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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

| 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

}}{{citation

| 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

Trigonometric functions

| 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}}

Other

| 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

}}{{citation

|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

}}

See also

References

{{reflist}}

{{DEFAULTSORT:Generalized Trigonometry}}

Category:Trigonometry