Generalized trigonometry

Ordinary trigonometry studies triangles in the Euclidean plane ⁠${\displaystyle \mathbb {R} ^{2}}$⁠. 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]}, 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.

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" (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 G₂(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. S2CID 125115660.
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