Generalized trigonometry

Short description: Study of triangles in other spaces than the Euclidean plane

Trigonometry

* Outline
* History
* Usage

* Functions (inverse)
* Generalized trigonometry

Reference

* Identities
* Exact constants
* Tables
* Unit circle

Laws and theorems

* Sines
* Cosines
* Tangents
* Cotangents

* Pythagorean theorem

Calculus

* Trigonometric substitution
* Integrals (inverse functions)
* Derivatives

* v
* t
* e

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, for example right-angled triangle definitions, unit circle definitions, series definitions[broken anchor], definitions via differential equations[broken anchor], 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.

## Contents

* 1 Trigonometry
* 2 Higher dimensions
* 3 Trigonometric functions
* 4 Other
* 5 See also
* 6 References

## 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[1]
* Spacetime trigonometries[2]
* Fuzzy qualitative trigonometry[3]
* Operator trigonometry[4]
* Lattice trigonometry[5]
* Trigonometry on symmetric spaces[6][7][8]

## Higher dimensions

* Schläfli orthoschemes \- right simplexes (right triangles generalized to n dimensions) - studied by Schoute who called the generalized trigonometry of n Euclidean dimensions polygonometry.
* Pythagorean theorems for n-simplices with an "orthogonal corner"
* Trigonometry of a tetrahedron[9]
* De Gua's theorem – a Pythagorean theorem for a tetrahedron with a cube corner
* A law of sines for tetrahedra
* Polar sine

## Trigonometric functions

* Trigonometric functions can be defined for fractional differential equations.[10]

* In time scale calculus, differential equations and difference equations are unified into dynamic equations on time scales which also includes q-difference equations. Trigonometric functions can be defined on an arbitrary time scale (a subset of the real numbers).
* The series definitions[broken anchor] of sin and cos define these functions on any algebra where the series converge such as complex numbers[broken anchor], p-adic numbers, matrices, and various Banach algebras.

## Other

* Polar/Trigonometric forms of hypercomplex numbers[11][12]
* Polygonometry – trigonometric identities for multiple distinct angles[13]
* The Lemniscate elliptic functions, sinlem and coslem

## See also

* The Pythagorean theorem in non-Euclidean geometry

## References

1. ↑ Thompson, K.; Dray, T. (2000), "Taxicab angles and trigonometry", Pi Mu Epsilon Journal 11 (2): 87–96, Bibcode: 2011arXiv1101.2917T, http://www.physics.orst.edu/~tevian/taxicab/taxicab.pdf, retrieved 2009-05-18
2. ↑ 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, doi:10.1088/0305-4470/33/24/309, Bibcode: 2000JPhA...33.4525H
3. ↑ Liu, Honghai; Coghill, George M. (2005), "Fuzzy Qualitative Trigonometry", 2005 IEEE International Conference on Systems, Man and Cybernetics, 2, pp. 1291–1296, archived from the original on 2011-07-25, https://web.archive.org/web/20110725170037/http://userweb.port.ac.uk/~liuh/Papers/LiuCoghill05c_SMC.pdf
4. ↑ Gustafson, K. E. (1999), "A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin", Вычислительные технологии 4 (3): 73–83, http://www.ict.nsc.ru/jct/getfile.php?id=159
5. ↑ Karpenkov, Oleg (2008), "Elementary notions of lattice trigonometry", Mathematica Scandinavica 102 (2): 161–205, doi:10.7146/math.scand.a-15058
6. ↑ 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
7. ↑ Leuzinger, Enrico (1992), "On the trigonometry of symmetric spaces", Commentarii Mathematici Helvetici 67 (2): 252–286, doi:10.1007/BF02566499
8. ↑ 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''G''2('''R'''''N'')&rft.jtitle=Rendiconti+del+Seminario+Matematico+Università+e+Politecnico+di+Torino&rft.aulast=Masala&rft.aufirst=G.&rft.au=Masala, G.&rft.date=1999&rft.volume=57&rft.issue=2&rft.pages=91–104&rfr_id=info:sid/en.wikibooks.org:Generalized_trigonometry">
9. ↑ Richardson, G. (1902-03-01). "The Trigonometry of the Tetrahedron". The Mathematical Gazette 2 (32): 149–158. doi:10.2307/3603090. https://zenodo.org/record/1449743.
10. ↑ 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
11. ↑ Harkin, Anthony A.; Harkin, Joseph B. (2004), "Geometry of generalized complex numbers", Mathematics Magazine 77 (2): 118–129, doi:10.1080/0025570X.2004.11953236
12. ↑ Yamaleev, Robert M. (2005), "Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics", Advances in Applied Clifford Algebras 15 (1): 123–150, doi:10.1007/s00006-005-0007-y, archived from the original on 2011-07-22, https://web.archive.org/web/20110722194119/http://www.clifford-algebras.org/v15/v151/YAMAL151.pdfn-order+polynomials+and+generalizations+of+trigonometry,+oscillator+model+and+Hamilton+dynamics&rft.jtitle=[[Advances+in+Applied+Clifford+Algebras]]&rft.aulast=Yamaleev&rft.aufirst=Robert+M.&rft.au=Yamaleev, Robert+M.&rft.date=2005&rft.volume=15&rft.issue=1&rft.pages=123–150&rft_id=info:doi/10.1007/s00006-005-0007-y&rft_id=https://web.archive.org/web/20110722194119/http://www.clifford-algebras.org/v15/v151/YAMAL151.pdf&rfr_id=info:sid/en.wikibooks.org:Generalized_trigonometry">
13. ↑ Antippa, Adel F. (2003), "The combinatorial structure of trigonometry", International Journal of Mathematics and Mathematical Sciences 2003 (8): 475–500, doi:10.1155/S0161171203106230, http://www.emis.de/journals/HOA/IJMMS/2003/8475.pdf

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