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Trigonometry |
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Reference |

Laws and theorems |

Calculus |

Ordinary trigonometry studies triangles in the Euclidean plane **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, 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.

- 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.

- 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]}

- Polar sine
- Trigonometry of a tetrahedron
^{[9]} - 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 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.

- Polar/Trigonometric forms of hypercomplex numbers
^{[11]}^{[12]}

- Polygonometry – trigonometric identities for multiple distinct angles
^{[13]}

- The Lemniscate elliptic functions, sinlem and coslem

**^**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