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Trigonometry () is a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

that studies relationships between side lengths and
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

s of
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

s. The field emerged in the
Hellenistic world In Classical antiquity, the Hellenistic period covers the time in History of the Mediterranean region, Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as sig ...
during the 3rd century BC from applications of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called
trigonometric functions In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

) such as . Throughout history, trigonometry has been applied in areas such as
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure ( geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equiv ...
,
surveying Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Two-dimensional space#In geometry, two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of ...

, celestial mechanics, and
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 navi ...

. Trigonometry is known for its many identities. These
trigonometric identities In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...
are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.

# History

Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and Early Bronze Age, early Bronze Ages between the sixth and fifth millennium BC. It is one of ...

ian astronomers studied angle measure, using a division of circles into 360 degrees. They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The used a similar method. In the 3rd century BC, Hellenistic mathematicians such as
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

and
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, ...

studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC,
Hipparchus Hipparchus (; el, wikt:Ἵππαρχος, Ἵππαρχος, ''Hipparkhos'';  BC) was a Ancient astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidenta ...
(from
Nicaea Nicaea, also known as Nicea or Nikaia (; ; grc-gre, wikt:Νίκαια, Νίκαια, ) was an ancient Greek city in Bithynia, where located in northwestern Anatolia and is primarily known as the site of the First Council of Nicaea, First and Se ...
, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and
spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
. In the 2nd century AD, the Greco-Egyptian astronomer
Ptolemy Claudius Ptolemy (; grc-gre, wikt:Πτολεμαῖος, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific Treatise, treatis ...
(from Alexandria, Egypt) constructed detailed trigonometric tables (
Ptolemy's table of chords The table of chords, created by the Greece, Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's ''Almagest'', a treatise on mathematical astron ...
) in Book 1, chapter 11 of his ''
Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it ca ...
''. Ptolemy used chord length to define his trigonometric functions, a minor difference from the convention we use today. (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval
Byzantine The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinople. It survi ...
,
Islamic Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic religions, Abrahamic Monotheism#Islam, monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God in Islam, God (or ...
, and, later, Western European worlds. The modern sine convention is first attested in the ''
Surya Siddhanta The ''Surya Siddhanta'' (; ) is a Sanskrit Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia ...
'', and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer
Aryabhata Aryabhata (ISO 15919, ISO: ) or Aryabhata I (476–550 Common Era, CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta era, Gupta Era and produced works su ...
. These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in
spherical geometry file:Spherical_triangle_3d.png, 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other ...
. The
polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...

Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right. He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his ''On the Sector Figure'', he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws. Knowledge of trigonometric functions and methods reached
Western Europe Western Europe is the western region of Europe. The region's countries and territories vary depending on context. The concept of "the West" appeared in Europe in juxtaposition to "the East" and originally applied to the ancient Mediterranean ...

via Latin translations of Ptolemy's Greek ''Almagest'' as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi. One of the earliest works on trigonometry by a northern European mathematician is ''De Triangulis'' by the 15th century German mathematician
Regiomontanus Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrumental ...

, who was encouraged to write, and provided with a copy of the ''Almagest'', by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years. At the same time, another translation of the ''Almagest'' from Greek into Latin was completed by the Cretan . Trigonometry was still so little known in 16th-century northern Europe that
Nicolaus Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic Church, Catholic cano ...

devoted two chapters of '''' to explain its basic concepts. Driven by the demands of
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 navi ...

and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his ''Trigonometria'' in 1595.
Gemma Frisius Gemma Frisius (; born Jemme Reinerszoon; December 9, 1508 – May 25, 1555) was a Frisian physician, mathematician, Cartography, cartographer, philosopher, and instrument maker. He created important globes, improved the mathematical instrume ...

described for the first time the method of
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...

still used today in surveying. It was
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

who fully incorporated
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 form a ...

s into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general
Taylor series In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
.

# Trigonometric ratios

Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given by the following
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s of the known angle ''A'', where ''a'', '' b'' and ''h'' refer to the lengths of the sides in the accompanying figure: * Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse. :: $\sin A=\frac=\frac.$ * Cosine function (cos), defined as the ratio of the adjacent side (right triangle), adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse. :: $\cos A=\frac=\frac.$ * Tangent (trigonometric function), Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg. ::$\tan A=\frac=\frac=\frac=\frac.$ The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle ''A''. The adjacent leg is the other side that is adjacent to angle ''A''. The opposite side is the side that is opposite to angle ''A''. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. See below under #Mnemonics, Mnemonics. Since any two right triangles with the same acute angle ''A'' are similar, the value of a trigonometric ratio depends only on the angle ''A''. The Multiplicative inverse, reciprocals of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively: :$\csc A=\frac=\frac=\frac ,$ :$\sec A=\frac=\frac=\frac ,$ :$\cot A=\frac=\frac=\frac=\frac .$ The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.

## Mnemonics

A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the ''sine'', ''cosine'', and ''tangent'' ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA: :Sine = Opposite ÷ Hypotenuse :Cosine = Adjacent ÷ Hypotenuse :Tangent = Opposite ÷ Adjacent One way to remember the letters is to sound them out phonetically (i.e. , similar to Krakatoa). Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".

## The unit circle and common trigonometric values

Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane. In this setting, the Angle#Positive and negative angles, terminal side of an angle ''A'' placed in Angle#Positive and negative angles, standard position will intersect the unit circle in a point (x,y), where $x = \cos A$ and $y = \sin A$. This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:

# Trigonometric functions of real or complex variables

Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments (see
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
).

## Graphs of trigonometric functions

The following table summarizes the properties of the graphs of the six main trigonometric functions:

## Inverse trigonometric functions

Because the six main trigonometric functions are periodic, they are not injective function, injective (or, 1 to 1), and thus are not invertible. By Restriction (mathematics), restricting the domain of a trigonometric function, however, they can be made invertible. The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:

## Power series representations

When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series. For instance, sine and cosine have the following representations: :$\begin \sin x & = x - \frac + \frac - \frac + \cdots \\ & = \sum_^\infty \frac \\ \end$ :$\begin \cos x & = 1 - \frac + \frac - \frac + \cdots \\ & = \sum_^\infty \frac. \end$ With these definitions the trigonometric functions can be defined for
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 form a ...

s. When extended as functions of real or complex variables, the following Euler's formula, formula holds for the complex exponential: : $e^ = e^x\left(\cos y + i \sin y\right).$ This complex exponential function, written in terms of trigonometric functions, is particularly useful.

## Calculating trigonometric functions

Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions. Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes Euler's formula, cis and their inverses). Most allow a choice of angle measurement methods: degree (angle), degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.

## Other trigonometric functions

In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord (), the versine () (which appeared in the earliest tables), the coversine (), the haversine (), the exsecant (), and the excosecant (). See List of trigonometric identities for more relations between these functions.

# Applications

## Astronomy

For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, predicting eclipses, and describing the orbits of the planets. In modern times, the technique of
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...

is used in astronomy to measure the distance to nearby stars, as well as in satellite navigation systems.

Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation. Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.

## Surveying

In land
surveying Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Two-dimensional space#In geometry, two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of ...

, trigonometry is used in the calculation of lengths, areas, and relative angles between objects. On a larger scale, trigonometry is used in geography to measure distances between landmarks.

## Periodic functions

The sine and cosine functions are fundamental to the theory of periodic functions, such as those that describe sound and light waves. Jean-Baptiste Joseph Fourier, Fourier discovered that every continuous function, continuous, periodic function could be described as an infinite series, infinite sum of trigonometric functions. Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics and telecommunication, communications, among other fields.

## Optics and acoustics

Trigonometry is useful in many physical sciences, including acoustics, and optics. In these areas, they are used to describe sound waves, sound and light waves, and to solve boundary- and transmission-related problems.

## Other applications

Other fields that use trigonometry or trigonometric functions include music theory,
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure ( geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equiv ...
, audio synthesis, architecture, electronics, biology, medical imaging (CT scans and ultrasound), chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, image compression, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

# Identities

Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. Identities involving only angles are known as ''trigonometric identities''. Other equations, known as ''triangle identities'', relate both the sides and angles of a given triangle.

## Triangle identities

In the following identities, ''A'', ''B'' and ''C'' are the angles of a triangle and ''a'', ''b'' and ''c'' are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).

### Law of sines

The law of sines (also known as the "sine rule") for an arbitrary triangle states: :$\frac = \frac = \frac = 2R = \frac,$ where $\Delta$ is the area of the triangle and ''R'' is the radius of the circumscribed circle of the triangle: :$R = \frac.$

### Law of cosines

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: :$c^2=a^2+b^2-2ab\cos C ,$ or equivalently: :$\cos C=\frac.$

### Law of tangents

The law of tangents, developed by François Viète, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables. It is given by: :$\frac=\frac$

### Area

Given two sides ''a'' and ''b'' and the angle between the sides ''C'', the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides: :$\mbox = \Delta = \fraca b\sin C$ Heron's formula is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths ''a'', ''b'', and ''c'', and if the semiperimeter is :$s=\frac\left(a+b+c\right),$ then the area of the triangle is: :$\mbox = \Delta = \sqrt = \frac$, where R is the radius of the circumcircle of the triangle.

## Trigonometric identities

### Pythagorean identities

The following trigonometric Identity (mathematics), identities are related to the Pythagorean theorem and hold for any value:Extract of page 856
/ref> :$\sin^2 A + \cos^2 A = 1 \$ :$\tan^2 A + 1 = \sec^2 A \$ :$\cot^2 A + 1 = \csc^2 A \$ The second and third equations are derived from dividing the first equation by $\cos^2$ and $\sin^2$, respectively.

### Euler's formula

Euler's formula, which states that $e^ = \cos x + i \sin x$, produces the following mathematical analysis, analytical identities for sine, cosine, and tangent in terms of ''e (mathematics), e'' and the imaginary unit ''i'': :$\sin x = \frac, \qquad \cos x = \frac, \qquad \tan x = \frac.$

### Other trigonometric identities

Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.

* Aryabhata's sine table * Generalized trigonometry * Lénárt sphere * List of triangle topics * List of trigonometric identities * Rational trigonometry * Skinny triangle * Small-angle approximation * Trigonometric functions * Unit circle * Uses of trigonometry

# Bibliography

* * *

* * Linton, Christopher M. (2004). ''From Eudoxus to Einstein: A History of Mathematical Astronomy''. Cambridge University Press. *

Khan Academy: Trigonometry, free online micro lectures

by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
Benjamin Banneker's Trigonometry Puzzle
a
Convergence

Dave's Short Course in Trigonometry
by David Joyce of Clark University