Generalized trigonometry

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, 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.
Rational trigonometry – a reformulation of trigonometry in terms of spread and quadrance rather than angle and length.^{[dubious – discuss]}
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[]

Polar sine
Trigonometry of a tetrahedron^[9]
- A law of sines for tetrahedra
Simplexes with an "orthogonal corner" – Pythagorean theorems for n-simplices
- De Gua's theorem – a Pythagorean theorem for a tetrahedron with a cube corner

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 of sin and cos define these functions on any algebra where the series converge such as complex numbers, 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

References[]

^ Thompson, K.; Dray, T. (2000), "Taxicab angles and trigonometry" (PDF), Pi Mu Epsilon Journal, 11 (2): 87–96, arXiv:1101.2917, Bibcode:2011arXiv1101.2917T
^ 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, arXiv:math-ph/9910041, Bibcode:2000JPhA...33.4525H, doi:10.1088/0305-4470/33/24/309, MR 1768742, S2CID 15313035
^ Liu, Honghai; Coghill, George M. (2005), "Fuzzy Qualitative Trigonometry", 2005 IEEE International Conference on Systems, Man and Cybernetics (PDF), vol. 2, pp. 1291–1296, archived from the original (PDF) on 2011-07-25
^ Gustafson, K. E. (1999), "A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin", Вычислительные технологии, 4 (3): 73–83
^ Karpenkov, Oleg (2008), "Elementary notions of lattice trigonometry", Mathematica Scandinavica, 102 (2): 161–205, arXiv:math/0604129, doi:10.7146/math.scand.a-15058, MR 2437186, S2CID 49911437
^ 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, S2CID 123684622
^ Masala, G. (1999), "Regular triangles and isoclinic triangles in the Grassmann manifolds $G 2 (R N)$ ", 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". 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, S2CID 7837108
^ 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, S2CID 121144869, 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

[1] Thompson, K.; Dray, T. (2000), "Taxicab angles and trigonometry" (PDF), Pi Mu Epsilon Journal, 11 (2): 87–96, arXiv:1101.2917, Bibcode:2011arXiv1101.2917T

[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, arXiv:math-ph/9910041, Bibcode:2000JPhA...33.4525H, doi:10.1088/0305-4470/33/24/309, MR 1768742, S2CID 15313035

[3] Liu, Honghai; Coghill, George M. (2005), "Fuzzy Qualitative Trigonometry", 2005 IEEE International Conference on Systems, Man and Cybernetics (PDF), vol. 2, pp. 1291–1296, archived from the original (PDF) on 2011-07-25

[4] Gustafson, K. E. (1999), "A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin", Вычислительные технологии, 4 (3): 73–83

[5] Karpenkov, Oleg (2008), "Elementary notions of lattice trigonometry", Mathematica Scandinavica, 102 (2): 161–205, arXiv:math/0604129, doi:10.7146/math.scand.a-15058, MR 2437186, S2CID 49911437

[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, CiteSeerX 10.1.1.160.1580, MR 1468236

[7] Leuzinger, Enrico (1992), "On the trigonometry of symmetric spaces", Commentarii Mathematici Helvetici, 67 (2): 252–286, doi:10.1007/BF02566499, MR 1161284, S2CID 123684622

[8] Masala, G. (1999), "Regular triangles and isoclinic triangles in the Grassmann manifolds $G 2 (R N)$ ", Rendiconti del Seminario Matematico Università e Politecnico di Torino., 57 (2): 91–104, MR 1974445

[9] Richardson, G. (1902-03-01). "The Trigonometry of the Tetrahedron". The Mathematical Gazette. 2 (32): 149–158. doi:10.2307/3603090. JSTOR 3603090.

[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, MR 1988873

[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, JSTOR 3219099, MR 1573734, S2CID 7837108

[12] 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, S2CID 121144869, archived from the original (PDF) on 2011-07-22

[13] 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

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]