In

periodic function
A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...

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
:$\backslash begin\; \backslash sin\; (x+2k\backslash pi)\; \&=\backslash sin\; x\backslash \backslash \; \backslash cos\; (x+2k\backslash pi)\; \&=\backslash cos\; x\backslash \backslash \; \backslash tan\; (x+k\backslash pi)\; \&=\backslash tan\; x\backslash \backslash \; \backslash cot\; (x+k\backslash pi)\; \&=\backslash cot\; x\backslash \backslash \; \backslash csc\; (x+2k\backslash pi)\; \&=\backslash csc\; x\backslash \backslash \; \backslash sec\; (x+2k\backslash pi)\; \&=\backslash sec\; x.\; \backslash end$

periodic function
A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...

s. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.
Under rather general conditions, a periodic function can be expressed as a sum of sine waves or cosine waves in a Fourier series. 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 can be written as the Fourier series
$$f\_\backslash text(t)\; =\; \backslash frac\; \backslash sum\_^\backslash infty\; .$$
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 are shown underneath.

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.

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

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 has no generally ...

, 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; ''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 that are related ...

, 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 under the action of force
In physics
Physics (from g ...

, celestial mechanics, 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 that repeats its values at regular intervals, for example, the trigonometric functions
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...

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, 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 oppos ...

, 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 ...

, 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 ...

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, an inverse function (or anti-function) is a function (mathematics), function that "reverses" another function: if the function applied to an input gives a result of , then applying its inverse function to gives the result , 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). ...

.
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 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). ...

, 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, an ...

(i.e., a circle with radius
In classical geometry, a radius of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin ''radius'', meaning ray but al ...

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 equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates ...

s. This allows extending the domain of sine and cosine functions to the whole complex plane
Image:Complex conjugate picture.svg, Geometric representation of ''z'' and its conjugate ''z̅'' in the complex plane. The distance along the light blue line from the origin to the point ''z'' is the ''modulus'' or ''absolute value'' of ''z''. The ...

, 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, thehypotenuse
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 tha ...

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 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 anangle
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 ''vertex (geometry), vertex'' of the angle.
Angles formed by two rays lie in the plane (ge ...

. 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 infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

and mathematical analysis, 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
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 ...

s, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function
In mathematics, an exponential function is a Function (mathematics), function of the form
where is a positive real number, and the argument occurs as an exponent. For real numbers and , a function of the form f(x)=ab^ is also an exponential ...

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

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 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). ...

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 angles, and is the standard unit of angular measure used in many areas of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

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, an ...

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 Cartesian coordinate system, coordinate values of points on theEuclidean plane
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 ...

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, an ...

, 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 a ...

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 \text, is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

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
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 $\backslash theta\; >\; 0,$ 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, 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 ...

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
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 ...

to the unit circle at the point , is perpendicular
In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two Line (geometry), lines which meet at a right angle (90 Degree (angle), degrees). The property extends to other related Mathematical ob ...

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
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 ...

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 hypotenuse
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 tha ...

. 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 periodic function
A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...

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 (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, and −2048 are integers, while 9 ...

. 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 $\backslash pi$, the points and already return to their original position, so that the tangent function and the cotangent function have a fundamental period of . 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 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:
:$\backslash sin\; 0\; =\; \backslash sin\; 0^\backslash circ\; \backslash quad=\; \backslash frac2\; =\; 0$ (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 ...

)
:$\backslash sin\; \backslash frac\backslash pi6\; =\; \backslash sin\; 30^\backslash circ\; =\; \backslash frac2\; =\; \backslash frac$
:$\backslash sin\; \backslash frac\backslash pi4\; =\; \backslash sin\; 45^\backslash circ\; =\; \backslash frac$
:$\backslash sin\; \backslash frac\backslash pi3\; =\; \backslash sin\; 60^\backslash circ\; =\; \backslash frac$
:$\backslash sin\; \backslash frac\backslash pi2\; =\; \backslash sin\; 90^\backslash circ\; =\; \backslash frac2\; =\; 1$ (right angle
In geometry and trigonometry, a right angle is an angle of exactly 90Degree (angle), ° (degrees), 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 adjacent an ...

)
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 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). ...

of a non-real complex 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). I ...

. Galois theory
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 ...

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, the sine and the cosine are algebraic number
An algebraic number is any complex 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 ...

s, which may be expressed in terms of th roots. This results from the fact that the Galois group
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 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 all ...

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 ...

.
*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

Animation for the approximation of cosine via Taylor polynomials. The modern trend in mathematics is to buildgeometry
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 that are related ...

from 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 zero. They do not ex ...

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 ...

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 that repeats its values at regular intervals, for example, the trigonometric functions
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...

s, and their primitive period 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 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 ...

. At each end point of these intervals, the tangent function has a vertical asymptote
250px, A curve intersecting an asymptote infinitely many times.
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 ...

.
In calculus, there are two equivalent definitions of trigonometric functions, either using power 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). It ...

or differential equation
In mathematics, a differential equation is an equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates ...

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 theinitial 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 ...

:
:$\backslash frac\backslash sin\; x=\; \backslash cos\; x,\backslash \; \backslash frac\backslash cos\; x=\; -\backslash sin\; x,\backslash \; \backslash sin(0)=0,\backslash \; \backslash cos(0)=1.$
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 containing one or more functions of one independent variable and the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial ...

:$y\text{'}\text{'}+y=0.$
Applying the quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let f(x)=g(x)/h(x), where both and are differentiable and h(x)\neq 0. The quotient rule states that the deriv ...

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

Applying the differential equations topower 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). It ...

with indeterminate coefficients, one may deduce recurrence relation
In mathematics, a recurrence relation is an equation that expresses the ''n''th term of a sequence (mathematics), sequence as a function (mathematics), function of the ''k'' preceding terms, for some fixed ''k'' (independent from ''n''), which is ca ...

s for the coefficients of the Taylor 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). ...

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 $& = \sum_^\infty \fracx^ \\\cos x & = 1 - \frac + \frac - \frac + \cdots \\& = \sum_^\infty \fracx^.
\end
The radius of convergence
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 ...

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 that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...

s (also called "sine" and "cosine"), which are (by definition) complex-valued function
of the function
.
Hue represents the argument
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also relat ...

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
Image:Complex conjugate picture.svg, Geometric representation of ''z'' and its conjugate ''z̅'' in the complex plane. The distance along the light blue line from the origin to the point ''z'' is the ''modulus'' or ''absolute value'' of ''z''. The ...

.
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 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 isolated p ...

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 of people formed on the basis of a combination of shared features such as language, history, ethnicity, culture ...

. 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 (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). ...

of the other trigonometric functions. These series have a finite radius of convergence
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 ...

. Their coefficients have a combinatorial
Combinatorics is an area of 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 (mathematic ...

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 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). ...

, and
: , is the th Euler 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). I ...

,
one has the following series expansions:
: $\backslash begin\; \backslash tan\; x\; \&\; =\; \backslash sum\_^\backslash infty\; \backslash fracx^\; \backslash \backslash $& = \sum_^\infty \fracx^ \\& = x + \fracx^3 + \fracx^5 + \fracx^7 + \cdots, \qquad \text , x, < \frac.
\end
: $\backslash begin\; \backslash csc\; x\; \&=\; \backslash sum\_^\backslash infty\; \backslash fracx^\; \backslash \backslash $&= x^ + \fracx + \fracx^3 + \fracx^5 + \cdots, \qquad \text 0 < , x, < \pi.
\end
: $\backslash begin\; \backslash sec\; x\; \&=\; \backslash sum\_^\backslash infty\; \backslash fracx^\; =\; \backslash sum\_^\backslash infty\; \backslash fracx^\; \backslash \backslash $&= 1 + \fracx^2 + \fracx^4 + \fracx^6 + \cdots, \qquad \text , x, < \frac.
\end
: $\backslash begin\; \backslash cot\; x\; \&=\; \backslash sum\_^\backslash infty\; \backslash fracx^\; \backslash \backslash $&= 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 translatedreciprocal 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 ...

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 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:
:$\backslash pi\; \backslash cot\; \backslash pi\; x\; =\; \backslash frac\; +\; 2x\backslash sum\_^\backslash infty\; \backslash frac.$
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
:$\backslash frac\; =\; \backslash frac\; +\; 2x\backslash sum\_^\backslash infty\; \backslash frac,$
:$\backslash frac\; =\; \backslash sum\_^\backslash infty\; (-1)^n\; \backslash frac,$
:$\backslash pi\; \backslash tan\; \backslash pi\; x\; =\; 2x\backslash sum\_^\backslash infty\; \backslash frac.$
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 oppos ...

. 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's ...

relates sine and cosine to the exponential function
In mathematics, an exponential function is a Function (mathematics), function of the form
where is a positive real number, and the argument occurs as an exponent. For real numbers and , a function of the form f(x)=ab^ is also an exponential ...

:
:$e^\; =\; \backslash cos\; x\; +\; i\backslash sin\; x.$
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 $\backslash fracf\_j(x)=\; if\_j(x)$ for . The quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let f(x)=g(x)/h(x), where both and are differentiable and h(x)\neq 0. The quotient rule states that the deriv ...

implies thus that $\backslash frac\backslash left(\backslash frac\backslash right)=0$. Therefore, $\backslash frac$ 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 $e^ &= \cos x - i\sin x.
\end
Solving this linear system
In systems theory
Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent parts that can be natural or man-made, human-made. Every system is bounded by space and time, influenced by its enviro ...

in sine and cosine, one can express them in terms of the exponential function:
: $\backslash begin\backslash sin\; x\; \&=\; \backslash frac\backslash \backslash $\cos x &= \frac.
\end
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 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 variousfunctional equationIn 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 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 o ...

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 acomplex 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). I ...

$z=x+iy$ can be expressed in terms of real sines, cosines, and hyperbolic function
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). ...

s as follows:
: $\backslash begin\backslash sin\; z\; \&=\; \backslash sin\; x\; \backslash cosh\; y\; +\; i\; \backslash cos\; x\; \backslash sinh\; y\backslash \backslash $\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 ...

, 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 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 ...

. 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 infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

, 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 areeven 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; 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:
:$\backslash begin\; \backslash sin(-x)\; \&=-\backslash sin\; x\backslash \backslash \; \backslash cos(-x)\; \&=\backslash cos\; x\backslash \backslash \; \backslash tan(-x)\; \&=-\backslash tan\; x\backslash \backslash \; \backslash cot(-x)\; \&=-\backslash cot\; x\backslash \backslash \; \backslash csc(-x)\; \&=-\backslash csc\; x\backslash \backslash \; \backslash sec(-x)\; \&=\backslash sec\; x.\; \backslash end$
Periods

All trigonometric functions arePythagorean identity

The Pythagorean identity, is the expression of thePythagorean theorem
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) ...

in terms of trigonometric functions. It is
:$\backslash sin^2\; x\; +\; \backslash cos^2\; x\; =\; 1\; .$
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 toPtolemy
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 Ancient Greek, Greek ...

. 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's ...

.
; 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 $\cos\left(x+y\right)&=\cos x \cos y - \sin x \sin y,\\\tan(x + y) &= \frac.
\end
; Difference
:$\backslash begin\; \backslash sin\backslash left(x-y\backslash right)\&=\backslash sin\; x\; \backslash cos\; y\; -\; \backslash cos\; x\; \backslash sin\; y,\backslash \backslash $\cos\left(x-y\right)&=\cos x \cos y + \sin x \sin y,\\\tan(x - y) &= \frac.
\end
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
:$\backslash begin\; \backslash sin\; 2x\; \&=\; 2\; \backslash sin\; x\; \backslash cos\; x\; =\; \backslash frac,\; \backslash \backslash $\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 $t=\backslash tan\; \backslash tfrac12\; \backslash theta,$ all trigonometric functions of $\backslash theta$ 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 $t$:
:$\backslash begin\; \backslash sin\; \backslash theta\; \&=\; \backslash frac,\; \backslash \backslash $\cos \theta &= \frac,\\\tan \theta &= \frac.
\end
Together with
:$d\backslash theta\; =\; \backslash frac\; \backslash ,\; 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, 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 integration. Along with Deriv ...

s and antiderivative
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 ...

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

Thederivative
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). ...

s of trigonometric functions result from those of sine and cosine by applying quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let f(x)=g(x)/h(x), where both and are differentiable and h(x)\neq 0. The quotient rule states that the deriv ...

. The values given for the antiderivative
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 ...

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 infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. Th ...

.
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 (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 ...

, so strictly speaking, they do not have an inverse function
In mathematics, an inverse function (or anti-function) is a function (mathematics), function that "reverses" another function: if the function applied to an input gives a result of , then applying its inverse function to gives the result , i ...

. However, on each interval on which a trigonometric function is monotonic
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 ...

, one can define an inverse function, and this defines inverse trigonometric functions as multivalued function
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). ...

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 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 ...

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 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 ...

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".
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 logarithms.
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's circumscribed circle, circumradius. 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'', 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
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) ...

:
$$c^2=a^2+b^2-2ab\backslash cos\; C,$$
or equivalently,
$$\backslash cos\; C=\backslash frac.$$
In this formula the angle at is opposite to the side . This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem
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 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's incircle, then ''rs'' is the triangle's area. Therefore Heron's formula 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 describe 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. Trigonometric functions also prove to be useful in the study of generalHistory

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 (geometry), chord function was discovered by Hipparchus of İznik, Nicaea (180–125 BCE) andPtolemy
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 Ancient Greek, Greek ...

of Egypt (Roman province), Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) can be traced back to the Jyā, koti-jyā and utkrama-jyā, ''jyā'' and ''koti-jyā'' functions used in Gupta period Indian astronomy (''Aryabhatiya'', ''Surya Siddhanta''), 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 by the 9th century, as was the 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 (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, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Georg Joachim Rheticus, Rheticus, and Rheticus' student Valentinus Otho.
Madhava of Sangamagrama (c. 1400) made early strides in the mathematical analysis, analysis of trigonometric functions in terms of series (mathematics), infinite series. (See Madhava series and Madhava's sine table.)
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, Leibniz proved that is not an algebraic function of . Though introduced as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions 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 function, 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's ...

", 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 (geometry), chord, the versine (which appeared in the earliest tables), the coversine, the haversine, the exsecant and the excosecant. The list of trigonometric identities shows more relations between these functions.
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Etymology

The word derives from Latin ''wikt:sinus, sinus'', meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "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 and Muḥammad ibn Mūsā al-Khwārizmī, al-Khwārizmī into Medieval Latin. The choice was based on a misreading of the Arabic written form ''j-y-b'' (), which itself originated as a 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 language, Ancient Greek "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 (function prefix), co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter'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 * 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 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 ...

* 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