Ordinary
trigonometry studies
triangles 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 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
-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:
*# 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
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 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 The trigonometry of a tetrahedron explains the relationships between the lengths and various types of angles of a general tetrahedron.
Trigonometric quantities
Classical trigonometric quantities
The following are trigonometric quantities gener ...
**
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
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 equations 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 equation Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while ''q'' sta ...
s. 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
*
Polygonometry – 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