Function	Abbreviation	Identities (using radians)
Sine	sin
Cosine	cos
Tangent	tan (or tg)
Cosecant	csc (or cosec)
Secant	sec
Cotangent	cot (or ctg or ctn)

In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. Look up sine in Wiktionary, the free dictionary. ... In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. ... In mathematics and physics, the radian is a unit of angle measure. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... This article is about angles in geometry. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... Illustration of a unit circle. ... In mathematics, a series is a sum of a sequence of terms. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

In modern usage, there are six basic trigonometric functions, which are tabulated here along with equations relating them to one another. Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically or by other means and then derive these relations.

History

Main article: History of trigonometric functions

The notion that there should be some standard correspondence between the length of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is just these ratios that the trigonometric functions express. The history of trigonometric functions may span about 4000 years. ...

Trigonometric functions were studied by Hipparchus of Nicaea (180-125 BC), Ptolemy of Egypt (90–180 AD), Aryabhata (476–550), Varahamihira, Brahmagupta, Muḥammad ibn Mūsā al-Ḵwārizmī, Abū al-Wafā' al-Būzjānī, Omar Khayyam, Bhāskara II, Nasir al-Din al-Tusi, Ghiyath al-Kashi (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentin Otho. For the Athenian tyrant, see Hipparchus (son of Pisistratus). ... Iznik ceramic pitcher with flower decoration from ca. ... This article is about the geographer, mathematician and astronomer Ptolemy. ... For other uses, see Aryabhata (disambiguation). ... Varahamihira (505 – 587) was an Indian astronomer, mathematician, and astrologer born in Ujjain. ... Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) was an Indian mathematician and astronomer. ... A stamp issued September 6, 1983 in the Soviet Union, commemorating al-KhwārizmÄ«s (approximate) 1200th anniversary. ... (940 – 997/8) was a Persian mathematician and astronomer. ... Tomb of Omar Khayam, Neishapur, Iran. ... Bhaskara (1114 – 1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician and astronomer. ... Tusi couple from Vat. ... Kashani, dubbed, the Second Ptolemy, was an outstanding Persian mathematician of the middle ages. ... Ulugh Beg, here depicted on a Soviet stamp, was one of Islams greatest astronomers during the Middle Ages. ... Johannes Müller von Königsberg (June 6, 1436 – July 6, 1476), known by his Latin pseudonym Regiomontanus, was an important German mathematician, astronomer and astrologer. ... Georg Joachim von Lauchen, also known as Rheticus (February 16, 1514 – December 4, 1574), was a mathematician, cartographer, navigational and other instrument maker, medical practitioner, and teacher. ...

Madhava of Sangamagramma (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series. Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec. Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician from Kerala, in South India. ... Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

A few functions were common historically (and appeared in the earliest tables), but are now seldom used, such as the chord (crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin²(θ/2)), the haversine (haversin(θ) = versin(θ) / 2 = sin²(θ/2)), the exsecant (exsec(θ) = sec(θ) − 1) and the excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). Many more relations between these functions are listed in the article about trigonometric identities. A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ... The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the... The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the... The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at O. The exsecant is the portion DE of the secant exterior to (ex) the circle. ... The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at O. The exsecant is the portion DE of the secant exterior to (ex) the circle. ... In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. ...

Right triangle definitions

A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A: For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

We use the following names for the sides of the triangle:

• The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
• The opposite side is the side opposite to the angle we are interested in, in this case a.
• The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

All triangles are taken to exist in the Euclidean plane so that the inside angles of each triangle sum to π radians (or 180°); therefore, for a right triangle the two non-right angles are between zero and π/2 radians (or 90°). The reader should note that the following definitions, strictly speaking, only define the trigonometric functions for angles in this range. We extend them to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. A right triangle and its hypotenuse, h, along with catheti, c1 and c2. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... Some common angles, measured in radians. ... This article describes the unit of angle. ... This article describes the unit of angle. ... Illustration of a unit circle. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...

1) The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

.

Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar. Several equivalence relations in mathematics are called similarity. ...

The set of zeroes of sine (i.e., the values of x for which sinx = 0) is

.

2) The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

.

The set of zeros of cosine is

.

3) The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case

.

The set of zeroes of tangent is

.

The same set of the sine function since

.

The remaining three functions are best defined using the above three functions.

4) The cosecant csc(A) is the multiplicative inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side: The reciprocal function: y = 1/x. ...

.

5) The secant sec(A) is the multiplicative inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side: The reciprocal function: y = 1/x. ...

.

6) The cotangent cot(A) is the multiplicative inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side: The reciprocal function: y = 1/x. ...

.

Slope definitions

Equivalent to the right-triangle definitions, the trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to some horizontal line. The slope is commonly taught as "rise over run" or rise/run. The three main trigonometric functions are commonly taught in the order sine, cosine, tangent. With a unit circle, the following correspondence of definitions exists: This article is about the mathematical term. ... Illustration of a unit circle. ...

1. Sine is first, rise is first. Sine takes an angle and tells the rise.
2. Cosine is second, run is second. Cosine takes an angle and tells the run.
3. Tangent is the slope formula that combines the rise and run. Tangent takes an angle and tells the slope.

This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line, i.e., angles and slopes. (Note that the arctangent or "inverse tangent" is not to be confused with the cotangent, which is cos divided by sin.)

While the radius of the circle makes no difference for the slope (the slope doesn't depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run, just multiply the sine and cosine by the radius. For instance, if the circle has radius 5, the run at an angle of 1° is 5 cos(1°)

Unit-circle definitions

The unit circle

The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trigonometric functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles. From the Pythagorean theorem the equation for the unit circle is: Image File history File links Unit_circle_angles. ... Image File history File links Unit_circle_angles. ... Illustration of a unit circle. ... Illustration of a unit circle. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

In the picture, some common angles, measured in radians, are given. Measurements in the counter clockwise direction are positive angles and measurements in the clockwise direction are negative angles. Let a line through the origin, making an angle of θ with the positive half of the x-axis intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.

The f(x) = sin(x) and f(x) = cos(x) functions graphed on the cartesian plane.

Trigonometric functions: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent

For angles greater than 2π or less than −2π, simply continue to rotate around the circle. In this way, sine and cosine become periodic functions with period 2π: Image File history File links Sine_cosine_plot. ... Image File history File links Sine_cosine_plot. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...

for any angle θ and any integer k. The integers are commonly denoted by the above symbol. ...

The smallest positive period of a periodic function is called the primitive period of the function. The primitive period of the sine, cosine, secant, or cosecant is a full circle, i.e. 2π radians or 360 degrees; the primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees. Above, only sine and cosine were defined directly by the unit circle, but the other four trigonometric functions can be defined by:

The f(x) = tan(x) function graphed on the Cartesian plane.

To the right is an image that displays a noticeably different graph of the trigonometric function f(θ)= tan(θ) graphed on the cartesian plane. Note that its x-intercepts correspond to that of sin(θ) while its undefined values correspond to the x-intercepts of the cos(θ). Observe that the function's results change slowly around angles of kπ, but change rapidly at angles close to (k + 1/2)π. The graph of the tangent function also has a vertical asymptote at θ = (k + 1/2)π. This is the case because the function approaches infinity as θ approaches (k + 1/2)π from the left and minus infinity as it approaches (k + 1/2)π from the right. Image File history File links Tan. ... Image File history File links Tan. ... An asymptote is a straight line or curve which a curve approaches as one moves along the curve. ...

Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (shown at right, near the top of the page), and similar such geometric definitions were used historically. In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India (see above). cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD. tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF. sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively. DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle). From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.) The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the... For other uses, see tangent (disambiguation). ... A secant line of a curve is a line that intersects two or more points on the curve. ... The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at O. The exsecant is the portion DE of the secant exterior to (ex) the circle. ...

Series definitions

The sine function (blue) is closely approximated by its Taylor polynomial of degree 5 (pink) for a full cycle centered on the origin.

Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the negative of sine. (Here, and generally in calculus, all angles are measured in radians; see also the significance of radians below.) One can then use the theory of Taylor series to show that the following identities hold for all real numbers x: Image File history File links Taylorsine. ... Image File history File links Taylorsine. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... For a non-technical overview of the subject, see Calculus. ... For other uses, see Calculus (disambiguation). ... Some common angles, measured in radians. ... As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

These identities are often taken as the definitions of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.g., in Fourier series), since the theory of infinite series can be developed from the foundations of the real number system, independent of any geometric considerations. The differentiability and continuity of these functions are then established from the series definitions alone. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

Other series can be found:^[1]

where

is the nth up/down number,

is the nth Bernoulli number, and

(below) is the nth Euler number.

When this is expressed in a form in which the denominators are the corresponding factorials, and the numerators, called the "tangent numbers", have a combinatorial interpretation: they enumerate alternating permutations of finite sets of odd cardinality. In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. ... In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. ... In mathematics, in the area of number theory, the Euler numbers are a sequence En of integers defined by the following Taylor series expansion: where cosh t is the hyperbolic cosine. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., n} is an arrangement of those numbers into an order c1, ..., cn such that no element ci is between ci âˆ’ 1 and ci + 1 for any value of i. ...

When this is expressed in a form in which the denominators are the corresponding factorials, the numerators, called the "secant numbers", have a combinatorial interpretation: they enumerate alternating permutations of finite sets of even cardinality. Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., n} is an arrangement of those numbers into an order c1, ..., cn such that no element ci is between ci âˆ’ 1 and ci + 1 for any value of i. ...

From a theorem in complex analysis, there is a unique analytic extension of this real function to the complex numbers. They have the same Taylor series, and so the trigonometric functions are defined on the complex numbers using the Taylor series above. Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ...

Relationship to exponential function and complex numbers

complex sine

complex cosine

It can be shown from the series definitions that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary: Image File history File links Metadata Size of this preview: 524 × 599 pixelsFull resolution (944 × 1079 pixel, file size: 288 KB, MIME type: image/jpeg) This is the color function used in the picture above File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev... Image File history File links Metadata Size of this preview: 524 × 599 pixelsFull resolution (944 × 1079 pixel, file size: 288 KB, MIME type: image/jpeg) This is the color function used in the picture above File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev... Image File history File links Metadata Size of this preview: 552 × 599 pixelsFull resolution (945 × 1026 pixel, file size: 284 KB, MIME type: image/jpeg) This is the color function used in the picture above File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev... Image File history File links Metadata Size of this preview: 552 × 599 pixelsFull resolution (945 × 1026 pixel, file size: 284 KB, MIME type: image/jpeg) This is the color function used in the picture above File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... The exponential function is one of the most important functions in mathematics. ...

.

This relationship was first noted by Euler, and the identity is called Euler's formula. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the complex plane, defined by e^ix, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... This article is about the Eulers formula in complex analysis. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:

where i² = −1. Also, for purely real x,

It is also known that exponential processes are intimately linked to periodic behavior.

Definitions via differential equations

Both the sine and cosine functions satisfy the differential equation Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...

y'' = − y.

That is to say, each is the negative of its own second derivative. Within the 2-dimensional function space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions. Further, the observation that sine and cosine satisfies y'' = − y means that they are eigenfunctions of the second-derivative operator. In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. ...

The tangent function is the unique solution of the nonlinear differential equation

y' = 1 + y²

satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation; see Needham's Visual Complex Analysis.^[2]

The significance of radians

Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only those sines and cosines which map radians to ratios satisfy the differential equations which classically describe them. If an argument to sine or cosine in radians is scaled by frequency,

then the derivatives will scale by amplitude.

.

Here, k is a constant that represents a mapping between units. If x is in degrees, then

.

This means that the second derivative of a sine in degrees satisfies not the differential equation

,

but rather

;

cosine's second derivative behaves similarly.

This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.

Identities

Main article: List of trigonometric identities.

Many identities exist which interrelate the trigonometric functions. Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is always 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. In symbolic form, the Pythagorean identity reads, In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

,

which is more commonly written with the exponent "two" next to the sine and cosine symbol:

.

In some cases the inner parentheses may be omitted.

Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments which go back to Ptolemy; one can also produce them algebraically using Euler's formula. This article is about the geographer, mathematician and astronomer Ptolemy. ...

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulas.

These identities can also be used to derive the product-to-sum identities that were used in antiquity to transform the product of two numbers in a sum of numbers and greatly speed operations, much like the logarithm function. In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

For integrals and derivatives of trigonometric functions, see the relevant sections of table of derivatives, table of integrals, and list of integrals of trigonometric functions. This article is about the concept of integrals in calculus. ... For a non-technical overview of the subject, see Calculus. ... The primary operation in differential calculus is finding a derivative. ... It has been suggested that this article or section be merged with List of integrals. ... The following is a list of integrals (antiderivative functions) of trigonometric functions. ...

Definitions using functional equations

In mathematical analysis, one can define the trigonometric functions using functional equations based on properties like the sum and difference formulas. Taking as given these formulas and the Pythagorean identity, for example, one can prove that only two real functions satisfy those conditions. Symbolically, we say that there exists exactly one pair of real functions sin and cos such that for all real numbers x and y, the following equations hold: Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

with the added condition that

.

Other derivations, starting from other functional equations, are also possible, and such derivations can be extended to the complex numbers. As an example, this derivation can be used to define trigonometry in Galois fields. To meet Wikipedias quality standards, this article or section may require cleanup. ...

Computation

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found. (Below, it suffices to consider a small range of angles, say 0 to π/2, since all other angles can be reduced to this range by the periodicity and symmetries of the trigonometric functions.) This article is about the machine. ... A basic arithmetic calculator. ...

Main article: Generating trigonometric tables

Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures. Such tables have been available for as long as trigonometric functions have been described (see History above), and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2)=1). Tables of trigonometric functions are useful in a number of areas. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... Rounding to n significant figures is a form of rounding. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

Modern computers use a variety of techniques.^[3] One common method, especially on higher-end processors with floating point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction.^[4] On simpler devices that lack hardware multipliers, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons. A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterising the errors introduced thereby. ... Padé approximant is the best approximation of a function by a rational function of given order. ... As the degree of the Taylor series rises, it approaches the correct function. ... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ... ALU redirects here. ... CORDIC (digit-by-digit method, Volders algorithm) (for COordinate Rotation DIgital Computer) is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions. ... In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ... Computer hardware is the physical part of a computer, including the digital circuitry, as distinguished from the computer software that executes within the hardware. ...

For very high precision calculations, when series expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral.^[5] In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...

Main article: Exact trigonometric constants

Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in the following examples. In fact, the sine, cosine and tangent of any integer multiple of π / 60 radians (3°) can be found exactly by hand. Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... In mathematics and physics, the radian is a unit of angle measure. ... Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. ...

Consider a right triangle where the two other angles are equal, and therefore are both π / 4 radians (45°). Then the length of side b and the length of side a are equal; we can choose a = b = 1. The values of sine, cosine and tangent of an angle of π / 4 radians (45°) can then be found using the Pythagorean theorem:

.

Therefore:

,

.

To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:

,

,

.

Inverse functions

Main article: inverse trigonometric function

The trigonometric functions are periodic, and hence not injective, so strictly they do not have an inverse function. Therefore to define an inverse function we must restrict their domains so that the trigonometric function is bijective. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as: In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

For inverse trigonometric functions, the notations sin⁻¹ and cos⁻¹ are often used for arcsin and arccos, etc. When this notation is used, the inverse functions could be confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion, though "arcsec" can be confused with "arcsecond". This article does not cite any references or sources. ...

Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. For example,

These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, for example, can be written as the following integral:

Analogous formulas for the other functions can be found at Inverse trigonometric function. Using the complex logarithm, one can generalize all these functions to complex arguments: In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Logarithms to various bases: is to base e, is to base , and is to base . ...

Properties and applications

Main article: Uses of trigonometry

The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results. Trigonometry has an enormous variety of applications. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with...

Law of sines

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C: In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ... A triangle. ...

also known as:

where R is the radius of the triangle's circumcircle. In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ...

A Lissajous curve, a figure formed with a trigonometry-based function.

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. Image File history File links Lissajous_curve_5by4. ... Image File history File links Lissajous_curve_5by4. ... Lissajous figure on an oscilloscope- the shape of the ABC logo Lissajous figure in three dimensions In mathematics, a Lissajous curve (Lissajous figure or Bowditch curve) is the graph of the system of parametric equations which describes complex harmonic motion. ... Triangulation can be used to find the distance from the shore to the ship. ...

Law of cosines

The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem: Fig. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

also known as:

In this formula the angle at C is opposite to the side c. This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem. In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

The law of cosines is mostly used to determine a side of a triangle if two sides and an angle are known, although in some cases there can be two positive solutions as in the SSA ambiguous case. And can also be used to find the cosine of an angle (and consequently the angle itself) if all the sides are known. An example of congruence. ...

Other useful properties

There is also a law of tangents: In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane. ...

Periodic functions

Animation of the additive synthesis of a square wave with an increasing number of harmonics

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe the 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 the uniform circular motion. Image File history File links Synthesis_square. ... Image File history File links Synthesis_square. ... A square wave is a kind of basic waveform. ... Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ... The realm of physics consists of two types of circular motion: uniform circular motion and non-uniform circular motion. ...

Trigonometric functions also prove to be useful in the study of general periodic functions. These functions have characteristic wave patterns as graphs, useful for modeling recurring phenomena such as sound or light waves. Every signal can be written as a (typically infinite) sum of sine and cosine functions of different frequencies; this is the basic idea of Fourier analysis, where trigonometric series are used to solve a variety of boundary-value problems in partial differential equations. For example the square wave, can be written as the Fourier series In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... Surface waves in water This article is about waves in the most general scientific sense. ... Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ... A square wave is a kind of basic waveform. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

.

In the animation on the right it can be seen that just a few terms already produce a fairly good approximation.

Notes

1. ^ Abramowitz; Weisstein.
2. ^ Needham, p. ix.
3. ^ Kantabutra.
4. ^ However, doing that while maintaining precision is nontrivial, and methods like Gal's accurate tables, Cody and Waite reduction, and Payne and Hanek reduction algorithms can be used.
5. ^ R. P. Brent, "Fast Multiple-Precision Evaluation of Elementary Functions", J. ACM 23, 242 (1976).

References

• Abramowitz, Milton and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York. (1964). ISBN 0-486-61272-4.
• Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.
• Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
• 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 (February 25, 2002): ISBN 0-691-09541-8.
• Needham, Tristan, "Preface"" to Visual Complex Analysis. Oxford University Press, (1999). ISBN 0-19-853446-9.
• O'Connor, J.J., and E.F. Robertson, "Trigonometric functions", 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.

Wikibooks has a book on the topic of

Trigonometry

Page 97 showing part of a table of common logarithms. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... It has been suggested that Penguin Modern Poets, Penguin Great Ideas be merged into this article or section. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... is the 21st day of the year in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ...

See also

• Generating trigonometric tables
• Hyperbolic function
• Pythagorean theorem
• Unit vector (explains direction cosines)
• Table of Newtonian series
• Proofs of trigonometric identities
• Euler's formula
• All Students Take Calculus (A mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane)

Tables of trigonometric functions are useful in a number of areas. ... A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) functions). ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... In mathematics, a Newtonian series is a sum over a sequence written in the form where is the binomial coefficient and is the rising factorial. ... It has been suggested that multiple sections of List of trigonometric identities be merged into this article or section. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... All Students Take Calculus (ASTC) is a mnemonic in math that is used to help people memorize the sign values of the trigonometric functions in the 2 dimensional Cartesian coordinate system. ...