Ordinary trigonometry

Trigonometry

Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

 studies triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

s in the euclidean

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 plane R². There are a number of ways of defining the ordinary euclidean geometric

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

 trigonometric functions on real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s: right-angled triangle definitions, unit-circle definitions, series definitions, definitions via differential equations, 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

Polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

 with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angle

Solid angle

The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

s and polytopes such as tetrahedron

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

s and n-simplices.

Trigonometry

• In spherical trigonometry
  Spherical trigonometry
  Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
  
  , 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 triangle
    Hyperbolic triangle
    In mathematics, the term hyperbolic triangle has more than one meaning.-Hyperbolic geometry:In hyperbolic geometry, a hyperbolic triangle is a figure in the hyperbolic plane, analogous to a triangle in Euclidean geometry, consisting of three sides and three angles...
    
    s in hyperbolic geometry
    Hyperbolic geometry
    In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
    
     with hyperbolic function
    Hyperbolic function
    In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...
    
    s.
  • 2) Hyperbolic function
    Hyperbolic function
    In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...
    
    s in Euclidean geometry: The unit-circle is parameterized by (cos t, sin t) whereas the equilateral hyperbola
    Hyperbola
    In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
    
     is parameterized by the points (cosh t, sinh t).
  • 3) Gyrotrigonometry: A form of trigonometry used in the gyrovector space approach to hyperbolic geometry
    Hyperbolic geometry
    In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
    
    , with applications to special relativity
    Special relativity
    Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
    
     and quantum computation.
  • 4) Universal hyperbolic trigonometry – an algebraic approach based on Rational trigonometry
    Rational trigonometry
    Rational trigonometry is a recently introduced approach to trigonometry that eschews all transcendental functions and all proportional measurements of angles. In place of angles, it characterizes the separation between lines by a quantity called the "spread", which is a rational function of their...
    
    .

• Rational trigonometry
  Rational trigonometry
  Rational trigonometry is a recently introduced approach to trigonometry that eschews all transcendental functions and all proportional measurements of angles. In place of angles, it characterizes the separation between lines by a quantity called the "spread", which is a rational function of their...
  
   – a reformulation of trigonometry in terms of spread and quadrance rather than angle and length.

• Trigonometry in Galois fields
  Trigonometry in Galois fields
  In mathematics, trigonometry analogies are supported by the theory of quadratic extensions of finite fields, also known as Galois fields. The main motivation to deal with a finite field trigonometry is the power of the discrete transforms, which play an important role in engineering and mathematics...
  
  

• Trigonometry for taxicab geometry
  Taxicab geometry
  Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their coordinates...
  
  

• Spacetime trigonometries

• Fuzzy qualitative trigonometry

• Operator trigonometry

• Lattice trigonometry

• Trigonometry on symmetric spaces

Higher-dimensions

• Polar sine
  Polar sine
  In mathematics, the polar sine of a vertex angle of a polytope is defined as follows. Let v1, ..., vn, n ≥ 2, be non-zero vectors from the vertex in the directions of the edges...
  
  
• A law of sines for tetrahedra
• Simplexes with an "orthogonal corner" - Pythagorean theorems for n-simplexes
  • 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.

• In time scale calculus
  Time scale calculus
  In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems...
  
  , differential equation
  Differential equation
  A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
  
  s 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
  Algebra over a field
  In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
  
   where the series converge such as complex numbers, p-adic numbers
  P-adic analysis
  In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....
  
  , matrices
  Matrix function
  In mathematics, a matrix function is a function which maps a matrix to another matrix.- Extending scalar functions to matrix functions :There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained...
  
  , and various Banach algebra
  Banach algebra
  In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
  
  s.

Other

• Polar/Trigonometric forms of hypercomplex number
  Hypercomplex number
  In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
  
  s

• Polygonometry - trigonometric identities for multiple distinct angles

See also

• The Pythagorean theorem in non-Euclidean geometry