Ordinary
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
studies
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
s in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
. There are a number of ways of defining the ordinary
Euclidean geometric trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
on
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, 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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
or
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
. A triangle is the
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
s and polygons:
solid angles and
polytopes
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
such as
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
s and
-simplices.
Trigonometry
*In
spherical trigonometry, triangles on the surface of a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
is parameterized by (cos ''t'', sin ''t'') whereas the equilateral
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
is parameterized by (cosh ''t'', sinh ''t'').
*#
Gyrotrigonometry
A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Fo ...
: A form of trigonometry used in the
gyrovector space
A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Fo ...
approach to
hyperbolic geometry, with applications to
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
and
quantum computation
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
.
*
Rational trigonometry
''Divine Proportions: Rational Trigonometry to Universal Geometry'' is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry, called rational trigonometry. The book advocat ...
– a reformulation of trigonometry in terms of ''spread'' and ''quadrance'' rather than ''angle'' and ''length''.
*Trigonometry for
taxicab geometry
*Spacetime trigonometries
*Fuzzy qualitative trigonometry
*Operator trigonometry
*Lattice trigonometry
*Trigonometry on symmetric spaces
Higher dimensions
*
Schläfli orthoscheme
In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges (v_0v_1), (v_1v_2), \dots, (v_v ...
s - right simplexes (right triangles generalized to ''n'' dimensions) - studied by
Schoute who called the generalized trigonometry of ''n'' Euclidean dimensions
polygonometry
Ordinary trigonometry studies triangles in the Euclidean plane . 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 defin ...
.
** Pythagorean theorems for
''n''-simplices with an "orthogonal corner"
*
Trigonometry of a tetrahedron
**
De Gua's theorem
__NOTOC__
In mathematics, De Gua's theorem is a three-dimensional analog of the Pythagorean theorem named after Jean Paul de Gua de Malves. It states that if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of th ...
– a Pythagorean theorem for a tetrahedron with a cube corner
**
A law of sines for tetrahedra
*
Polar sine In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin.
Definition
''n'' vectors in ''n''-dimensional space
Let v1, ..., v''n'' (''n'' ≥ 2) be non-zero ...
Trigonometric functions
*Trigonometric functions can be defined for
fractional differential equation
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
:D f(x) = \frac f(x)\,,
and of the integration ...
s.
*In
time scale calculus,
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s and
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s 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 () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
where the
series converge such as
complex numbers,
''p''-adic numbers,
matrices, and various
Banach algebras.
Other
*Polar/Trigonometric forms of
hypercomplex numbers
*
Polygonometry
Ordinary trigonometry studies triangles in the Euclidean plane . 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 defin ...
– trigonometric identities for multiple distinct angles
*The
Lemniscate elliptic functions, sinlem and coslem
See also
*
The Pythagorean theorem in non-Euclidean geometry
References
{{DEFAULTSORT:Generalized Trigonometry
Trigonometry