Trigonometry (from Greek
'', "triangle" and ''metron
'', "measure") is a branch of mathematics
that studies relationships between side lengths and angle
s of triangle
s. The field emerged in the Hellenistic world
during the 3rd century BC from applications of geometry
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
) such as sine
Throughout history, trigonometry has been applied in areas such as geodesy
, celestial mechanics
, and navigation
Trigonometry is known for its many identities
are commonly used for rewriting trigonometrical expression
s with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation
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 ancient Nubians
used a similar method.
In the 3rd century BC, Hellenistic mathematicians
such as Euclid
studied the properties of chords
and inscribed angle
s 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
, 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
. In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy
(from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords
) in Book 1, chapter 11 of his ''Almagest
Ptolemy used chord
length to define his trigonometric functions, a minor difference from the sine
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
, and, later, Western European worlds.
The modern sine convention is first attested in the ''Surya Siddhanta
'', and its properties were further documented by the 5th century (AD) 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
The Persian polymath Nasir al-Din al-Tusi
has been described as the creator of trigonometry as a mathematical discipline in its own right. Nasīr al-Dīn al-Tūsī
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
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
, 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 George of Trebizond
. Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus
devoted two chapters of ''De revolutionibus orbium coelestium
'' to explain its basic concepts.
Driven by the demands of navigation
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
described for the first time the method of triangulation
still used today in surveying. It was Leonhard Euler
who fully incorporated complex number
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
Trigonometric ratios are the ratios between edges of a right triangle. 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:
function (sin), defined as the ratio of the side opposite the angle to the hypotenuse
function (cos), defined as the ratio of the adjacent
leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.
function (tan), defined as the ratio of the opposite leg to the adjacent leg.
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
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''.
of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
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.
A common use of mnemonic
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 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., ''SOH-CAH-TOA'', which is pronounced 'so-ka-toe-uh' ). 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 terminal side
of an angle ''A'' placed in standard position
will intersect the unit circle in a point (x,y), where
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
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
(or, 1 to 1), and thus are not invertible. By 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:
With these definitions the trigonometric functions can be defined for complex number
When extended as functions of real or complex variables, the following formula
holds for the complex exponential:
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 table
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.
s had special scales for trigonometric functions.
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 gradians
. 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 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.
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
is used in astronomy
to measure the distance to nearby stars,
as well as in satellite navigation system
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 vehicle
In land surveying
, 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.
The sine and cosine functions are fundamental to the theory of periodic function
such as those that describe sound and light
discovered that every continuous
, periodic function
could be described as an 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
among other fields.
Optics and acoustics
Trigonometry is useful in many physical science
In these areas, they are used to describe sound
and light wave
s, and to solve boundary- and transmission-related problems.
Other fields that use trigonometry or trigonometric functions include music theory
, audio synthesis
s and ultrasound
(and hence cryptology
, mechanical engineering
, civil engineering
and game development
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.
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:
is the area of the triangle and ''R'' is the radius of the circumscribed circle
of the triangle:
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:
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:
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:
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
then the area of the triangle is:
where R is the radius of the circumcircle
of the triangle.
The following trigonometric identities
are related to the Pythagorean theorem
and hold for any value:
[Extract of page 856]
Euler's formula, which states that , produces the following analytical identities for sine, cosine, and tangent in terms of ''e'' and the imaginary unit ''i'':
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
* 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
Dave's Short Course in Trigonometry
by David Joyce of Clark University
Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License