HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
s which relate an angle of a
right-angled triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...
to ratios of two side lengths. They are widely used in all sciences that are related to
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, such as
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navig ...
,
solid mechanics Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature chang ...
,
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
,
geodesy Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...
, and many others. They are among the simplest
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
s, and as such are also widely used for studying periodic phenomena through
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
. 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 opposite th ...
, the
cosine 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 opposite that ...
, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
, and an analog among the
hyperbolic functions In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for
acute angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...
s. To extend the sine and cosine functions to functions whose domain is the whole
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
, geometrical definitions using the standard
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
(i.e., a circle with
radius In classical geometry, a radius (: radii or radiuses) 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 radius of a regular polygon is th ...
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, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
, 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 "" for sine, "" for cosine, "" or "" for tangent, "" for secant, "" or "" for cosecant, and "" or "" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
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 In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called ...
, for example . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression \sin x+y would typically be interpreted to mean (\sin x)+y, so parentheses are required to express \sin (x+y). A
positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
appearing as a superscript after the symbol of the function denotes
exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
, not
function composition In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \c ...
. For example \sin^2 x and \sin^2 (x) denote (\sin x)^2, not \sin(\sin x). This differs from the (historically later) general functional notation in which f^2(x) = (f \circ f)(x) = f(f(x)). In contrast, the superscript -1 is commonly used to denote the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
, not the reciprocal. For example \sin^x and \sin^(x) denote the
inverse trigonometric function In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specific ...
alternatively written \arcsin x\,. The equation \theta = \sin^x implies \sin \theta = x, not \theta \cdot \sin x = 1. In this case, the superscript ''could'' be considered as denoting a composed or
iterated function In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some ...
, 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
hypotenuse In geometry, a hypotenuse is the side of a right triangle opposite to the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called '' catheti'' or ''legs''. Every rectangle can be divided ...
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. Various mnemonics can be used to remember these definitions. In a right-angled triangle, the sum of the two acute angles is a right angle, that is, or . Therefore \sin(\theta) and \cos(90^\circ - \theta) represent the same ratio, and thus are equal. This identity and 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
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
. For this purpose, any
angular unit In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...
is convenient. One common unit is degrees, in which a right angle is 90° and a complete turn is 360° (particularly in
elementary mathematics Elementary mathematics, also known as primary or secondary school mathematics, is the study of mathematics topics that are commonly taught at the primary or secondary school levels around the world. It includes a wide range of mathematical c ...
). However, in
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
and
mathematical 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 ( ...
, the trigonometric functions are generally regarded more abstractly as functions of real or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, rather than angles. In fact, the functions and can be defined for all complex numbers in terms of the exponential function, 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 (, , , ) can be defined as quotients and reciprocals of and , 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 in radians. Moreover, these definitions result in simple expressions for the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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. It is defined such that one radian is the angle subtended at ...
s (rad) are employed, the angle is given as the length of the arc of the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
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 To turn is to rotate, either continuously like a wheel turns on its axle, or in a finite motion changing an object's orientation. Turn may also refer to: Sports and games * Turn (game), a segment of a game * Turn (poker), the fourth of five co ...
(360°) is an angle of 2 (≈ 6.28) rad. For real number ''x'', the notation , , etc. refers 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 (, , etc.). Using this standard notation, the argument ''x'' for the trigonometric functions satisfies the relationship ''x'' = (180''x''/)°, so that, for example, when we take ''x'' = . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = /180 ≈ 0.0175.


Unit-circle definitions

The six trigonometric functions can be defined as coordinate values of points on the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
that are related to the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, 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. The distance between any point of the circle and the centre is cal ...
of radius one centered at the origin of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between and \frac
radian The radian, denoted by the symbol 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. It is defined such that one radian is the angle subtended at ...
s the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers. Let \mathcal L be the ray obtained by rotating by an angle the positive half of the -axis (
counterclockwise Two-dimensional rotation can occur in two possible directions or senses of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to ...
rotation for \theta > 0, and clockwise rotation for \theta < 0). This ray intersects the unit circle at the point \mathrm = (x_\mathrm,y_\mathrm). The ray \mathcal L, extended to a line if necessary, intersects the line of equation x=1 at point \mathrm = (1,y_\mathrm), and the line of equation y=1 at point \mathrm = (x_\mathrm,1). The
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the unit circle at the point , is
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to \mathcal L, and intersects the - and -axes at points \mathrm = (0,y_\mathrm) and \mathrm = (x_\mathrm,0). The
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...
of these points give the values of all trigonometric functions for any arbitrary real value of in the following manner. The trigonometric functions and are defined, respectively, as the ''x''- and ''y''-coordinate values of point . That is, :\cos \theta = x_\mathrm \quad and \quad \sin \theta = y_\mathrm. In the range 0 \le \theta \le \pi/2, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius as
hypotenuse In geometry, a hypotenuse is the side of a right triangle opposite to the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called '' catheti'' or ''legs''. Every rectangle can be divided ...
. And since the equation x^2+y^2=1 holds for all points \mathrm = (x,y) on the unit circle, this definition of cosine and sine also satisfies the
Pythagorean identity The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations ...
. :\cos^2\theta+\sin^2\theta=1. The other trigonometric functions can be found along the unit circle as :\tan \theta = y_\mathrm \quad and \quad\cot \theta = x_\mathrm, :\csc \theta\ = y_\mathrm \quad and \quad\sec \theta = x_\mathrm. By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is : \tan \theta =\frac,\quad \cot\theta=\frac,\quad \sec\theta=\frac,\quad \csc\theta=\frac. Since a rotation of an angle of \pm2\pi does not change the position or size of a shape, the points , , , , and are the same for two angles whose difference is an integer multiple of 2\pi. Thus trigonometric functions are
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
s with period 2\pi. That is, the equalities : \sin\theta = \sin\left(\theta + 2 k \pi \right)\quad and \quad \cos\theta = \cos\left(\theta + 2 k \pi \right) hold for any angle and any
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. 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\pi is the smallest value for which they are periodic (i.e., 2\pi is the fundamental period of these functions). However, after a rotation by an angle \pi, the points and already return to their original position, so that the tangent function and the cotangent function have a fundamental period of \pi. That is, the equalities : \tan\theta = \tan(\theta + k\pi) \quad and \quad \cot\theta = \cot(\theta + k\pi) hold for any angle and any integer .


Algebraic values

The
algebraic expression In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number pow ...
s for the most important angles are as follows: :\sin 0 = \sin 0^\circ \quad= \frac2 = 0 ( zero angle) :\sin \frac\pi6 = \sin 30^\circ = \frac2 = \frac :\sin \frac\pi4 = \sin 45^\circ = \frac = \frac :\sin \frac\pi3 = \sin 60^\circ = \frac :\sin \frac\pi2 = \sin 90^\circ = \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 In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
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 ...
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 that has the given number as its third power; that is y^3=x. The number of cube roots of a number depends on the number system that is considered. Every real number has exactly one real cub ...
of a non-real
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
.
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable. *For an angle which, expressed in degrees, is a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
, the sine and the cosine are
algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
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 associated with the field extension. The study of field extensions and their relationship to the pol ...
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 roots are all ''n''th prim ...
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 real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . ...
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 logarithms of algebraic numbers. Nearly fifteen years earlier, Alexander Gelfond had considered the pr ...
, proved in 1966. *If the sine of an angle is a rational number then the cosine is not necessarily a rational number, and vice-versa. However if the tangent of an angle is rational then both the sine and cosine of the double angle will be rational.


Simple algebraic values

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


Definitions in analysis

Graphs of sine, cosine and tangent
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
noted in his 1908 work ''
A Course of Pure Mathematics ''A Course of Pure Mathematics'' is a classic textbook on introductory mathematical analysis, written by G. H. Hardy. It is recommended for people studying calculus. First published in 1908, it went through ten editions (up to 1952) and several re ...
'' that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry. Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include: * Using the "geometry" of the unit circle, which requires formulating the arc length of a circle (or area of a sector) analytically. * By a power series, which is particularly well-suited to complex variables.Whittaker, E. T., & Watson, G. N. (1920). A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. University press. * By using an infinite product expansion. * By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions. * As solutions of a differential equation.Bartle, R. G., & Sherbert, D. R. (2000). Introduction to real analysis (3rd ed). Wiley.


Definition by differential equations

Sine and cosine can be defined as the unique solution to the
initial 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 at a given point in the domain. Modeling a system in physics or ...
: :\frac\sin x= \cos x,\ \frac\cos x= -\sin x,\ \sin(0)=0,\ \cos(0)=1. Differentiating again, \frac\sin x = \frac\cos x = -\sin x and \frac\cos x = -\frac\sin x = -\cos x, so both sine and cosine are solutions of the same
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
:y''+y=0\,. Sine is the unique solution with and ; cosine is the unique solution with and . One can then prove, as a theorem, that solutions \cos,\sin are periodic, having the same period. Writing this period as 2\pi is then a definition of the real number \pi which is independent of geometry. Applying the
quotient rule In calculus, the quotient rule is a method of finding the derivative of a function (mathematics), function that is the ratio of two differentiable functions. Let h(x)=\frac, where both and are differentiable and g(x)\neq 0. The quotient rule sta ...
to the tangent \tan x = \sin x / \cos x, :\frac\tan x = \frac = 1+\tan^2 x\,, so the tangent function satisfies the ordinary differential equation :y' = 1 + y^2\,. It is the unique solution with .


Power series expansion

The basic trigonometric functions can be defined by the following power series expansions. These series are also known as the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
or
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
of these trigonometric functions: : \begin \sin x & = x - \frac + \frac - \frac + \cdots \\ mu& = \sum_^\infty \fracx^ \\ pt\cos x & = 1 - \frac + \frac - \frac + \cdots \\ mu& = \sum_^\infty \fracx^. \end 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 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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic g ...
s that are defined and
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...
on the whole
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation. 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 that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. ...
s, that is functions that are holomorphic in the whole complex plane, except some isolated points called
poles Pole or poles may refer to: People *Poles (people), another term for Polish people, from the country of Poland * Pole (surname), including a list of people with the name * Pole (musician) (Stefan Betke, born 1967), German electronic music artist ...
. Here, the poles are the numbers of the form (2k+1)\frac \pi 2 for the tangent and the secant, or k\pi for the cotangent and the cosecant, where is an arbitrary integer. Recurrences relations may also be computed for the coefficients of the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
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 as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
interpretation: they enumerate alternating permutations of finite sets. More precisely, defining : , the th up/down number, : , the th
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent function ...
, and : , is the th
Euler number Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, one has the following series expansions: : \begin \tan x & = \sum_^\infty \fracx^ \\ mu& = \sum_^\infty \fracx^ \\ mu& = x + \fracx^3 + \fracx^5 + \fracx^7 + \cdots, \qquad \text , x, < \frac. \end : \begin \csc x &= \sum_^\infty \fracx^ \\ mu&= x^ + \fracx + \fracx^3 + \fracx^5 + \cdots, \qquad \text 0 < , x, < \pi. \end : \begin \sec x &= \sum_^\infty \fracx^ = \sum_^\infty \fracx^ \\ mu&= 1 + \fracx^2 + \fracx^4 + \fracx^6 + \cdots, \qquad \text , x, < \frac. \end : \begin \cot x &= \sum_^\infty \fracx^ \\ mu&= x^ - \fracx - \fracx^3 - \fracx^5 - \cdots, \qquad \text 0 < , x, < \pi. \end


Continued fraction expansion

The following
continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
s are valid in the whole complex plane: : \sin x = \cfrac : \cos x = \cfrac :\tan x = \cfrac=\cfrac The last one was used in the historically first
proof that π is irrational In the 1760s, Johann Heinrich Lambert was the first to prove that the number is irrational, meaning it cannot be expressed as a fraction a/b, where a and b are both integers. In the 19th century, Charles Hermite found a proof that requires no p ...
.


Partial fraction expansion

There is a series representation as partial fraction expansion where just translated
reciprocal function In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
s are summed up, such that the poles of the cotangent function and the reciprocal functions match: : \pi \cot \pi x = \lim_\sum_^N \frac. This identity can be proved with the Herglotz trick. Combining the th with the th term lead to
absolutely convergent In mathematics, an 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 or complex series \textstyle\sum_^\infty a_n is said ...
series: : \pi \cot \pi x = \frac + 2x\sum_^\infty \frac. Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: : \pi\csc\pi x = \sum_^\infty \frac=\frac + 2x\sum_^\infty \frac, :\pi^2\csc^2\pi x=\sum_^\infty \frac, : \pi\sec\pi x = \sum_^\infty (-1)^n \frac, : \pi \tan \pi x = 2x\sum_^\infty \frac.


Infinite product expansion

The following infinite product for the sine is due to
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, and is of great importance in complex analysis: :\sin z = z \prod_^\infty \left(1-\frac\right), \quad z\in\mathbb C. This may be obtained from the partial fraction decomposition of \cot z given above, which is the logarithmic derivative of \sin z. From this, it can be deduced also that :\cos z = \prod_^\infty \left(1-\frac\right), \quad z\in\mathbb C.


Euler's formula and the exponential function

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for ...
relates sine and cosine to the exponential function: : e^ = \cos x + i\sin x. This formula is commonly considered for real values of , but it remains true for all complex values. ''Proof'': Let f_1(x)=\cos x + i\sin x, and f_2(x)=e^. One has df_j(x)/dx= if_j(x) for . The
quotient rule In calculus, the quotient rule is a method of finding the derivative of a function (mathematics), function that is the ratio of two differentiable functions. Let h(x)=\frac, where both and are differentiable and g(x)\neq 0. The quotient rule sta ...
implies thus that d/dx\, (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 :\begin e^ &= \cos x + i\sin x\\ pte^ &= \cos x - i\sin x. \end Solving this
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...
in sine and cosine, one can express them in terms of the exponential function: : \begin\sin x &= \frac\\ pt\cos x &= \frac. \end When is real, this may be rewritten as : \cos x = \operatorname\left(e^\right), \qquad \sin x = \operatorname\left(e^\right). Most
trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
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. Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
s. The set U of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group \mathbb R/\mathbb Z, via an isomorphism e:\mathbb R/\mathbb Z\to U. In pedestrian terms e(t) = \exp(2\pi i t), and this isomorphism is unique up to taking complex conjugates. For a nonzero real number a (the ''base''), the function t\mapsto e(t/a) defines an isomorphism of the group \mathbb R/a\mathbb Z\to U. The real and imaginary parts of e(t/a) are the cosine and sine, where a is used as the base for measuring angles. For example, when a=2\pi, we get the measure in radians, and the usual trigonometric functions. When a=360, we get the sine and cosine of angles measured in degrees. Note that a=2\pi is the unique value at which the derivative \frac e(t/a) becomes a
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
with positive imaginary part at t=0. This fact can, in turn, be used to define the constant 2\pi.


Definition via integration

Another way to define the trigonometric functions in analysis is using integration. For a real number t, put \theta(t) = \int_0^t \frac=\arctan t where this defines this inverse tangent function. Also, \pi is defined by \frac12\pi = \int_0^\infty \frac a definition that goes back to
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
. On the interval -\pi/2<\theta<\pi/2, the trigonometric functions are defined by inverting the relation \theta = \arctan t. Thus we define the trigonometric functions by \tan\theta = t,\quad \cos\theta = (1+t^2)^,\quad \sin\theta = t(1+t^2)^ where the point (t,\theta) is on the graph of \theta=\arctan t and the positive square root is taken. This defines the trigonometric functions on (-\pi/2,\pi/2). The definition can be extended to all real numbers by first observing that, as \theta\to\pi/2, t\to\infty, and so \cos\theta = (1+t^2)^\to 0 and \sin\theta = t(1+t^2)^\to 1. Thus \cos\theta and \sin\theta are extended continuously so that \cos(\pi/2)=0,\sin(\pi/2)=1. Now the conditions \cos(\theta+\pi)=-\cos(\theta) and \sin(\theta+\pi)=-\sin(\theta) define the sine and cosine as periodic functions with period 2\pi, for all real numbers. Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First, \arctan s + \arctan t = \arctan \frac holds, provided \arctan s+\arctan t\in(-\pi/2,\pi/2), since \arctan s + \arctan t= \int_^t\frac=\int_0^\frac after the substitution \tau \to \frac. In particular, the limiting case as s\to\infty gives \arctan t + \frac = \arctan(-1/t),\quad t\in (-\infty,0). Thus we have \sin\left(\theta + \frac\right) = \frac = \frac = -\cos(\theta) and \cos\left(\theta + \frac\right) = \frac = \frac = \sin(\theta). So the sine and cosine functions are related by translation over a quarter period \pi/2.


Definitions using functional equations

One can also define the trigonometric functions using various
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
s. For example, the sine and the cosine form the unique pair of
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s that satisfy the difference formula : \cos(x- y) = \cos x\cos y + \sin x\sin y\, and the added condition : 0 < x\cos x < \sin x < x\quad\text\quad 0 < x < 1.


In the complex plane

The sine and cosine of a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
z=x+iy can be expressed in terms of real sines, cosines, and
hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
s as follows: : \begin\sin z &= \sin x \cosh y + i \cos x \sinh y\\ pt\cos z &= \cos x \cosh y - i \sin x \sinh y\end By taking advantage of
domain coloring In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the complex plane to different colors and brightness, do ...
, 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.


Periodicity and asymptotes

The sine and cosine functions are periodic, with period 2\pi, which is the smallest positive period: \sin(z+2\pi) = \sin(z),\quad \cos(z+2\pi) = \cos(z). Consequently, the cosecant and secant also have 2\pi as their period. The functions sine and cosine also have semiperiods \pi, and \sin(z+\pi)=-\sin(z),\quad \cos(z+\pi)=-\cos(z) and consequently \tan(z+\pi) = \tan(z),\quad \cot(z+\pi) = \cot(z). Also, \sin(x+\pi/2)=\cos(x),\quad \cos(x+\pi/2) = -\sin(x) (see
Complementary angles In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
). The function \sin(z) has a unique zero (at z=0) in the strip -\pi < \real(z) <\pi. The function \cos(z) has the pair of zeros z=\pm\pi/2 in the same strip. Because of the periodicity, the zeros of sine are \pi\mathbb Z = \left\\subset\mathbb C. There zeros of cosine are \frac + \pi\mathbb Z = \left\\subset\mathbb C. All of the zeros are simple zeros, and both functions have derivative \pm 1 at each of the zeros. The tangent function \tan(z)=\sin(z)/\cos(z) has a simple zero at z=0 and vertical asymptotes at z=\pm\pi/2, where it has a simple pole of residue -1. Again, owing to the periodicity, the zeros are all the integer multiples of \pi and the poles are odd multiples of \pi/2, all having the same residue. The poles correspond to vertical asymptotes \lim_\tan(x) = +\infty,\quad \lim_\tan(x) = -\infty. The cotangent function \cot(z)=\cos(z)/\sin(z) has a simple pole of residue 1 at the integer multiples of \pi and simple zeros at odd multiples of \pi/2. The poles correspond to vertical asymptotes \lim_\cot(x) = -\infty,\quad \lim_\cot(x) = +\infty.


Basic identities

Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see
List of trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
. 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 is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.


Parity

The cosine and the secant are
even function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...
s; the other trigonometric functions are
odd function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...
s. That is: :\begin \sin(-x) &=-\sin x\\ \cos(-x) &=\cos x\\ \tan(-x) &=-\tan x\\ \cot(-x) &=-\cot x\\ \csc(-x) &=-\csc x\\ \sec(-x) &=\sec x. \end


Periods

All trigonometric functions are
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
s of period . This is the smallest period, except for the tangent and the cotangent, which have as smallest period. This means that, for every integer , one has :\begin \sin(x+&2k\pi) &=\sin x \\ \cos(x+&2k\pi) &=\cos x \\ \tan(x+&k\pi) &=\tan x \\ \cot(x+&k\pi) &=\cot x \\ \csc(x+&2k\pi) &=\csc x \\ \sec(x+&2k\pi) &=\sec x. \end See Periodicity and asymptotes.


Pythagorean identity

The Pythagorean identity, is the expression of 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 ...
in terms of trigonometric functions. It is :\sin^2 x + \cos^2 x = 1. Dividing through by either \cos^2 x or \sin^2 x gives :\tan^2 x + 1 = \sec^2 x :1 + \cot^2 x = \csc^2 x and :\sec^2 x + \csc^2 x = \sec^2 x \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 to
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
(see Angle sum and difference identities). One can also produce them algebraically using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for ...
. ; Sum :\begin \sin\left(x+y\right)&=\sin x \cos y + \cos x \sin y,\\ mu\cos\left(x+y\right)&=\cos x \cos y - \sin x \sin y,\\ mu\tan(x + y) &= \frac. \end ; Difference :\begin \sin\left(x-y\right)&=\sin x \cos y - \cos x \sin y,\\ mu\cos\left(x-y\right)&=\cos x \cos y + \sin x \sin y,\\ mu\tan(x - y) &= \frac. \end When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae. :\begin \sin 2x &= 2 \sin x \cos x = \frac, \\ mu\cos 2x &= \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x = \frac,\\ mu\tan 2x &= \frac. \end These identities can be used to derive the product-to-sum identities. By setting t=\tan \tfrac12 \theta, all trigonometric functions of \theta can be expressed as
rational fraction In algebra, an algebraic fraction is a 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 arithmetic fractions. A ration ...
s of t: :\begin \sin \theta &= \frac, \\ mu\cos \theta &= \frac,\\ mu\tan \theta &= \frac. \end Together with :d\theta = \frac \, dt, this is the
tangent half-angle substitution In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of x into an ordinary rational function of t by setting t = \tan \tfr ...
, which reduces the computation of
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s and
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...
s of trigonometric functions to that of rational fractions.


Derivatives and antiderivatives

The
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s of trigonometric functions result from those of sine and cosine by applying the
quotient rule In calculus, the quotient rule is a method of finding the derivative of a function (mathematics), function that is the ratio of two differentiable functions. Let h(x)=\frac, where both and are differentiable and g(x)\neq 0. The quotient rule sta ...
. The values given for the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...
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 of all antiderivatives of f(x)), on a connecte ...
. Note: For 0 the integral of \csc x can also be written as -\operatorname(\cot x), and for the integral of \sec x for -\pi/2 as \operatorname(\tan x), where \operatorname is the inverse hyperbolic sine. Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule: : \begin \frac &= \frac\sin(\pi/2-x)=-\cos(\pi/2-x)=-\sin x \, , \\ \frac &= \frac\sec(\pi/2 - x) = -\sec(\pi/2 - x)\tan(\pi/2 - x) = -\csc x \cot x \, , \\ \frac &= \frac\tan(\pi/2 - x) = -\sec^2(\pi/2 - x) = -\csc^2 x \, . \end


Inverse functions

The trigonometric functions are periodic, and hence not
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
, so strictly speaking, they do not have an
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
. However, on each interval on which a trigonometric function is
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
, one can define an inverse function, and this defines inverse trigonometric functions as
multivalued function In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional p ...
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, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
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 (mathematical analysis), branch of that Function (mathematics), function, so that it is Single-valued function, ...
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 (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...
". 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 : \frac = \frac = \frac = \frac, where is the area of the triangle, or, equivalently, \frac = \frac = \frac = 2R, where is the triangle's 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 m ...
'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.


Law of cosines

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the
Pythagorean theorem In mathematics, 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\cos C, or equivalently, \cos C=\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: :\frac = \frac.


Law of cotangents

If ''s'' is the triangle's semiperimeter, (''a'' + ''b'' + ''c'')/2, and ''r'' is the radius of the triangle's
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter ...
, then ''rs'' is the triangle's area. Therefore
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century ...
implies that: : r = \sqrt. The law of cotangents says that: :\cot = \frac It follows that :\frac=\frac=\frac=\frac.


Periodic functions

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe
simple harmonic motion In mechanics and physics, simple harmonic motion (sometimes abbreviated as ) is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from ...
, 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, circular motion is movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate o ...
. Trigonometric functions also prove to be useful in the study of general
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
s. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light
wave In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
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 an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
. Denoting the sine or cosine basis functions by , the expansion of the periodic function takes the form: f(t) = \sum _^\infty c_k \varphi_k(t). For example, the
square wave Square wave may refer to: *Square wave (waveform) 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 ...
can be written as the
Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
f_\text(t) = \frac \sum_^\infty . In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a
sawtooth wave The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ...
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 defined by
Hipparchus Hipparchus (; , ;  BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hippar ...
of
Nicaea Nicaea (also spelled Nicæa or Nicea, ; ), also known as Nikaia (, Attic: , Koine: ), was an ancient Greek city in the north-western Anatolian region of Bithynia. It was the site of the First and Second Councils of Nicaea (the first and seve ...
(180–125 BCE) and
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
of
Roman Egypt Roman Egypt was an imperial province of the Roman Empire from 30 BC to AD 642. The province encompassed most of modern-day Egypt except for the Sinai. It was bordered by the provinces of Crete and Cyrenaica to the west and Judaea, ...
(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 ''Aryabhatia'',''jyā'' and ''koti-jyā'' functions used in
Gupta period The Gupta Empire was an Indian empire during the classical period of the Indian subcontinent which existed from the mid 3rd century to mid 6th century CE. At its zenith, the dynasty ruled over an empire that spanned much of the northern Indian ...
Indian astronomy Astronomy has a long history in the Indian subcontinent, stretching from History of India, pre-historic to History of India (1947–present), modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valle ...
(''
Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Indian astronomy, Sanskrit astronomical treatise, is the ''Masterpiece, magnum opus'' and only known surviving work of the 5th century Indian mathematics, Indian mathematician Aryabhata. Philos ...
'', ''
Surya Siddhanta The ''Surya Siddhanta'' (; ) is a Sanskrit treatise in Indian astronomy dated to 4th to 5th century,Menso Folkerts, Craig G. Fraser, Jeremy John Gray, John L. Berggren, Wilbur R. Knorr (2017)Mathematics Encyclopaedia Britannica, Quote: "(...) i ...
''), 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 In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\ ...
, used in
solving triangles Solution of triangles () is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring tr ...
.
Al-Khwārizmī Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
(c. 780–850) produced tables of sines and cosines. Circa 860, Habash al-Hasib al-Marwazi defined the tangent and the cotangent, and produced their tables.Jacques Sesiano, "Islamic mathematics", p. 157, in Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) defined 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 Nīshābūrī (18 May 1048 – 4 December 1131) ( Persian: غیاث الدین ابوالفتح عمر بن ابراهیم خیام نیشابورﻯ), commonly known as Omar Khayyam (), was ...
,
Bhāskara II Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian people, Indian polymath, Indian mathematicians, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferre ...
,
Nasir al-Din al-Tusi Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī (1201 – 1274), also known as Naṣīr al-Dīn al-Ṭūsī (; ) or simply as (al-)Tusi, was a Persians, Persian polymath, architect, Early Islamic philosophy, philosopher, Islamic medicine, phy ...
,
Jamshīd al-Kāshī Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxiana) was a Persian astronomer and mathematician during the reign of Tamerlane. ...
(14th century),
Ulugh Beg Mīrzā Muhammad Tarāghāy bin Shāhrukh (; ), better known as Ulugh Beg (; 22 March 1394 – 27 October 1449), was a Timurid sultan, as well as an astronomer and mathematician. Ulugh Beg was notable for his work in astronomy-related ma ...
(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 instrument ...
(1464), Rheticus, and Rheticus' student Valentinus Otho.
Madhava of Sangamagrama Mādhava of Sangamagrāma (Mādhavan) Availabl/ref> () was an Indian mathematician and astronomer who is considered to be the founder of the Kerala school of astronomy and mathematics in the Late Middle Ages. Madhava made pioneering contributio ...
(c. 1400) made early strides in the
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
of trigonometric functions in terms of
infinite series In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
. (See
Madhava series In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent function (mathematics), functions discovered in 14th or 15th century in Kerala, India by the mathematician and astronomer Madhava o ...
and Madhava's sine table.) The tangent function was brought to Europe by Giovanni Bianchini 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 Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...
proved that is not an
algebraic function In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operati ...
of . Though defined as ratios of sides of a
right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...
, and thus appearing to be
rational function In mathematics, a rational function is any 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 polynomials need not be ...
s, Leibnitz result established that they are actually
transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction ...
s of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his ''
Introduction to the Analysis of the Infinite ''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...
'' (1748). His method was to show that the sine and cosine functions are
alternating series In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . Like an ...
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 mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for ...
", 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,
versine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',),
haversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',coversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',exsecant.
List of trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
shows more relations between these functions. : \begin \operatorname\theta &= 2 \sin\tfrac12\theta, \\ mu\operatorname\theta&=1-\cos \theta = 2\sin^2\tfrac12\theta, \\ mu\operatorname\theta&=\tfrac\operatorname\theta = \sin^2\tfrac12\theta, \\ mu\operatorname\theta&=1-\sin\theta = \operatorname\bigl(\tfrac12\pi - \theta\bigr), \\ mu\operatorname\theta &= \sec\theta - 1. \end Historically, trigonometric functions were often combined with
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
s in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.


Etymology

The word derives from
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
'' 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 historical tra ...
", "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 Al-Battani (before 858929), archaically Latinized as Albategnius, was a Muslim astronomer, astrologer, geographer and mathematician, who lived and worked for most of his life at Raqqa, now in Syria. He is considered to be the greatest and mos ...
and
al-Khwārizmī Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
into
Medieval Latin Medieval Latin was the form of Literary Latin used in Roman Catholic Church, Roman Catholic Western Europe during the Middle Ages. It was also the administrative language in the former Western Roman Empire, Roman Provinces of Mauretania, Numidi ...
. The choice was based on a misreading of the Arabic written form ''j-y-b'' (), which itself originated as a
transliteration Transliteration is a type of conversion of a text from one script to another that involves swapping letters (thus '' trans-'' + '' liter-'') in predictable ways, such as Greek → and → the digraph , Cyrillic → , Armenian → or L ...
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 ...
"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 q ...
's ''Canon triangulorum'' (1620), which defines the ''cosinus'' as an abbreviation of the ''sinus complementi'' (sine of the complementary angle) and proceeds to define the ''cotangens'' similarly.


See also

*
Bhāskara I's sine approximation formula In mathematics, Bhāskara 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 ...
*
Small-angle approximation For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations: : \begin \sin \theta &\approx \tan \theta \approx \theta, \\ mu\cos \theta &\approx 1 - \t ...
* 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 w ...
* 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 ...
*
Polar sine In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin. Definition ''n'' vectors in ''n''-dimensional space Let v1, ..., v''n'' (''n'' ≥ 1) be non-zero ...
– a generalization to vertex angles *
Sinc function In mathematics, physics and engineering, the sinc function ( ), denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatorname(x) = \frac. Alternatively, ...


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 Limited is a Germany, German-owned English 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 ...
, 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 ''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 ...
'', 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 as . Typically, mathematicians are interested in ''q'' ...
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 ...

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 as . Typically, mathematicians are interested in ''q'' ...
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 ...
{{DEFAULTSORT:Trigonometric Functions Analytic functions Angle Ratios