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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are
real function In mathematical analysis, and applications in geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...
s which relate an angle of a
right-angled triangle A right triangle (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, American Eng ...
to ratios of two side lengths. They are widely used in all sciences that are related to
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, such as navigation,
solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature change ...
,
celestial mechanics Celestial mechanics is the branch of astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and cel ...
,
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...
, and many others. They are among the simplest
periodic function A periodic function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logica ...

periodic function
s, and as such are also widely used for studying periodic phenomena through
Fourier analysis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
. The trigonometric functions most widely used in modern mathematics are the
sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

sine
, the
cosine 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 al ...

cosine
, and the tangent. Their
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...

reciprocal
s are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
, and an analog among the
hyperbolic functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

hyperbolic functions
. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for
acute angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...
s. To extend the sine and cosine functions to functions whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
is the whole
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, geometrical definitions using the standard
unit circle measure. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, a ...

unit circle
(i.e., a circle with
radius In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

radius
1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as
infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
or as solutions of
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

differential equation
s. This allows extending the domain of sine and cosine functions to the whole
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.


Right-angled triangle definitions

If the acute angle is given, then any right triangles that have an angle of are similar to each other. This means that the ratio of any two side lengths depends only on . Thus these six ratios define six functions of , which are the trigonometric functions. In the following definitions, the
hypotenuse In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

hypotenuse
is the length of the side opposite the right angle, ''opposite'' represents the side opposite the given angle , and ''adjacent'' represents the side between the angle and the right angle. In a right-angled triangle, the sum of the two acute angles is a right angle, that is, or . Therefore \sin(\theta) and \cos(90^\circ - \theta) represent the same ratio, and thus are equal. This identity and analagous relationships between the other trigonometric functions are summarized in the following table.


Radians versus degrees

In geometric applications, the argument of a trigonometric function is generally the measure of an
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

angle
. For this purpose, any
angular unit Throughout history, angles have been measure (mathematics), measured in many different unit (measurement), units. These are known as angular units, with the most contemporary units being the degree (angle), degree ( ° ), the radian (rad), and the ...
is convenient, and angles are most commonly measured in conventional units of
degrees Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
in which a right angle is 90° and a complete turn is 360° (particularly in
elementary mathematics 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
). However, in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
and
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, the trigonometric functions are generally regarded more abstractly as functions of
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of a ...

exponential function
via power series or as solutions to
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

differential equation
s given particular initial values (''see below''), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions ''if'' ''the argument is regarded as an angle given in radians''. Moreover, these definitions result in simple expressions for the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
s and
indefinite integrals In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures. When
Radian The radian, denoted by the symbol \text, is the SI unit for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

Radian
s (rad) are employed, the angle is given as the length of the arc of the
unit circle measure. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, a ...

unit circle
subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete
turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...
(360°) is an angle of 2 (≈ 6.28) rad. For real number ''x'', the notations sin ''x'', cos ''x'', etc. refer to the value of the trigonometric functions evaluated at an angle of ''x'' rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., sin ''x°'', cos ''x°'', etc.). Using this standard notation, the argument ''x'' for the trigonometric functions satisfies the relationship ''x'' = (180''x''/)°, so that, for example, sin = sin 180° when we take ''x'' = . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = /180 ≈ 0.0175.


Unit-circle definitions

The six trigonometric functions can be defined as coordinate values of points on the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
that are related to the
unit circle measure. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, a ...

unit circle
, which is the
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
of radius one centered at the origin of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between and \frac
radian The radian, denoted by the symbol \text, is the SI unit for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

radian
the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers. Let \mathcal L be the
ray Ray may refer to: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ...
obtained by rotating by an angle the positive half of the -axis (
counterclockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite sen ...
rotation for \theta > 0, and clockwise rotation for \theta < 0). This ray intersects the unit circle at the point \mathrm = (x_\mathrm,y_\mathrm). The ray \mathcal L, extended to a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
if necessary, intersects the line of equation x=1 at point \mathrm = (1,y_\mathrm), and the line of equation y=1 at point \mathrm = (x_\mathrm,1). The
tangent line In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

tangent line
to the unit circle at the point , is
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

perpendicular
to \mathcal L, and intersects the - and -axes at points \mathrm = (0,y_\mathrm) and \mathrm = (x_\mathrm,0). The
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

coordinates
of these points give the values of all trigonometric functions for any arbitrary real value of in the following manner. The trigonometric functions and are defined, respectively, as the ''x''- and ''y''-coordinate values of point . That is, :\cos \theta = x_\mathrm \quad and \quad \sin \theta = y_\mathrm. In the range 0 \le \theta \le \pi/2, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius as
hypotenuse In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

hypotenuse
. And since the equation x^2+y^2=1 holds for all points \mathrm = (x,y) on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity. :\cos^2\theta+\sin^2\theta=1. The other trigonometric functions can be found along the unit circle as :\tan \theta = y_\mathrm \quad and \quad\cot \theta = x_\mathrm, :\csc \theta\ = y_\mathrm \quad and \quad\sec \theta = x_\mathrm. By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is : \tan \theta =\frac,\quad \cot\theta=\frac,\quad \sec\theta=\frac,\quad \csc\theta=\frac. Since a rotation of an angle of \pm2\pi does not change the position or size of a shape, the points , , , , and are the same for two angles whose difference is an integer multiple of 2\pi. Thus trigonometric functions are
periodic function A periodic function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logica ...

periodic function
s with period 2\pi. That is, the equalities : \sin\theta = \sin\left(\theta + 2 k \pi \right)\quad and \quad \cos\theta = \cos\left(\theta + 2 k \pi \right) hold for any angle and any
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that is the smallest value for which they are periodic (i.e., is the of these functions). However, after a rotation by an angle \pi, the points and already return to their original position, so that the tangent function and the cotangent function have a fundamental period of . That is, the equalities : \tan\theta = \tan(\theta + k\pi) \quad and \quad \cot\theta = \cot(\theta + k\pi) hold for any angle and any integer .


Algebraic values

The
algebraic expressionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s for the most important angles are as follows: :\sin 0 = \sin 0^\circ \quad= \frac2 = 0 (
straight angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...
) :\sin \frac\pi6 = \sin 30^\circ = \frac2 = \frac :\sin \frac\pi4 = \sin 45^\circ = \frac :\sin \frac\pi3 = \sin 60^\circ = \frac :\sin \frac\pi2 = \sin 90^\circ = \frac2 = 1 (
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

right angle
) Writing the numerators as
square roots In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square (algebra), square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of ...
of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. Such simple expressions generally do not exist for other angles which are rational multiples of a right angle. *For an angle which, measured in degrees, is a multiple of three, the exact trigonometric values of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by ruler and compass. *For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the
cube root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

cube root
of a non-real
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
.
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable. *For an angle which, expressed in degrees, is a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
, the sine and the cosine are
algebraic number An algebraic number is any complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...
s, which may be expressed in terms of th roots. This results from the fact that the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s of the
cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its root of a function, roots are al ...
s are
cyclic Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...

cyclic
. *For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are
transcendental number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. This is a corollary of
Baker's theorem In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
, proved in 1966.


Simple algebraic values

The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.


In calculus

Graphs of sine, cosine and tangent
Graphs</a> of sine, cosine and tangent
The modern trend in mathematics is to build
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
from
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
rather than the converse. Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus. Trigonometric functions are
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
and analytic at every point where they are defined; that is, everywhere for the sine and the cosine, and, for the tangent, everywhere except at for every integer . The trigonometric function are
periodic function A periodic function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logica ...

periodic function
s, and their is for the sine and the cosine, and for the tangent, which is
increasing Image:Monotonicity example3.png, Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that pres ...
in each
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. At each end point of these intervals, the tangent function has a vertical
asymptote In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέ ...

asymptote
. In calculus, there are two equivalent definitions of trigonometric functions, either using
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
or
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

differential equation
s. These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.


Definition by differential equations

Sine and cosine can be defined as the unique solution to the
initial value problem In multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infi ...
: :\frac\sin x= \cos x,\ \frac\cos x= -\sin x,\ \sin(0)=0,\ \cos(0)=1. Differentiating again, \frac\sin x = \frac\cos x = -\sin x and \frac\cos x = -\frac\sin x = -\cos x, so both sine and cosine are solutions of the
ordinary differential equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
:y''+y=0. Applying the
quotient rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of g ...
to the tangent \tan x = \sin x / \cos x, we derive :\frac\tan x = \frac = 1+\tan^2 x.


Power series expansion

Applying the differential equations to
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
with indeterminate coefficients, one may deduce
recurrence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s for the coefficients of the
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions : \begin \sin x & = x - \frac + \frac - \frac + \cdots \\& = \sum_^\infty \fracx^ \\\cos x & = 1 - \frac + \frac - \frac + \cdots \\& = \sum_^\infty \fracx^. \end The
radius of convergence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of these series is infinite. Therefore, the sine and the cosine can be extended to
entire function In complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...
s (also called "sine" and "cosine"), which are (by definition)
complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
s that are defined and
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
on the whole
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. Being defined as fractions of entire functions, the other trigonometric functions may be extended to
meromorphic function In the mathematical field of complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Anci ...
s, that is functions that are holomorphic in the whole complex plane, except some isolated points called
poles The Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a nation A nation is a community A community is a social unitThe term "level of analysis" is used in the social sciences to point to the loc ...
. Here, the poles are the numbers of the form (2k+1)\frac \pi 2 for the tangent and the secant, or k\pi for the cotangent and the cosecant, where is an arbitrary integer. Recurrences relations may also be computed for the coefficients of the
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of the other trigonometric functions. These series have a finite
radius of convergence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. Their coefficients have a
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other are ...
interpretation: they enumerate
alternating permutation In combinatorics, combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set is a permutation (arrangement) of those numbers so that each entry is alternately greater or less than the preceding entry. For example, ...
s of finite sets. More precisely, defining : , the th
up/down number In combinatorics, combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set is a permutation (arrangement) of those numbers so that each entry is alternately greater or less than the preceding entry. For example, ...
, : , the th
Bernoulli number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, and : , is the th
Euler number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, one has the following series expansions: : \begin \tan x & = \sum_^\infty \fracx^ \\& = \sum_^\infty \fracx^ \\& = x + \fracx^3 + \fracx^5 + \fracx^7 + \cdots, \qquad \text , x, < \frac. \end : \begin \csc x &= \sum_^\infty \fracx^ \\&= x^ + \fracx + \fracx^3 + \fracx^5 + \cdots, \qquad \text 0 < , x, < \pi. \end : \begin \sec x &= \sum_^\infty \fracx^ = \sum_^\infty \fracx^ \\&= 1 + \fracx^2 + \fracx^4 + \fracx^6 + \cdots, \qquad \text , x, < \frac. \end : \begin \cot x &= \sum_^\infty \fracx^ \\&= x^ - \fracx - \fracx^3 - \fracx^5 - \cdots, \qquad \text 0 < , x, < \pi. \end


Partial fraction expansion

There is a series representation as partial fraction expansion where just translated
reciprocal function Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

reciprocal function
s are summed up, such that the
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
s of the cotangent function and the reciprocal functions match: : \pi \cot \pi x = \lim_\sum_^N \frac. This identity can be proven with the Herglotz trick. Combining the th with the th term lead to
absolutely convergent In mathematics, an Series (mathematics), infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a Real number, real or Complex number, ...
series: : \pi \cot \pi x = \frac + 2x\sum_^\infty \frac. Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: : \frac = \frac + 2x\sum_^\infty \frac, : \frac = \sum_^\infty (-1)^n \frac, : \pi \tan \pi x = 2x\sum_^\infty \frac.


Infinite product expansion

The following infinite product for the sine is of great importance in complex analysis: :\sin z = z \prod_^\infty \left(1-\frac\right), \quad z\in\mathbb C. For the proof of this expansion, see
Sine In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. From this, it can be deduced that :\cos z = \prod_^\infty \left(1-\frac\right), \quad z\in\mathbb C.


Relationship to exponential function (Euler's formula)

Euler's formula Euler's formula, named after Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

Euler's formula
relates sine and cosine to the
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of a ...

exponential function
: : e^ = \cos x + i\sin x. This formula is commonly considered for real values of , but it remains true for all complex values. ''Proof'': Let f_1(x)=\cos x + i\sin x, and f_2(x)=e^. One has \fracf_j(x)= if_j(x) for . The
quotient rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of g ...
implies thus that \frac\left(\frac\right)=0. Therefore, \frac is a constant function, which equals , as f_1(0)=f_2(0)=1. This proves the formula. One has :\begin e^ &= \cos x + i\sin x\\e^ &= \cos x - i\sin x. \end Solving this
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or ...
in sine and cosine, one can express them in terms of the exponential function: : \begin\sin x &= \frac\\\cos x &= \frac. \end When is real, this may be rewritten as : \cos x = \operatorname\left(e^\right), \qquad \sin x = \operatorname\left(e^\right). Most
trigonometric identities In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity e^=e^ae^b for simplifying the result.


Definitions using functional equations

One can also define the trigonometric functions using various
functional equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
s. For example, the sine and the cosine form the unique pair of
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s that satisfy the difference formula : \cos(x- y) = \cos x\cos y + \sin x\sin y\, and the added condition : 0 < x\cos x < \sin x < x\quad\text\quad 0 < x < 1.


In the complex plane

The sine and cosine of a
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
z=x+iy can be expressed in terms of real sines, cosines, and
hyperbolic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

hyperbolic function
s as follows: : \begin\sin z &= \sin x \cosh y + i \cos x \sinh y\\\cos z &= \cos x \cosh y - i \sin x \sinh y\end By taking advantage of
domain coloring In complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investiga ...

domain coloring
, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of z becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.


Basic identities

Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see
List of trigonometric identities In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval , see Proofs of trigonometric identities). For non-geometrical proofs using only tools of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.


Parity

The cosine and the secant are
even function The cosine function and all of its Taylor polynomials are even functions. This image shows \cos(x) and its Taylor approximation of degree 4. In mathematics, even functions and odd functions are function (mathematics), functions which satisfy par ...

even function
s; the other trigonometric functions are
odd function The cosine function and all of its Taylor polynomials are even functions. This image shows \cos(x) and its Taylor approximation of degree 4. In mathematics, even functions and odd functions are function (mathematics), functions which satisfy par ...
s. That is: :\begin \sin(-x) &=-\sin x\\ \cos(-x) &=\cos x\\ \tan(-x) &=-\tan x\\ \cot(-x) &=-\cot x\\ \csc(-x) &=-\csc x\\ \sec(-x) &=\sec x. \end


Periods

All trigonometric functions are
periodic function A periodic function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logica ...

periodic function
s of period . This is the smallest period, except for the tangent and the cotangent, which have as smallest period. This means that, for every integer , one has :\begin \sin (x+2k\pi) &=\sin x\\ \cos (x+2k\pi) &=\cos x\\ \tan (x+k\pi) &=\tan x\\ \cot (x+k\pi) &=\cot x\\ \csc (x+2k\pi) &=\csc x\\ \sec (x+2k\pi) &=\sec x. \end


Pythagorean identity

The Pythagorean identity, is the expression of the
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

Pythagorean theorem
in terms of trigonometric functions. It is :\sin^2 x + \cos^2 x = 1 .


Sum and difference formulas

The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to
Ptolemy Claudius Ptolemy (; grc-koi, Κλαύδιος Πτολεμαῖος, , ; la, Claudius Ptolemaeus; AD) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes ...
. One can also produce them algebraically using
Euler's formula Euler's formula, named after Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

Euler's formula
. ; Sum :\begin \sin\left(x+y\right)&=\sin x \cos y + \cos x \sin y,\\\cos\left(x+y\right)&=\cos x \cos y - \sin x \sin y,\\\tan(x + y) &= \frac. \end ; Difference :\begin \sin\left(x-y\right)&=\sin x \cos y - \cos x \sin y,\\\cos\left(x-y\right)&=\cos x \cos y + \sin x \sin y,\\\tan(x - y) &= \frac. \end When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae. :\begin \sin 2x &= 2 \sin x \cos x = \frac, \\\cos 2x &= \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x = \frac,\\\tan 2x &= \frac. \end These identities can be used to derive the product-to-sum identities. By setting t=\tan \tfrac12 \theta, all trigonometric functions of \theta can be expressed as
rational fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s of t: :\begin \sin \theta &= \frac, \\\cos \theta &= \frac,\\\tan \theta &= \frac. \end Together with :d\theta = \frac \, dt, this is the
tangent half-angle substitution In integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integr ...
, which reduces the computation of
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

integral
s and
antiderivative In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study ...
s of trigonometric functions to that of rational fractions.


Derivatives and antiderivatives

The
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
s of trigonometric functions result from those of sine and cosine by applying
quotient rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of g ...
. The values given for the
antiderivative In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study ...
s in the following table can be verified by differentiating them. The number  is a
constant of integration In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ge ...

constant of integration
. Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule: : \begin \frac &= \frac\sin(\pi/2-x)=-\cos(\pi/2-x)=-\sin x \, , \\ \frac &= \frac\sec(\pi/2 - x) = -\sec(\pi/2 - x)\tan(\pi/2 - x) = -\csc x \cot x \, , \\ \frac &= \frac\tan(\pi/2 - x) = -\sec^2(\pi/2 - x) = -\csc^2 x \, . \end


Inverse functions

The trigonometric functions are periodic, and hence not
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

injective
, so strictly speaking, they do not have an
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
. However, on each interval on which a trigonometric function is
monotonic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, one can define an inverse function, and this defines inverse trigonometric functions as
multivalued function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

multivalued function
s. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijective
from this interval to its image by the function. The common choice for this interval, called the set of
principal value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function. The notations , , etc. are often used for and , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "
arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of Angular unit, angular measurement equal to of one Degree (angle), degree. Since one degree is of a turn (geometry), turn (or complete rotati ...
". Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of
complex logarithm of the color is used to show the ''arg Arg or ARG may refer to: Places *''Arg'' () means "citadel" in Persian, and may refer to: **Arg, Iran, a village in Fars Province, Iran **Arg (Kabul), presidential palace in Kabul, Afghanistan **Arg, South ...
s.


Applications


Angles and sides of a triangle

In this section , , denote the three (interior) angles of a triangle, and , , denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.


Law of sines

The law of sines states that for an arbitrary triangle with sides , , and and angles opposite those sides , and : \frac = \frac = \frac = \frac, where is the area of the triangle, or, equivalently, \frac = \frac = \frac = 2R, where is the triangle's
circumradius In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
. It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in ''
triangulation In trigonometry Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...

triangulation
'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.


Law of cosines

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

Pythagorean theorem
: c^2=a^2+b^2-2ab\cos C, or equivalently, \cos C=\frac. In this formula the angle at is opposite to the side . This theorem can be proven by dividing the triangle into two right ones and using the
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

Pythagorean theorem
. The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.


Law of tangents

The law of tangents says that: :\frac = \frac.


Law of cotangents

If ''s'' is the triangle's semiperimeter, (''a'' + ''b'' + ''c'')/2, and ''r'' is the radius of the triangle's
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle ...

incircle
, then ''rs'' is the triangle's area. Therefore
Heron's formula In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

Heron's formula
implies that: : r = \sqrt. The law of cotangents says that: :\cot = \frac It follows that :\frac=\frac=\frac=\frac.


Periodic functions

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe
simple harmonic motion In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approxima ...

simple harmonic motion
, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of
uniform circular motion In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...

uniform circular motion
. Trigonometric functions also prove to be useful in the study of general
periodic function A periodic function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logica ...

periodic function
s. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light
wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

wave
s. Under rather general conditions, a periodic function can be expressed as a sum of sine waves or cosine waves in a
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Denoting the sine or cosine
basis functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
by , the expansion of the periodic function takes the form: f(t) = \sum _^\infty c_k \varphi_k(t). For example, the
square wave A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Tim ...

square wave
can be written as the
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
f_\text(t) = \frac \sum_^\infty . In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a
sawtooth wave The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw A saw is a tool A tool is an object that can extend an individual's ability to modify fe ...

sawtooth wave
are shown underneath.


History

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The
chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord (ast ...
function was discovered by
Hipparchus Hipparchus of Nicaea (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discov ...
of
Nicaea Nicaea or Nicea (; el, Νίκαια, ''Níkaia'') was an ancient Greek city in northwestern Anatolia Anatolia,, tr, Anadolu Yarımadası), and the Anatolian plateau. also known as Asia Minor, is a large peninsula in Western Asia and t ...
(180–125 BCE) and
Ptolemy Claudius Ptolemy (; grc-koi, Κλαύδιος Πτολεμαῖος, , ; la, Claudius Ptolemaeus; AD) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes ...
of
Roman Egypt , conventional_long_name = Roman Egypt , common_name = Egypt , subdivision = Roman province, Province , nation = the Roman Empire , era = Late antiquity , capital = Alexandria , title_leader = Praefectus Augustalis , image_ ...
(90–165 CE). The functions of sine and
versine The versine or versed sine is a trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

versine
(1 – cosine) can be traced back to the ''jyā'' and ''koti-jyā'' functions used in
Gupta period The Gupta Empire was an empire which existed from the early 4th century CE to late 6th century CE. At its zenith, from approximately 319 to 467 CE, it covered much of the . This period is considered as the by historians. The ruling dynasty of ...
Indian astronomy (''
Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Indian astronomy, Sanskrit astronomical treatise, is the ''Masterpiece, magnum opus'' and only known surviving work of the 5th century Indian mathematics, Indian mathematician Aryabhata. Philos ...
'', ''
Surya Siddhanta The ''Surya Siddhanta'' (; ) is a Sanskrit treatise in Indian astronomy in fourteen chapters.Plofkerpp. 71–72 The ''Surya Siddhanta'' describes rules to calculate the motions of various planets and the moon relative to various constellations, a ...
''), via translation from Sanskrit to Arabic and then from Arabic to Latin. (See Aryabhata's sine table.) All six trigonometric functions in current use were known in
Islamic mathematics Mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
by the 9th century, as was the
law of sines In trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks foc ...

law of sines
, used in solving triangles. With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant.
Al-Khwārizmī Muḥammad ibn Mūsā al-Khwārizmī ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), Arabization, Arabized as al-Khwarizmi and formerly Latinisation of names, Latinized as ''Algorithmi'', was a Persians, Persian ...
(c. 780–850) produced tables of sines, cosines and tangents. Circa 830,
Habash al-Hasib al-Marwazi Ahmad ibn 'Abdallah Habash Hasib Marwazi (766 - d. after 869 in Samarra Samarra ( ar, سَامَرَّاء, ') is a city in Iraq Iraq ( ar, ٱلْعِرَاق, '; ku, عێراق '), officially the Republic of Iraq ( ar, جُمْهُو ...
discovered the cotangent, and produced tables of tangents and cotangents.Jacques Sesiano, "Islamic mathematics", p. 157, in
Muhammad ibn Jābir al-Harrānī al-Battānī Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī ( ar, محمد بن جابر بن سنان البتاني) (Latinization (literature), Latinized as Albategnius, Albategni or Albatenius) (c. 8 ...
(853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. The trigonometric functions were later studied by mathematicians including
Omar Khayyám Omar Khayyam (; fa, عمر خیّام ; 18 May 1048 – 4 December 1131) was a Persians, Persian polymath, Mathematics in medieval Islam, mathematician, Astronomy in the medieval Islamic world, astronomer, philosopher, and Persian poetry, ...
,
Bhāskara II Bhāskara (c. 1114–1185) also known as Bhāskarācārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian people, Indian Indian mathematicians, mathematician and astronomer. He was born in Bij ...
,
Nasir al-Din al-Tusi Muhammad ibn Muhammad ibn al-Hasan al-Tūsī ( fa, محمد ابن محمد ابن حسن طوسی 18 February 1201 – 26 June 1274), better known as Nasir al-Din al-Tusi ( fa, نصیر الدین طوسی, links=no; or simply Tusi in the ...
, Jamshīd al-Kāshī (14th century),
Ulugh Beg Mīrzā Muhammad Tāraghay bin Shāhrukh ( chg, میرزا محمد طارق بن شاہ رخ, fa, میرزا محمد تراغای بن شاہ رخ), better known as Ulugh Beg () (22 March 1394 – 27 October 1449), was a TimuridTimurid refe ...
(14th century),
Regiomontanus Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) inc ...

Regiomontanus
(1464),
Rheticus Georg Joachim de Porris, also known as Rheticus (Help:IPA/English, /ˈrɛtɪkəs/; 16 February 1514 – 4 December 1574), was a mathematician, astronomer, cartographer, navigational-instrument maker, medical practitioner, and teacher. He is perhaps ...
, and Rheticus' student
Valentinus OthoValentinus Otho (also Valentin Otto; born around 1545/46 possibly in Magdeburg Magdeburg (; nds, label= Low Saxon, Meideborg ) is the capital and second-largest city of the German States of Germany, state of Saxony-Anhalt, after Halle (Saale). ...
.
Madhava of Sangamagrama Iriññāttappiḷḷi Mādhavan Nampūtiri known as Mādhava of Sangamagrāma () was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes t ...
(c. 1400) made early strides in the
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
of trigonometric functions in terms of
infinite series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. (See
Madhava series In mathematics, a Madhava series or Leibniz series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of ...
and
Madhava's sine table Madhava's sine table is the table (information), table of sine, trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-f ...
.) The terms ''tangent'' and ''secant'' were first introduced by the Danish mathematician
Thomas Fincke Thomas Fincke (6 January 1561 – 24 April 1656) was a Danish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...
in his book ''Geometria rotundi'' (1583). The 17th century French mathematician
Albert Girard Albert Girard () (1595 in Saint-Mihiel Saint-Mihiel () is a Communes of France, commune in the Meuse (department), Meuse Departments of France, department in Grand Est in north-eastern France. Geography Saint-Mihiel lies on the banks of the riv ...
made the first published use of the abbreviations ''sin'', ''cos'', and ''tan'' in his book ''Trigonométrie''. In a paper published in 1682,
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, ...
proved that is not an
algebraic functionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of . Though introduced as ratios of sides of a
right triangle A right triangle (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of variety (linguistics), varieties of the English language native to the United States. ...

right triangle
, and thus appearing to be
rational function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

rational function
s, Leibnitz result established that they are actually
transcendental function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his ''
Introduction to the Analysis of the Infinite ''Introductio in analysin infinitorum'' (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Throug ...
'' (1748). His method was to show that the sine and cosine functions are
alternating series Alternating may refer to: Mathematics * Alternating algebra, an algebra in which odd-grade elements square to zero * Alternating form, a function formula in algebra * Alternating group, the group of even permutations of a finite set * Alternati ...
formed from the even and odd terms respectively of the . He presented "
Euler's formula Euler's formula, named after Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

Euler's formula
", as well as near-modern abbreviations (''sin.'', ''cos.'', ''tang.'', ''cot.'', ''sec.'', and ''cosec.''). A few functions were common historically, but are now seldom used, such as the
chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord (ast ...
, the
versine The versine or versed sine is a trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

versine
(which appeared in the earliest tables), the
coversine The versine or versed sine is a trigonometric function found in some of the earliest (Vedic Aryabhatia I) trigonometric tables. The versine of an angle is 1 minus its cosine. There are several related functions, most notably the coversine and ha ...

coversine
, the
haversine The versine or versed sine is a trigonometric function found in some of the earliest (Vedic Aryabhatia I) trigonometric tables. The versine of an angle is 1 minus its cosine. There are several related functions, most notably the coversine and ...
, the
exsecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant (trigonometry), secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, ci ...
and the
excosecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, str ...
. The
list of trigonometric identities In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
shows more relations between these functions. * * * * * *


Etymology

The word derives from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

Latin
'' sinus'', meaning "bend; bay", and more specifically "the hanging fold of the upper part of a
toga The toga (, ), a distinctive garment of ancient Rome In historiography Historiography is the study of the methods of historian ( 484– 425 BC) was a Greek historian who lived in the 5th century BC and one of the earliest histori ...
", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word ''jaib'', meaning "pocket" or "fold" in the twelfth-century translations of works by
Al-Battani Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī ( ar, محمد بن جابر بن سنان البتاني) ( Latinized as Albategnius, Albategni or Albatenius) (c. 858 – 929) was a Syrian A ...

Al-Battani
and
al-Khwārizmī Muḥammad ibn Mūsā al-Khwārizmī ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), Arabization, Arabized as al-Khwarizmi and formerly Latinisation of names, Latinized as ''Algorithmi'', was a Persians, Persian ...

al-Khwārizmī
into
Medieval Latin Medieval Latin was the form of Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share ...
. The choice was based on a misreading of the Arabic written form ''j-y-b'' (), which itself originated as a
transliteration Transliteration is a type of conversion of a text from one script Script may refer to: Writing systems * Script, a distinctive writing system, based on a repertoire of specific elements or symbols, or that repertoire * Script (styles of h ...

transliteration
from Sanskrit ', which along with its synonym ' (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from
Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...
"string". The word ''tangent'' comes from Latin ''tangens'' meaning "touching", since the line ''touches'' the circle of unit radius, whereas ''secant'' stems from Latin ''secans''—"cutting"—since the line ''cuts'' the circle.Oxford English Dictionary The prefix " co-" (in "cosine", "cotangent", "cosecant") is found in
Edmund Gunter Edmund Gunter (158110 December 1626), was an English clergyman, mathematician, geometer A geometer is a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ...
's ''Canon triangulorum'' (1620), which defines the ''cosinus'' as an abbreviation for the ''sinus complementi'' (sine of the
complementary angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...
) and proceeds to define the ''cotangens'' similarly.


See also

* – a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane * Bhaskara I's sine approximation formula * Differentiation of trigonometric functions * Generalized trigonometry * Generating trigonometric tables * Hyperbolic function * List of integrals of trigonometric functions * List of periodic functions *
List of trigonometric identities In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
* Polar sine – a generalization to vertex angles * Proofs of trigonometric identities * Versine – for several less used trigonometric functions


Notes


References

* * Lars Ahlfors, ''Complex Analysis: an introduction to the theory of analytic functions of one complex variable'', second edition, McGraw-Hill Book Company, New York, 1966. * Carl Benjamin Boyer, Boyer, Carl B., ''A History of Mathematics'', John Wiley & Sons, Inc., 2nd edition. (1991). . * Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (1991). * Joseph, George G., ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd ed. Penguin Books, London. (2000). . * Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," ''IEEE Trans. Computers'' 45 (3), 328–339 (1996). * Maor, Eli,
Trigonometric Delights
', Princeton Univ. Press. (1998). Reprint edition (2002): . * Needham, Tristan
"Preface"
to
Visual Complex Analysis
'. Oxford University Press, (1999). . * * O'Connor, J. J., and E. F. Robertson

''MacTutor History of Mathematics archive''. (1996). * O'Connor, J. J., and E. F. Robertson
"Madhava of Sangamagramma"
''MacTutor History of Mathematics archive''. (2000). * Pearce, Ian G.
"Madhava of Sangamagramma"
''MacTutor History of Mathematics archive''. (2002). * * Weisstein, Eric W.
"Tangent"
from ''MathWorld'', accessed 21 January 2006.


External links

*
Visionlearning Module on Wave Mathematics

GonioLab
Visualization of the unit circle, trigonometric and hyperbolic functions

Article about the q-analog of sin at MathWorld
q-Cosine
Article about the q-analog of cos at MathWorld {{DEFAULTSORT:Trigonometric Functions Trigonometric functions, Angle Trigonometry Elementary special functions Analytic functions Ratios Dimensionless numbers