generalized trigonometry
{{Short description|Study of triangles in other spaces than the Euclidean plane}}
{{Trigonometry}}
Ordinary trigonometry studies triangles in the Euclidean plane {{tmath|\mathbb{R}^2}}. 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|date=2024-09-29|bot=User:Cewbot/log/20201008/configuration|target_link=trigonometric functions#Series definitions|reason= The anchor (Series definitions) has been deleted.}}, definitions via differential equations{{Broken anchor|date=2024-09-29|bot=User:Cewbot/log/20201008/configuration|target_link=trigonometric functions#Definitions via differential equations|reason= The anchor (Definitions via differential equations) has been deleted.}}, 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-simplex.
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:
- # Study of hyperbolic triangles in hyperbolic geometry with hyperbolic functions.
- # 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).
- # 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{{citation|url=http://www.physics.orst.edu/~tevian/taxicab/taxicab.pdf|title=Taxicab angles and trigonometry|first1=K.|last1=Thompson|first2=T.|last2=Dray|journal=Pi Mu Epsilon Journal|year=2000|volume=11|issue=2|pages=87–96|arxiv=1101.2917|bibcode=2011arXiv1101.2917T}}
- 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{{citation|url=http://userweb.port.ac.uk/~liuh/Papers/LiuCoghill05c_SMC.pdf |contribution=Fuzzy Qualitative Trigonometry |first1=Honghai |last1=Liu |first2=George M. |last2=Coghill |title=2005 IEEE International Conference on Systems, Man and Cybernetics |year=2005 |volume=2 |pages=1291–1296 |url-status=dead |archiveurl=https://web.archive.org/web/20110725170037/http://userweb.port.ac.uk/~liuh/Papers/LiuCoghill05c_SMC.pdf |archivedate=2011-07-25 }}
- Operator trigonometry{{citation|url=http://www.ict.nsc.ru/jct/getfile.php?id=159|title=A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin|first=K. E.|last=Gustafson|author-link=Karl Edwin Gustafson|journal=Вычислительные технологии|volume=4|issue=3|pages=73–83|year=1999}}
- 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 {{math|G2(RN)}}
| volume = 57
| year = 1999}}
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{{Cite journal|title = The Trigonometry of the Tetrahedron|jstor = 3603090|journal = The Mathematical Gazette|date = 1902-03-01|pages = 149–158|volume = 2|issue = 32|doi = 10.2307/3603090|first = G.|last = Richardson| s2cid=125115660 |url = https://zenodo.org/record/1449743}}
- 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.{{citation
| last1 = West | first1 = Bruce J.
| last2 = Bologna | first2 = Mauro
| last3 = Grigolini | first3 = Paolo
| isbn = 0-387-95554-2
| location = New York
| mr = 1988873
| page = 101
| publisher = Springer-Verlag
| series = Institute for Nonlinear Science
| title = Physics of fractal operators
| year = 2003 | doi=10.1007/978-0-387-21746-8}}
- 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|date=2024-09-29|bot=User:Cewbot/log/20201008/configuration|target_link=trigonometric functions#Series definitions|reason= The anchor (Series definitions) has been deleted.}} of sin and cos define these functions on any algebra where the series converge such as complex numbers{{Broken anchor|date=2024-09-29|bot=User:Cewbot/log/20201008/configuration|target_link=trigonometric functions#Relationship to exponential function (Euler's formula)|reason= The anchor (Relationship to exponential function (Euler's formula)) has been deleted.}}, p-adic numbers, matrices, and various Banach algebras.
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 {{mvar|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