolic functions.
    2. Hyperbolic functions in Euclidean geometry: The unit circle is parameterized by (cos t, sin t) whereas the equilateral hyperbola is parameterized by (cosh t, sinh t).
    3. Gyrotrigonometry: A form of trigonometry used in the gyrovector space approach to hyperbolic geometry, with applications to special relativity and quantum computation.
  • Trigonometry for taxicab geometry
  • Spacetime trigonometries{{citation | last1 = Herranz | first1 = Francisco J. | last2 = Ortega | first2 = Ramón | last3 = Santander | first3 = Mariano | arxiv = math-ph/9910041 | doi = 10.1088/0305-4470/33/24/309 | issue = 24 | journal = Journal of Physics A | mr = 1768742 | pages = 4525–4551 | title = Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry | volume = 33 | year = 2000| bibcode = 2000JPhA...33.4525H| s2cid = 15313035
  • Fuzzy qualitative trigonometry
  • Operator trigonometry
  • Lattice trigonometry{{citation | last = Karpenkov | first = Oleg | arxiv = math/0604129 | issue = 2 | journal = Mathematica Scandinavica | mr = 2437186 | pages = 161–205 | title = Elementary notions of lattice trigonometry | volume = 102 | year = 2008 | doi=10.7146/math.scand.a-15058| s2cid = 49911437
  • Trigonometry on symmetric spaces{{citation | last1 = Aslaksen | first1 = Helmer | last2 = Huynh | first2 = Hsueh-Ling | contribution = Laws of trigonometry in symmetric spaces | location = Berlin | mr = 1468236 | pages = 23–36 | publisher = de Gruyter | title = Geometry from the Pacific Rim (Singapore, 1994) | citeseerx = 10.1.1.160.1580 | year = 1997}}{{citation | last = Leuzinger | first = Enrico | doi = 10.1007/BF02566499 | issue = 2 | journal = Commentarii Mathematici Helvetici | mr = 1161284 | pages = 252–286 | title = On the trigonometry of symmetric spaces | volume = 67 | year = 1992| s2cid = 123684622 | last = Masala | first = G. | issue = 2 | journal = Rendiconti del Seminario Matematico Università e Politecnico di Torino | mr = 1974445 | pages = 91–104 | title = Regular triangles and isoclinic triangles in the Grassmann manifolds G2(RN) | volume = 57 | year = 1999}}

Higher dimensions

Trigonometric functions

Other

  • Polar/Trigonometric forms of hypercomplex numbers{{citation | last1 = Harkin | first1 = Anthony A. | last2 = Harkin | first2 = Joseph B. | issue = 2 | journal = Mathematics Magazine | jstor = 3219099 | mr = 1573734 | pages = 118–129 | title = Geometry of generalized complex numbers | volume = 77 | year = 2004| doi = 10.1080/0025570X.2004.11953236 | s2cid = 7837108 |last=Yamaleev |first=Robert M. |doi=10.1007/s00006-005-0007-y |issue=1 |journal=Advances in Applied Clifford Algebras |mr=2236628 |pages=123–150 |title=Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics |url=http://www.clifford-algebras.org/v15/v151/YAMAL151.pdf |volume=15 |year=2005 |s2cid=121144869 |url-status=dead |archiveurl=https://web.archive.org/web/20110722194119/http://www.clifford-algebras.org/v15/v151/YAMAL151.pdf |archivedate=2011-07-22
  • Polygonometry – trigonometric identities for multiple distinct angles{{citation | last = Antippa | first = Adel F. | doi = 10.1155/S0161171203106230 | issue = 8 | journal = International Journal of Mathematics and Mathematical Sciences | volume = 2003 | mr = 1967890 | pages = 475–500 | title = The combinatorial structure of trigonometry | url = http://www.emis.de/journals/HOA/IJMMS/2003/8475.pdf | year = 2003| doi-access = free
  • The Lemniscate elliptic functions, sinlem and coslem

References

References

  1. (2000). "Taxicab angles and trigonometry". Pi Mu Epsilon Journal.
  2. (2005). "2005 IEEE International Conference on Systems, Man and Cybernetics".
  3. Gustafson, K. E.. (1999). "A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin". Вычислительные технологии.
  4. Richardson, G.. (1902-03-01). "The Trigonometry of the Tetrahedron". The Mathematical Gazette.

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trigonometry