In

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 sciences that are related to , such as , solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest s, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the , the cosine, and the tangent. Their 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, and an analog among the hyperbolic functions.
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with 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 or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Notation

Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular s or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression $\backslash sin\; x+y$ would typically be interpreted to mean $\backslash sin\; (x)+y,$ so parentheses are required to express $\backslash sin\; (x+y).$ A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example $\backslash sin^2\; x$ and $\backslash sin^2\; (x)$ denote $\backslash sin(x)\; \backslash cdot\; \backslash sin(x),$ not $\backslash sin(\backslash sin\; x).$ This differs from the (historically later) general functional notation in which $f^2(x)\; =\; (f\; \backslash circ\; f)(x)\; =\; f(f(x)).$ However, the exponent $$ is commonly used to denote the inverse function, not the reciprocal. For example $\backslash sin^x$ and $\backslash sin^(x)$ denote the inverse trigonometric function alternatively written $\backslash arcsin\; x\backslash colon$ The equation $\backslash theta\; =\; \backslash sin^x$ implies $\backslash sin\; \backslash theta\; =\; x,$ not $\backslash theta\; \backslash cdot\; \backslash sin\; x\; =\; 1.$ In this case, the superscript ''could'' be considered as denoting a composed or iterated function, but negative superscripts other than $$ are not in common use.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 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 $\backslash sin(\backslash theta)$ and $\backslash cos(90^\backslash circ\; -\; \backslash theta)$ represent the same ratio, and thus are equal. This identity and analogous 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 . For this purpose, any angular unit is convenient, and angles are most commonly measured in conventional units of degrees in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics). However, in andmathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, the trigonometric functions are generally regarded more abstractly as functions of real or s, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the via power series or as solutions to differential equations 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 is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

s and indefinite integrals 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 rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

s (rad) are employed, the angle is given as the length of the arc of the 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 (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 theEuclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometry, geometric setting in which two real number, real quantities are required to determine the position (geometry), position of each point (mathematics), ...

that are related to the unit circle, which is the circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...

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 $\backslash frac$ radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

s the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let $\backslash mathcal\; L$ be the ray 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 s ...

rotation for $\backslash theta\; >\; 0,$ and clockwise rotation for $\backslash theta\; <\; 0$). This ray intersects the unit circle at the point $\backslash mathrm\; =\; (x\_\backslash mathrm,y\_\backslash mathrm).$ The ray $\backslash mathcal\; L,$ extended to a line
Line most often refers to:
* Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclide ...

if necessary, intersects the line of equation $x=1$ at point $\backslash mathrm\; =\; (1,y\_\backslash mathrm),$ and the line of equation $y=1$ at point $\backslash mathrm\; =\; (x\_\backslash mathrm,1).$ The tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...

to the unit circle at the point , is perpendicular
In elementary 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 i ...

to $\backslash mathcal\; L,$ and intersects the - and -axes at points $\backslash mathrm\; =\; (0,y\_\backslash mathrm)$ and $\backslash mathrm\; =\; (x\_\backslash mathrm,0).$ The coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space ...

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,
:$\backslash cos\; \backslash theta\; =\; x\_\backslash mathrm\; \backslash quad$ and $\backslash quad\; \backslash sin\; \backslash theta\; =\; y\_\backslash mathrm.$
In the range $0\; \backslash le\; \backslash theta\; \backslash le\; \backslash pi/2$, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius as . And since the equation $x^2+y^2=1$ holds for all points $\backslash mathrm\; =\; (x,y)$ on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity.
:$\backslash cos^2\backslash theta+\backslash sin^2\backslash theta=1.$
The other trigonometric functions can be found along the unit circle as
:$\backslash tan\; \backslash theta\; =\; y\_\backslash mathrm\; \backslash quad$ and $\backslash quad\backslash cot\; \backslash theta\; =\; x\_\backslash mathrm,$
:$\backslash csc\; \backslash theta\backslash \; =\; y\_\backslash mathrm\; \backslash quad$ and $\backslash quad\backslash sec\; \backslash theta\; =\; x\_\backslash mathrm.$
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
: $\backslash tan\; \backslash theta\; =\backslash frac,\backslash quad\; \backslash cot\backslash theta=\backslash frac,\backslash quad\; \backslash sec\backslash theta=\backslash frac,\backslash quad\; \backslash csc\backslash theta=\backslash frac.$
Since a rotation of an angle of $\backslash pm2\backslash pi$ 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 $2\backslash pi$. Thus trigonometric functions are s with period $2\backslash pi$. That is, the equalities
: $\backslash sin\backslash theta\; =\; \backslash sin\backslash left(\backslash theta\; +\; 2\; k\; \backslash pi\; \backslash right)\backslash quad$ and $\backslash quad\; \backslash cos\backslash theta\; =\; \backslash cos\backslash left(\backslash theta\; +\; 2\; k\; \backslash pi\; \backslash right)$
hold for any angle and any integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

. 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 $2\backslash pi$ is the smallest value for which they are periodic (i.e., $2\backslash pi$ is the fundamental period of these functions). However, after a rotation by an angle $\backslash pi$, the points and already return to their original position, so that the tangent function and the cotangent function have a fundamental period of $\backslash pi$. That is, the equalities
: $\backslash tan\backslash theta\; =\; \backslash tan(\backslash theta\; +\; k\backslash pi)\; \backslash quad$ and $\backslash quad\; \backslash cot\backslash theta\; =\; \backslash cot(\backslash theta\; +\; k\backslash pi)$
hold for any angle and any integer .
Algebraic values

Thealgebraic expression In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

s for the most important angles are as follows:
:$\backslash sin\; 0\; =\; \backslash sin\; 0^\backslash circ\; \backslash quad=\; \backslash frac2\; =\; 0$ ( zero angle)
:$\backslash sin\; \backslash frac\backslash pi6\; =\; \backslash sin\; 30^\backslash circ\; =\; \backslash frac2\; =\; \backslash frac$
:$\backslash sin\; \backslash frac\backslash pi4\; =\; \backslash sin\; 45^\backslash circ\; =\; \backslash frac\; =\; \backslash frac$
:$\backslash sin\; \backslash frac\backslash pi3\; =\; \backslash sin\; 60^\backslash circ\; =\; \backslash frac$
:$\backslash sin\; \backslash frac\backslash pi2\; =\; \backslash sin\; 90^\backslash circ\; =\; \backslash frac2\; =\; 1$ (right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

)
Writing the numerators as square roots 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
In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values f ...

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, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. Fo ...

of a non-real . Galois theory
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

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 a number that is a root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plan ...

s, which may be expressed in terms of th roots. This results from the fact that the Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group (mathematics), group associated with the field extension. The study of field extensions and their rel ...

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.
*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, a transcendental number is a number that is not algebraic number, algebraic—that is, not the Zero of a function, root of a non-zero polynomial of finite degree with rational number, rational coefficients. The best known transcen ...

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, the logarithm is the inverse function to exponentiation. That means th ...

, 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 The modern trend in mathematics is to build from rather than the converse. Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus. Trigonometric functions aredifferentiable
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

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 s, and their primitive period is for the sine and the cosine, and for the tangent, which is increasing in each open interval
In mathematics, a (real) interval is a set (mathematics), set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers ...

. At each end point of these intervals, the tangent function has a vertical asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...

.
In calculus, there are two equivalent definitions of trigonometric functions, either using power series
In mathematics, a power series (in one variable (mathematics), 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 + \dots
where ''an'' represents the coefficient of the ''n''th te ...

or differential equations. 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 theinitial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function (mathematics), function at a given point in the domain of a functio ...

:
:$\backslash frac\backslash sin\; x=\; \backslash cos\; x,\backslash \; \backslash frac\backslash cos\; x=\; -\backslash sin\; x,\backslash \; \backslash sin(0)=0,\backslash \; \backslash cos(0)=1.$
Differentiating again, $\backslash frac\backslash sin\; x\; =\; \backslash frac\backslash cos\; x\; =\; -\backslash sin\; x$ and $\backslash frac\backslash cos\; x\; =\; -\backslash frac\backslash sin\; x\; =\; -\backslash cos\; x$, so both sine and cosine are solutions of the ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable (mathematics), variable and involves the derivatives of those functions. The term ''ordinary ...

:$y\text{'}\text{'}+y=0.$
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 o ...

to the tangent $\backslash tan\; x\; =\; \backslash sin\; x\; /\; \backslash cos\; x$, we derive
:$\backslash frac\backslash tan\; x\; =\; \backslash frac\; =\; 1+\backslash tan^2\; x\; =\; \backslash sec^2\; x.$
Power series expansion

Applying the differential equations topower series
In mathematics, a power series (in one variable (mathematics), 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 + \dots
where ''an'' represents the coefficient of the ''n''th te ...

with indeterminate coefficients, one may deduce recurrence relation
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 for the coefficients of the Taylor series
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions
:$\backslash begin\; \backslash sin\; x\; \&\; =\; x\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\; \backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
The radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest Disk (mathematics), disk at the Power series, center of the series in which the series Convergent series, converges. It is either a non-negative real number o ...

of these series is infinite. Therefore, the sine and the cosine can be extended to entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued Function (mathematics), function that is holomorphic function, holomorphic on the whole complex plane. Typical examples of entire functions are polynomia ...

s (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open set, open subset ''D'' of the complex plane is a function (mathematics), function that is holomorphic function, holomorphic on all of ''D'' ''except'' for a set of is ...

s, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavs, West Slavic nation and ethnic group, who share a common History of Poland, history, Culture of Poland, culture, the Polish language and ...

. Here, the poles are the numbers of the form $(2k+1)\backslash frac\; \backslash pi\; 2$ for the tangent and the secant, or $k\backslash pi$ 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 is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

of the other trigonometric functions. These series have a finite radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest Disk (mathematics), disk at the Power series, center of the series in which the series Convergent series, converges. It is either a non-negative real number o ...

. 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 ar ...

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,
: , the th Bernoulli number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

, and
: , is the th Euler number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

,
one has the following series expansions:
: $\backslash begin\; \backslash tan\; x\; \&\; =\; \backslash sum\_^\backslash infty\; \backslash fracx^\; \backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
: $\backslash begin\; \backslash csc\; x\; \&=\; \backslash sum\_^\backslash infty\; \backslash fracx^\; \backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
: $\backslash begin\; \backslash sec\; x\; \&=\; \backslash sum\_^\backslash infty\; \backslash fracx^\; =\; \backslash sum\_^\backslash infty\; \backslash fracx^\; \backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
: $\backslash begin\; \backslash cot\; x\; \&=\; \backslash sum\_^\backslash infty\; \backslash fracx^\; \backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
Continued fraction expansion

The following expansions are valid in the whole complex plane: :$\backslash sin\; x\; =\; \backslash cfrac$ :$\backslash cos\; x\; =\; \backslash cfrac$ :$\backslash tan\; x\; =\; \backslash cfrac=\backslash cfrac$ The last one was used in the historically first proof that π is irrational.Partial fraction expansion

There is a series representation as partial fraction expansion where just translatedreciprocal function
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...

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:
: $\backslash pi\; \backslash cot\; \backslash pi\; x\; =\; \backslash lim\_\backslash sum\_^N\; \backslash frac.$
This identity can be proved 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:
:$\backslash pi\; \backslash cot\; \backslash pi\; x\; =\; \backslash frac\; +\; 2x\backslash sum\_^\backslash infty\; \backslash frac.$
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
:$\backslash pi\backslash csc\backslash pi\; x\; =\; \backslash sum\_^\backslash infty\; \backslash frac=\backslash frac\; +\; 2x\backslash sum\_^\backslash infty\; \backslash frac,$
:$\backslash pi^2\backslash csc^2\backslash pi\; x=\backslash sum\_^\backslash infty\; \backslash frac,$
:$\backslash pi\backslash sec\backslash pi\; x\; =\; \backslash sum\_^\backslash infty\; (-1)^n\; \backslash frac,$
:$\backslash pi\; \backslash tan\; \backslash pi\; x\; =\; 2x\backslash sum\_^\backslash infty\; \backslash frac.$
Infinite product expansion

The following infinite product for the sine is of great importance in complex analysis: :$\backslash sin\; z\; =\; z\; \backslash prod\_^\backslash infty\; \backslash left(1-\backslash frac\backslash right),\; \backslash quad\; z\backslash in\backslash mathbb\; C.$ For the proof of this expansion, seeSine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

. From this, it can be deduced that
:$\backslash cos\; z\; =\; \backslash prod\_^\backslash infty\; \backslash left(1-\backslash frac\backslash right),\; \backslash quad\; z\backslash in\backslash mathbb\; C.$
Relationship to exponential function (Euler's formula)

Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler ...

relates sine and cosine to the :
:$e^\; =\; \backslash cos\; x\; +\; i\backslash sin\; x.$
This formula is commonly considered for real values of , but it remains true for all complex values.
''Proof'': Let $f\_1(x)=\backslash cos\; x\; +\; i\backslash sin\; x,$ and $f\_2(x)=e^.$ One has $df\_j(x)/dx=\; if\_j(x)$ 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 o ...

implies thus that $d/dx\backslash ,\; (f\_1(x)/f\_2(x))=0$. Therefore, $f\_1(x)/f\_2(x)$ is a constant function, which equals , as $f\_1(0)=f\_2(0)=1.$ This proves the formula.
One has
:$\backslash begin\; e^\; \&=\; \backslash cos\; x\; +\; i\backslash sin\; x\backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$
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:
: $\backslash begin\backslash sin\; x\; \&=\; \backslash frac\backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$
When is real, this may be rewritten as
: $\backslash cos\; x\; =\; \backslash operatorname\backslash left(e^\backslash right),\; \backslash qquad\; \backslash sin\; x\; =\; \backslash operatorname\backslash left(e^\backslash right).$
Most trigonometric identities
In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...

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 variousfunctional equation
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

s.
For example, the sine and the cosine form the unique pair of continuous function
In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...

s that satisfy the difference formula
: $\backslash cos(x-\; y)\; =\; \backslash cos\; x\backslash cos\; y\; +\; \backslash sin\; x\backslash sin\; y\backslash ,$
and the added condition
: $0\; <\; x\backslash cos\; x\; <\; \backslash sin\; x\; <\; x\backslash quad\backslash text\backslash quad\; 0\; <\; x\; <\; 1.$
In the complex plane

The sine and cosine of a $z=x+iy$ can be expressed in terms of real sines, cosines, andhyperbolic function
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s as follows:
: $\backslash begin\backslash sin\; z\; \&=\; \backslash sin\; x\; \backslash cosh\; y\; +\; i\; \backslash cos\; x\; \backslash sinh\; y\backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$
By taking advantage of 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, seeList of trigonometric identities
In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...

. 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
There are several equivalent ways for defining trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-an ...

). For non-geometrical proofs using only tools of , 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 functions; the other trigonometric functions are odd functions. That is: :$\backslash begin\; \backslash sin(-x)\; \&=-\backslash sin\; x\backslash \backslash \; \backslash cos(-x)\; \&=\backslash cos\; x\backslash \backslash \; \backslash tan(-x)\; \&=-\backslash tan\; x\backslash \backslash \; \backslash cot(-x)\; \&=-\backslash cot\; x\backslash \backslash \; \backslash csc(-x)\; \&=-\backslash csc\; x\backslash \backslash \; \backslash sec(-x)\; \&=\backslash sec\; x.\; \backslash end$Periods

All trigonometric functions are 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 :$\backslash begin\; \backslash sin\; (x+2k\backslash pi)\; \&=\backslash sin\; x\backslash \backslash \; \backslash cos\; (x+2k\backslash pi)\; \&=\backslash cos\; x\backslash \backslash \; \backslash tan\; (x+k\backslash pi)\; \&=\backslash tan\; x\backslash \backslash \; \backslash cot\; (x+k\backslash pi)\; \&=\backslash cot\; x\backslash \backslash \; \backslash csc\; (x+2k\backslash pi)\; \&=\backslash csc\; x\backslash \backslash \; \backslash sec\; (x+2k\backslash pi)\; \&=\backslash sec\; x.\; \backslash end$Pythagorean identity

The Pythagorean identity, is the expression of thePythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

in terms of trigonometric functions. It is
:$\backslash sin^2\; x\; +\; \backslash cos^2\; x\; =\; 1$.
Dividing through by either $\backslash cos^2\; x$ or $\backslash sin^2\; x$ gives
:$\backslash tan^2\; x\; +\; 1\; =\; \backslash sec^2\; x$
and
:$1\; +\; \backslash cot^2\; x\; =\; \backslash csc^2\; x$.
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 toPtolemy
Claudius Ptolemy (; grc-gre, wikt:Πτολεμαῖος, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific Treatise, treatis ...

. One can also produce them algebraically using Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler ...

.
; Sum
:$\backslash begin\; \backslash sin\backslash left(x+y\backslash right)\&=\backslash sin\; x\; \backslash cos\; y\; +\; \backslash cos\; x\; \backslash sin\; y,\backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
; Difference
:$\backslash begin\; \backslash sin\backslash left(x-y\backslash right)\&=\backslash sin\; x\; \backslash cos\; y\; -\; \backslash cos\; x\; \backslash sin\; y,\backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
:$\backslash begin\; \backslash sin\; 2x\; \&=\; 2\; \backslash sin\; x\; \backslash cos\; x\; =\; \backslash frac,\; \backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
These identities can be used to derive the product-to-sum identities.
By setting $t=\backslash tan\; \backslash tfrac12\; \backslash theta,$ all trigonometric functions of $\backslash theta$ can be expressed as rational fraction
In algebra, an algebraic fraction is a fraction (mathematics), fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmet ...

s of $t$:
:$\backslash begin\; \backslash sin\; \backslash theta\; \&=\; \backslash frac,\; \backslash \backslash ;\; href="/html/ALL/s/mu.html"\; ;"title="mu">mu$
Together with
:$d\backslash theta\; =\; \backslash frac\; \backslash ,\; dt,$
this is the tangent half-angle substitution, which reduces the computation of integral
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

s and antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This ca ...

s of trigonometric functions to that of rational fractions.
Derivatives and antiderivatives

Thederivative
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

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 o ...

. The values given for the antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This ca ...

s in the following table can be verified by differentiating them. The number is a constant of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the Set (mathematics), set of all antiderivatives of f(x) ...

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

The trigonometric functions are periodic, and hence notinjective
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

, so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic
In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...

, one can define an inverse function, and this defines inverse trigonometric functions as multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function (mathematics), function, but may associate several values to each input. More precisely, a multivalued funct ...

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, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...

from this interval to its image by the function. The common choice for this interval, called the set of principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch (mathematics), branch of that Function (mathematics), function, so that it is Single-valued function, single-val ...

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 rotat ...

".
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
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to be ...

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 : $$\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac,$$ where is the area of the triangle, or, equivalently, $$\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; 2R,$$ where is the triangle'scircumradius
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertex (geometry), vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius ...

.
It can be proved 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 and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle me ...

'', 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 thePythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

:
$$c^2=a^2+b^2-2ab\backslash cos\; C,$$
or equivalently,
$$\backslash cos\; C=\backslash frac.$$
In this formula the angle at is opposite to the side . This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

.
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: :$\backslash frac\; =\; \backslash frac$.Law of cotangents

If ''s'' is the triangle's semiperimeter, (''a'' + ''b'' + ''c'')/2, and ''r'' is the radius of the triangle'sincircle
In 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 o ...

, then ''rs'' is the triangle's area. Therefore Heron's formula
In 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 ...

implies that:
:$r\; =\; \backslash sqrt$.
The law of cotangents says that:
:$\backslash cot\; =\; \backslash frac$
It follows that
:$\backslash frac=\backslash frac=\backslash frac=\backslash frac.$
Periodic functions

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describesimple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of Periodic function, periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the b ...

, 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 is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...

.
Trigonometric functions also prove to be useful in the study of general s. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. Waves can be Periodic function, periodic, in which case those quantities ...

s.
Under rather general conditions, a periodic function can be expressed as a sum of sine waves or cosine waves in a Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

. Denoting the sine or cosine basis functions by , the expansion of the periodic function takes the form:
$$f(t)\; =\; \backslash sum\; \_^\backslash infty\; c\_k\; \backslash varphi\_k(t).$$
For example, the square wave
A square wave is a non-sinusoidal waveform, non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square ...

can be written as the Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

$$f\_\backslash text(t)\; =\; \backslash frac\; \backslash sum\_^\backslash infty\; .$$
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 with a zero Rake (angle), rake angle. A single sawtooth, or an intermittently triggered sawtooth ...

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 function was discovered byHipparchus
Hipparchus (; el, wikt:Ἵππαρχος, Ἵππαρχος, ''Hipparkhos''; BC) was a Ancient astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidenta ...

of Nicaea
Nicaea, also known as Nicea or Nikaia (; ; grc-gre, wikt:Νίκαια, Νίκαια, ) was an ancient Greek city in Bithynia, where located in northwestern Anatolia and is primarily known as the site of the First Council of Nicaea, First and Se ...

(180–125 BCE) and Ptolemy
Claudius Ptolemy (; grc-gre, wikt:Πτολεμαῖος, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific Treatise, treatis ...

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 found in some of the earliest (Sanskrit Āryabhaṭa's sine table , ''Aryabhatia'',

(1 – cosine) can be traced back to the ''jyā'' and ''koti-jyā'' functions used in Gupta period
The Gupta Empire was an Outline of ancient India, ancient Indian 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 Indian subcontinent. This period ...

Indian astronomy
Astronomy has long history in Indian subcontinent stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley civilisation or earlier. Astronomy later developed as a dis ...

(''Aryabhatiya
''Aryabhatiya'' ( IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the '' magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata
Aryabhata (ISO 15919, ISO: ) or Aryabhata I (47 ...

'', ''Surya Siddhanta
The ''Surya Siddhanta'' (; ) is a Sanskrit
Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia ...

''), 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 during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius of Perga, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress wa ...

by the 9th century, as was the law of sines
In trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic period, Hellenistic world during the 3rd century BC from applications ...

, 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 (c. 780–850) produced tables of sines, cosines and tangents. Circa 830, Habash al-Hasib al-Marwazi discovered the cotangent, and produced tables of tangents and cotangents.Jacques Sesiano, "Islamic mathematics", p. 157, in Muhammad ibn Jābir al-Harrānī al-Battānī (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
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath
A polymath ( el, πολυμαθής, , "having learned much"; l ...

, Bhāskara II, 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 West ...

, 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 Timurid sultan
...

(14th century), Regiomontanus
Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrumental ...

(1464), Rheticus
Georg Joachim de Porris, also known as Rheticus ( /ˈrɛtɪkəs/; 16 February 1514 – 5 December 1576), was a mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathemat ...

, and Rheticus' student Valentinus Otho.
Madhava of Sangamagrama (c. 1400) made early strides in the analysis
Analysis (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...

of trigonometric functions in terms of infinite series
In mathematics, a series is, roughly speaking, a description of the operation of addition, adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalizat ...

. (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 an Indian Mathematician and Astronomer Madhava of Sangamagrama (c.&nbs ...

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 ...

.)
The tangent function was brought to Europe by Giovanni Bianchini
Giovanni Bianchini (in Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) aroun ...

in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.
The terms ''tangent'' and ''secant'' were first introduced by the Danish mathematician Thomas Fincke in his book ''Geometria rotundi'' (1583).
The 17th century French mathematician Albert Girard made the first published use of the abbreviations ''sin'', ''cos'', and ''tan'' in his book ''Trigonométrie''.
In a paper published in 1682, Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...

proved that is not an algebraic function In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

of . Though introduced as ratios of sides of a right triangle
A right triangle (American English) or right-angled triangle (British English, British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one an ...

, and thus appearing to be rational function
In mathematics, a rational function is any function (mathematics), function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the poly ...

s, Leibnitz result established that they are actually transcendental function
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

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'' (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series. He presented "Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler ...

", 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, the versine
The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Āryabhaṭa's sine table , ''Aryabhatia'',

(which appeared in the earliest tables), the coversine, the haversine
The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Āryabhaṭa's sine table , ''Aryabhatia'',

, the exsecant
The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in ...

and the excosecant. The list of trigonometric identities
In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...

shows more relations between these functions.
*
*
*
*
*
*
Etymology

The word derives fromLatin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...

'' 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, was a roughly semicircular cloth, between in length, draped over the shoulders and around the body. It was usually woven from white wool, and was worn over a tunic. In Roman historiography, ...

", "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) (Latinization (literature), Latinized as Albategnius, Albategni or Albatenius) (c. ...

and al-Khwārizmī into Medieval Latin
Medieval Latin was the form of Literary Latin used in Roman Catholic Church, Roman Catholic Western Europe during the Middle Ages. In this region it served as the primary written language, though local languages were also written to varying deg ...

.
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 writing system, script to another that involves swapping Letter (alphabet), letters (thus ''wikt:trans-#Prefix, trans-'' + ''wikt:littera#Latin, liter-'') in predictable ways, such as ...

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 used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek Dark ...

"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 and astronomer of Welsh descent. He is best remembered for his mathematical contributions which include the invention of the Gunter's chain, the #Gunter's qu ...

's ''Canon triangulorum'' (1620), which defines the ''cosinus'' as an abbreviation for the ''sinus complementi'' (sine of the complementary angle) and proceeds to define the ''cotangens'' similarly.
See also

* All Students Take Calculus – a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane *Bhaskara I's sine approximation formula
In mathematics, Bhaskara I's sine approximation formula is a rational fraction, rational expression in one Variable (mathematics), variable for the computation of the approximation, approximate values of the sine, trigonometric sines discovered by ...

* Differentiation of trigonometric functions
* Generalized trigonometry
* Generating trigonometric tables
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables wa ...

* Hyperbolic function
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

* List of integrals of trigonometric functions
* List of periodic functions
This is a list of some well-known periodic functions. The constant function , where is independent of , is periodic with any period, but lacks a ''fundamental period''. A definition is given for some of the following functions, though each funct ...

* List of trigonometric identities
In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...

* Polar sine In 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 ge ...

– a generalization to vertex angles
* Proofs of trigonometric identities
There are several equivalent ways for defining trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-an ...

* Versine
The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Āryabhaṭa's sine table , ''Aryabhatia'',

– 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. * 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
Penguin Books is a British publishing, publishing house. It was co-founded in 1935 by Allen Lane with his brothers Richard and John, as a line of the publishers The Bodley Head, only becoming a separate company the following year.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

''

"Madhava of Sangamagramma"

''

"Madhava of Sangamagramma"

, ''

"Tangent"

from ''

Visionlearning Module on Wave Mathematics

GonioLab

Visualization of the unit circle, trigonometric and hyperbolic functions

Article about the

q-Cosine

Article about the

', 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
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biography, biographies on many historical and contemporar ...

''. (1996).
* O'Connor, J. J., and E. F. Robertson"Madhava of Sangamagramma"

''

MacTutor History of Mathematics archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biography, biographies on many historical and contemporar ...

''. (2000).
* Pearce, Ian G."Madhava of Sangamagramma"

, ''

MacTutor History of Mathematics archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biography, biographies on many historical and contemporar ...

''. (2002).
*
* Weisstein, Eric W."Tangent"

from ''

MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...

'', 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
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit (mathematics), limit as . Typically, mathematicians are ...

of sin at MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...

q-Cosine

Article about the

q-analog
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit (mathematics), limit as . Typically, mathematicians are ...

of cos at MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...

{{DEFAULTSORT:Trigonometric Functions
Angle
Trigonometry
Elementary special functions
Analytic functions
Ratios
Dimensionless numbers