Trigonometry
Overview
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
that studies triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
s and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
used extensively for astronomical studies. It is also the foundation of the practical art of surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or threedimensional position of points and the distances and angles between them...
.
Trigonometry basics are often taught in middle school
Middle school
Middle School and Junior High School are levels of schooling between elementary and high schools. Most school systems use one term or the other, not both. The terms are not interchangeable...
or junior high school as part of the basic math curriculum.
Unanswered Questions
Discussions
Encyclopedia
Trigonometry is a branch of mathematics
that studies triangle
s and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry
used extensively for astronomical studies. It is also the foundation of the practical art of surveying
.
Trigonometry basics are often taught in middle school
or junior high school as part of the basic math curriculum. More in depth trigonometry is sometimes taught in high school
either as a separate course or as part of a precalculus
course. The trigonometric functions are pervasive in parts of pure mathematics
and applied mathematics
such as Fourier analysis and the wave equation
, which are in turn essential to many branches of science and technology. Spherical trigonometry
studies triangles on sphere
s, surfaces of constant positive curvature
, in elliptic geometry
. It is fundamental to astronomy
and navigation
. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry
.
ian astronomers introduced angle measure, using a division of circles into 360 degrees. They and their successors 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 ancient Nubians
used a similar methodology. The ancient Greeks transformed trigonometry into an ordered science.
Classical Greek mathematicians
(such as Euclid
and Archimedes
) studied the properties of chords
and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemy
expanded upon Hipparchus
' Chords in a Circle in his Almagest
. The modern sine function was first defined in the Surya Siddhanta
, and its properties were further documented by the 5th century Indian mathematician
and astronomer Aryabhata
. 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
. At about the same time, Chinese
mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir alDin alTusi
. One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German
mathematician Regiomontanus
. Trigonometry was still so little known in 16th century Europe that Nicolaus Copernicus
devoted two chapters of De revolutionibus orbium coelestium
to explaining its basic concepts.
Driven by the demands of navigation
and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus
was the first to use the word, publishing his Trigonometria in 1595. Gemma Frisius
described for the first time the method of triangulation
still used today in surveying. It was Leonhard Euler
who fully incorporated complex numbers into trigonometry. The works of 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
.
of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles
. The shape
of a triangle is completely determined, except for similarity
, by the angles. Once the angles are known, the ratio
s of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric function
s of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
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. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOHCAHTOA (see below under Mnemonics).
The reciprocals
of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively. The inverse functions
are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. 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. These laws are useful in all branches of geometry, since every polygon
may be described as a finite combination of triangles.
s) only. Using the unit circle
, one can extend them to all positive and negative arguments (see trigonometric function
). The trigonometric functions are periodic
, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus
and infinite series. With these definitions the trigonometric functions can be defined for complex number
s. The complex exponential function is particularly useful.
See Euler's
and De Moivre's
formulas.
s 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 as strings of letters. For instance, a mnemonic for English speakers is SOHCAHTOA:
One way to remember the letters is to sound them out phonetically (i.e. "SOHCAHTOA", which is pronounced 'sokətow'uh'). Another method is to expand the letters into a sentence, such as "Some Old Hippy Caught Another Hippy Trippin' On Acid".
s. 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 rule
s had special scales for trigonometric functions.
Today scientific calculator
s have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad
. Most computer programming language
s provide function libraries that include the trigonometric functions. The floating point unit
hardware incorporated into the microprocessor chips used in most personal computers have builtin instructions for calculating trigonometric functions.
There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation
is used in astronomy
to measure the distance to nearby stars, in geography
to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic function
s such as those that describe sound and light
waves.
Fields that use trigonometry or trigonometric functions include astronomy
(especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation
(on the oceans, in aircraft, and in space), music theory
, acoustics
, optics
, analysis of financial markets, electronics
, probability theory
, statistics
, biology
, medical imaging
(CAT scans and ultrasound
), pharmacy
, chemistry
, number theory
(and hence cryptology), seismology
, meteorology
, oceanography
, many physical science
s, land surveying
and geodesy
, architecture
, phonetics
, economics
, electrical engineering
, mechanical engineering
, civil engineering
, computer graphics
, cartography
, crystallography
and game development
.
are those equations that hold true for any value.
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.
(also known as the "sine rule") for an arbitrary triangle states:
where R is the radius of the circumscribed circle
of the triangle:
Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:
(known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem
to arbitrary triangles:
or equivalently:
:
, which states that , produces the following analytical
identities for sine, cosine, and tangent in terms of e and the imaginary unit
i:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
that studies triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
s and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
used extensively for astronomical studies. It is also the foundation of the practical art of surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or threedimensional position of points and the distances and angles between them...
.
Trigonometry basics are often taught in middle school
Middle school
Middle School and Junior High School are levels of schooling between elementary and high schools. Most school systems use one term or the other, not both. The terms are not interchangeable...
or junior high school as part of the basic math curriculum. More in depth trigonometry is sometimes taught in high school
High school
High school is a term used in parts of the English speaking world to describe institutions which provide all or part of secondary education. The term is often incorporated into the name of such institutions....
either as a separate course or as part of a precalculus
Precalculus
In American mathematics education, precalculus , an advanced form of secondary school algebra, is a foundational mathematical discipline. It is also called Introduction to Analysis. In many schools, precalculus is actually two separate courses: Algebra and Trigonometry...
course. The trigonometric functions are pervasive in parts of pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
and applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
such as Fourier analysis and the wave equation
Wave equation
The wave equation is an important secondorder linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
, which are in turn essential to many branches of science and technology. Spherical trigonometry
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
studies triangles on sphere
Sphere
A sphere is a perfectly round geometrical object in threedimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
s, surfaces of constant positive curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
, in elliptic geometry
Elliptic geometry
Elliptic geometry is a nonEuclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
. It is fundamental to astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
and navigation
Navigation
Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...
. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a nonEuclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
History
SumerSumer
Sumer was a civilization and historical region in southern Mesopotamia, modern Iraq during the Chalcolithic and Early Bronze Age....
ian astronomers introduced angle measure, using a division of circles into 360 degrees. They and their successors 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 ancient Nubians
Nubia
Nubia is a region along the Nile river, which is located in northern Sudan and southern Egypt.There were a number of small Nubian kingdoms throughout the Middle Ages, the last of which collapsed in 1504, when Nubia became divided between Egypt and the Sennar sultanate resulting in the Arabization...
used a similar methodology. The ancient Greeks transformed trigonometry into an ordered science.
Classical Greek mathematicians
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
(such as Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
and Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
) studied the properties of chords
Chord (geometry)
A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
expanded upon Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
' Chords in a Circle in his Almagest
Almagest
The Almagest is a 2ndcentury mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
. The modern sine function was first defined in the Surya Siddhanta
Surya Siddhanta
The Surya Siddhanta is one of the earliest siddhanta in archeoastronomy of the Hindus by an unknown author. It describes the archeoastronomy theories, principles and methods of the ancient Hindus. This siddhanta is supposed to be the knowledge that the Sun god gave to an Asura called Maya. Asuras...
, and its properties were further documented by the 5th century Indian mathematician
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
and astronomer Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematicianastronomers from the classical age of Indian mathematics and Indian astronomy...
. 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
Spherical geometry
Spherical geometry is the geometry of the twodimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
. At about the same time, Chinese
Chinese mathematics
Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry....
mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir alDin alTusi
Nasir alDin alTusi
Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī , better known as Naṣīr alDīn alṬūsī , was a Persian polymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed...
. One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician Regiomontanus
Regiomontanus
Johannes Müller von Königsberg , today best known by his Latin toponym Regiomontanus, was a German mathematician, astronomer, astrologer, translator and instrument maker....
. Trigonometry was still so little known in 16th century Europe that Nicolaus Copernicus
Nicolaus Copernicus
Nicolaus Copernicus was a Renaissance astronomer and the first person to formulate a comprehensive heliocentric cosmology which displaced the Earth from the center of the universe....
devoted two chapters of De revolutionibus orbium coelestium
De revolutionibus orbium coelestium
De revolutionibus orbium coelestium is the seminal work on the heliocentric theory of the Renaissance astronomer Nicolaus Copernicus...
to explaining its basic concepts.
Driven by the demands of navigation
Navigation
Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...
and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus
Bartholomaeus Pitiscus
Bartholomaeus Pitiscus was a 16th century German trigonometrist, astronomer and theologian who first coined the word Trigonometry....
was the first to use the word, publishing his Trigonometria in 1595. Gemma Frisius
Gemma Frisius
Gemma Frisius , was a physician, mathematician, cartographer, philosopher, and instrument maker...
described for the first time the method of triangulation
Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...
still used today in surveying. It was Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
who fully incorporated complex numbers into trigonometry. The works of James Gregory
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions. Biography :The...
in the 17th century and Colin Maclaurin
Colin Maclaurin
Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor
Brook Taylor
Brook Taylor FRS was an English mathematician who is best known for Taylor's theorem and the Taylor series. Life and work :...
defined the general Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
.
Overview
If one angleAngle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles
Complementary angles
In geometry, complementary angles are angles whose measures sum to 90°. If the two complementary angles are adjacent their nonshared sides form a right angle....
. The shape
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
of a triangle is completely determined, except for similarity
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
, by the angles. Once the angles are known, the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
 SineSineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
function (sin), defined as the ratio of the side opposite the angle to the hypotenuseHypotenuseIn geometry, a hypotenuse is the longest side of a rightangled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the...
.


 Cosine function (cos), defined as the ratio of the adjacentAdjacentAdjacent is an adjective meaning contiguous, adjoining or abuttingIn geometry, adjacent is when sides meet to make an angle.In graph theory adjacent nodes in a graph are linked by an edge....
leg to the hypotenuse.
 Cosine function (cos), defined as the ratio of the adjacent

 TangentTangentIn geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
function (tan), defined as the ratio of the opposite leg to the adjacent leg.
 Tangent

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. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOHCAHTOA (see below under Mnemonics).
The reciprocals
Multiplicative inverse
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/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively. The inverse functions
Inverse trigonometric function
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions with suitably restricted domains .The notations sin−1, cos−1, etc...
are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. 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
Law of sines
In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...
and the law of cosines
Law of cosines
In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...
. 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. These laws are useful in all branches of geometry, since every polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
may be described as a finite combination of triangles.
Extending the definitions
The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radianRadian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s) only. Using the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
, one can extend them to all positive and negative arguments (see trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
). The trigonometric functions are periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
and infinite series. With these definitions the trigonometric functions can be defined for complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. The complex exponential function is particularly useful.
See Euler's
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
and De Moivre's
De Moivre's formula
In mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
formulas.
Mnemonics
A common use of mnemonicMnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...
s 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 as strings of letters. For instance, a mnemonic for English speakers is SOHCAHTOA:
 Sine = Opposite ÷ Hypotenuse
 Cosine = Adjacent ÷ Hypotenuse
 Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e. "SOHCAHTOA", which is pronounced 'sokətow'uh'). Another method is to expand the letters into a sentence, such as "Some Old Hippy Caught Another Hippy Trippin' On Acid".
Calculating trigonometric functions
Trigonometric functions were among the earliest uses for mathematical tableMathematical table
Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation...
s. 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 rule
Slide rule
The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...
s had special scales for trigonometric functions.
Today scientific calculator
Scientific calculator
A scientific calculator is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science, engineering, and mathematics...
s have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad
Grad (angle)
The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...
. Most computer programming language
Programming language
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....
s provide function libraries that include the trigonometric functions. The floating point unit
Floating point unit
A floatingpoint unit is a part of a computer system specially designed to carry out operations on floating point numbers. Typical operations are addition, subtraction, multiplication, division, and square root...
hardware incorporated into the microprocessor chips used in most personal computers have builtin instructions for calculating trigonometric functions.
Applications of trigonometry
There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation
Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...
is used in astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
to measure the distance to nearby stars, in geography
Geography
Geography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...
to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic function
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
s such as those that describe sound and light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
waves.
Fields that use trigonometry or trigonometric functions include astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
(especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation
Navigation
Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...
(on the oceans, in aircraft, and in space), music theory
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
, acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
, optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, analysis of financial markets, electronics
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
, probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of nondeterministic events or measured quantities that may either be single...
, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
, medical imaging
Medical imaging
Medical imaging is the technique and process used to create images of the human body for clinical purposes or medical science...
(CAT scans and ultrasound
Ultrasound
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...
), pharmacy
Pharmacy
Pharmacy is the health profession that links the health sciences with the chemical sciences and it is charged with ensuring the safe and effective use of pharmaceutical drugs...
, chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
(and hence cryptology), seismology
Seismology
Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planetlike bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...
, meteorology
Meteorology
Meteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...
, oceanography
Oceanography
Oceanography , also called oceanology or marine science, is the branch of Earth science that studies the ocean...
, many physical science
Physical science
Physical science is an encompassing term for the branches of natural science and science that study nonliving systems, in contrast to the life sciences...
s, land surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or threedimensional position of points and the distances and angles between them...
and geodesy
Geodesy
Geodesy , also named geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a threedimensional timevarying space. Geodesists also study geodynamical phenomena such as crustal...
, architecture
Architecture
Architecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...
, phonetics
Phonetics
Phonetics is a branch of linguistics that comprises the study of the sounds of human speech, or—in the case of sign languages—the equivalent aspects of sign. It is concerned with the physical properties of speech sounds or signs : their physiological production, acoustic properties, auditory...
, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, mechanical engineering
Mechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...
, civil engineering
Civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including works like roads, bridges, canals, dams, and buildings...
, computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
, cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...
, crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
and game development
Game development
Game development is the software development process by which a video game is developed. Development is undertaken by a game developer, which may range from a single person to a large business. Mainstream games are normally funded by a publisher and take several years to develop. Indie games can...
.
Standard identities
IdentitiesIdentity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
are those equations that hold true for any value.
Angle transformation formulae
Common formulas
Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.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.
Law of sines
The law of sinesLaw of sines
In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...
(also known as the "sine rule") for an arbitrary triangle states:
where R is the radius of the circumscribed circle
Circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter....
of the triangle:
Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:
Law of cosines
The law of cosinesLaw of cosines
In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...
(known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
to arbitrary triangles:
or equivalently:
Law of tangents
The law of tangentsLaw of tangents
In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposite sides....
:
Euler's formula
Euler's formulaEuler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
, which states that , produces the following analytical
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
identities for sine, cosine, and tangent in terms of e and the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
i:
See also
 Generalized trigonometryGeneralized trigonometryOrdinary trigonometry studies triangles in the euclidean plane R2. There are a number of ways of defining the ordinary euclidean geometric trigonometric functions on real numbers: rightangled triangle definitions, unitcircle definitions, series definitions, definitions via differential equations,...
 List of triangle topics
 Trigonometric functions
 Aryabhata's sine tableĀryabhaṭa's sine tableĀryabhaṭa's sine table is a set of twentyfour of numbers given in the astronomical treatise Āryabhaṭiya composed by the fifth century Indian mathematician and astronomer Āryabhaṭa , for the computation of the halfchords of certain set of arcs of a circle...
 List of trigonometric identities
 Rational trigonometryRational trigonometryRational trigonometry is a recently introduced approach to trigonometry that eschews all transcendental functions and all proportional measurements of angles. In place of angles, it characterizes the separation between lines by a quantity called the "spread", which is a rational function of their...
 Trigonometry in Galois fieldsTrigonometry in Galois fieldsIn mathematics, trigonometry analogies are supported by the theory of quadratic extensions of finite fields, also known as Galois fields. The main motivation to deal with a finite field trigonometry is the power of the discrete transforms, which play an important role in engineering and mathematics...
 Unit circleUnit circleIn mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
 Uses of trigonometryUses of trigonometryAmongst the lay public of nonmathematicians and nonscientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still other uses are more technical, such as in number theory...
 Smallangle approximation
 Skinny triangleSkinny triangleA skinny triangle in trigonometry is a triangle whose height is much greater than its base. The solution of such triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to the angle in radians...
External links
 Trigonometric Delights, by Eli MaorEli MaorEli Maor, an Israelborn historian of mathematics, is the author of several books about the history of mathematics. Eli Maor received his PhD at the Technion – Israel Institute of Technology. He teaches the history of mathematics at Loyola University Chicago...
, Princeton University Press, 1998. Ebook version, in PDF format, full text presented.  Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
 Benjamin Banneker's Trigonometry Puzzle at Convergence
 Dave's Short Course in Trigonometry by David Joyce of Clark UniversityClark UniversityClark University is a private research university and liberal arts college in Worcester, Massachusetts.Founded in 1887, it is the oldest educational institution founded as an allgraduate university. Clark now also educates undergraduates...