# Dictionary Definition

trigonometry n : the mathematics of triangles and
trigonometric functions [syn: trig]

# User Contributed Dictionary

## English

### Noun

- The branch of mathematics that deals with the relationships between the sides and the angles of triangles and the calculations based on them, particularly the trigonometric functions.

#### Related terms

#### Translations

branch of matematics

- Chinese: 三角函数
- Czech: trigonometrie
- Finnish: trigonometria
- French: trigonométrie
- German: Trigonometrie
- Irish: triantánacht
- Italian: trigonometria
- Japanese: 三角法 (sankakuhō)
- Portuguese: trigonometria
- Swedish: trigonometri
- Telugu: త్రికోణమితి (trikONamiti)

# Extensive Definition

main History
of trigonometry Trigonometry was probably developed for use in
sailing as a navigation
method used with astronomy. The origins of
trigonometry can be traced to the civilizations of ancient
Egypt, Mesopotamia and
the Indus
Valley, more than 4000 years ago. The common practice of
measuring angles in degrees, minutes and seconds comes from the
Babylonian's
base sixty system of
numeration. The Sulba Sutras
written in India, between 800 BC and 500 BC, correctly computes the
sine of \frac (=45°) as
\frac in a procedure for "circling the square" (i.e., constructing
the inscribed
circle).

The first recorded use of trigonometry came from
the Hellenistic
mathematician Hipparchushttp://www.math.rutgers.edu/~cherlin/History/Papers2000/hunt.html
circa 150 BC, who compiled a trigonometric table
using the sine for solving
triangles. Ptolemy further
developed trigonometric calculations circa 100 AD.

The ancient Sinhalese
in Sri
Lanka, when constructing reservoirs in the Anuradhapura
kingdom, used trigonometry to calculate the gradient of the water flow.
Archeological research also provides evidence of trigonometry used
in other unique hydrological structures dating back to 4 BC.

The Indian mathematician Aryabhata in 499,
gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for
sine, kotizya for cosine, and otkram zya for inverse sine, and also
introduced the versine.
Another Indian mathematician, Brahmagupta in
628, used an interpolation formula to
compute values of sines, up to the second order of the Newton-Stirling
interpolation formula.

In the 10th century, the Persian mathematician
and astronomer Abul Wáfa
introduced the tangent
function and improved methods of calculating trigonometry
tables. He established the angle addition identities, e.g. sin (a +
b), and discovered the sine formula for spherical geometry:

- \frac = \frac = \frac.

Also in the late 10th and early 11th centuries,
the Egyptian astronomer Ibn Yunus
performed many careful trigonometric calculations and demonstrated
the formula

- \cos a \cos b = \frac..

Indian
mathematicians were the pioneers of variable computations
algebra for use in
astronomical calculations along with trigonometry. Lagadha (circa
1350-1200 BC) is the first person thought to have used geometry and
trigonometry for astronomy, in his Vedanga
Jyotisha.

Persian
mathematician
Omar
Khayyám (1048-1131) combined trigonometry and approximation
theory to provide methods of solving algebraic equations by
geometrical means. Khayyam solved the cubic equation x^3 + 200 x =
20 x^2 + 2000 and found a positive root of this cubic by
considering the intersection of a rectangular hyperbola and a circle. An
approximate numerical solution was then found by interpolation in
trigonometric tables.

Detailed methods for constructing a table of
sines for any angle were given by the Indian mathematician Bhaskara in 1150,
along with some sine and cosine formulae. Bhaskara also developed
spherical
trigonometry.

The 13th century Persian
mathematician
Nasir
al-Din Tusi, along with Bhaskara, was probably the first to
treat trigonometry as a distinct mathematical discipline. Nasir
al-Din Tusi in his Treatise on the Quadrilateral was the first to
list the six distinct cases of a right angled triangle in spherical
trigonometry.

In the 14th century, Persian mathematician
al-Kashi
and Timurid
mathematician Ulugh Beg
(grandson of Timur) produced
tables of trigonometric functions as part of their studies of
astronomy.

The mathematician Bartholemaeus
Pitiscus published an influential work on trigonometry in 1595
which may have coined the word "trigonometry".

## Overview

If one angle of a right 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 right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:- The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

- \sin A=\frac=\frac\,.

- The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

- \cos A=\frac=\frac\,.

- The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

- \tan A=\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. Many people find it easy to remember
what sides of the right triangle are equal to sine, cosine, or
tangent, by memorizing the word SOH-CAH-TOA (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.

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 an 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.

### Extending the definitions

The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) 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 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
numbers. The complex function cis is particularly useful

- \operatorname\,x = \cos x + i\sin x \! = e^.

See Euler's
and De
Moivre's formulas.

### Mnemonics

Students often use mnemonics 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, as in SOH-CAH-TOA.

- Sine = Opposite ÷ Hypotenuse
- Cosine = Adjacent ÷ Hypotenuse
- Tangent = Opposite ÷ Adjacent

Alternatively, one can devise sentences which
consist of words beginning with the letters to be remembered. For
example, to recall that Tan = Opposite/Adjacent, the letters T-O-A
must be remembered. Any memorable phrase constructed of words
beginning with the letters T-O-A will serve.

Another type of mnemonic describes facts in a
simple, memorable way, such as "Plus to the right, minus to the
left; positive height, negative depth," which refers to
trigonometric functions generated by a revolving line.

### 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.Today scientific
calculators 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
languages provide function libraries that include the
trigonometric functions. The floating
point unit hardware incorporated into the microprocessor chips
used in most personal computers have built in instructions for
calculating trigonometric functions.

## Applications of trigonometry

There are an enormous number of applications
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
functions such as those that describe sound and light waves.

Fields which make use of trigonometry or
trigonometric functions include astronomy (especially, for
locating the 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
sciences, land surveying and geodesy, architecture, phonetics, economics, electrical
engineering, mechanical
engineering, civil
engineering, computer
graphics, cartography, crystallography and
game
development.

## Common formulae

- main Trigonometric identity
- main Trigonometric function

### Trigonometric identities

#### Pythagorean identities

- \begin

#### Sum and product identities

##### Sum to product:

- \begin

##### Product to sum:

- \begin

##### Sine, cosine, and tangent of a sum

Detailed, diagramed proofs of the first two of
these formulas are available for download as a four-page PDF
document at 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
sines (also know as the "sine rule") for an arbitrary triangle
states:

- \frac = \frac = \frac = 2R,

#### Law of cosines

The law of
cosines (also 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:

- \frac=\frac

## See also

## References

## External links

sisterlinks Trigonometry- Trigonometric Delights, by Eli Maor, 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.
- Trigonometry on PlainMath.net Trigonometry Articles from PlainMath.Net
- Trigonometry on Mathwords.com index of trigonometry entries on Mathwords.com
- Benjamin Banneker's Trigonometry Puzzle at Convergence
- Trigonometry
- Dave's Short Course in Trigonometry by David Joyce of Clark University

trigonometry in Afrikaans: Driehoeksmeting

trigonometry in Arabic: حساب مثلثات

trigonometry in Bengali: ত্রিকোণমিতি

trigonometry in Min Nan: Saⁿ-kak-hoat

trigonometry in Belarusian: Трыганаметрыя

trigonometry in Belarusian (Tarashkevitsa):
Трыганамэтрыя

trigonometry in Bosnian: Trigonometrija

trigonometry in Bulgarian: Тригонометрия

trigonometry in Catalan: Trigonometria

trigonometry in Czech: Trigonometrie

trigonometry in Welsh: Trigonometreg

trigonometry in Danish: Trigonometri

trigonometry in German: Trigonometrie

trigonometry in Modern Greek (1453-):
Τριγωνομετρία

trigonometry in Emiliano-Romagnolo:
Trigonometrî

trigonometry in Spanish: Trigonometría

trigonometry in Esperanto: Trigonometrio

trigonometry in Basque: Trigonometria

trigonometry in Persian: مثلثات

trigonometry in French: Trigonométrie

trigonometry in Galician: Trigonometría

trigonometry in Korean: 삼각법

trigonometry in Hindi: त्रिकोणमिति

trigonometry in Croatian: Trigonometrija

trigonometry in Indonesian: Trigonometri

trigonometry in Icelandic: Hornafall

trigonometry in Italian: Trigonometria

trigonometry in Hebrew: טריגונומטריה

trigonometry in Georgian: ტრიგონომეტრია

trigonometry in Kurdish: Trigonometri

trigonometry in Lao: ໄຕມຸມ

trigonometry in Latin: Trigonometria

trigonometry in Latvian: Trigonometrija

trigonometry in Lithuanian: Trigonometrija

trigonometry in Hungarian: Trigonometria

trigonometry in Macedonian: Тригонометрија

trigonometry in Malayalam: ത്രികോണമിതി

trigonometry in Malay (macrolanguage):
Trigonometri

trigonometry in Dutch: Goniometrie

trigonometry in Japanese: 三角法

trigonometry in Norwegian: Trigonometri

trigonometry in Norwegian Nynorsk:
Trigonometri

trigonometry in Uzbek: Trigonometriya

trigonometry in Panjabi: ਤਿਕੋਣਮਿਤੀ

trigonometry in Polish: Trygonometria

trigonometry in Portuguese: Trigonometria

trigonometry in Romanian: Trigonometrie

trigonometry in Quechua: Wamp'artupuykama

trigonometry in Russian: Тригонометрия

trigonometry in Sicilian: Trigunomitrìa

trigonometry in Simple English:
Trigonometry

trigonometry in Slovak: Trigonometria

trigonometry in Slovenian: Trigonometrija

trigonometry in Serbian: Тригонометрија

trigonometry in Serbo-Croatian:
Trigonometrija

trigonometry in Finnish: Trigonometria

trigonometry in Swedish: Trigonometri

trigonometry in Tamil: முக்கோணவியல்

trigonometry in Telugu: త్రికోణమితి

trigonometry in Thai: ตรีโกณมิติ

trigonometry in Vietnamese: Lượng giác

trigonometry in Turkish: Trigonometri

trigonometry in Ukrainian: Тригонометрія

trigonometry in Contenese: 三角學

trigonometry in Chinese: 三角学

# Synonyms, Antonyms and Related Words

Boolean algebra, Euclidean geometry, Fourier
analysis, Lagrangian function, algebra, algebraic geometry,
analysis, analytic
geometry, arithmetic,
associative algebra, binary arithmetic, calculus, circle geometry,
descriptive geometry, differential calculus, division algebra,
equivalent algebras, game theory, geodesy, geometry, goniometry, graphic algebra,
group theory, higher algebra, higher arithmetic, hyperbolic
geometry, infinitesimal calculus, integral calculus, intuitional
geometry, invariant subalgebra, inverse geometry, line geometry,
linear algebra, mathematical physics, matrix algebra, metageometry, modular
arithmetic, n-tuple linear algebra, natural geometry, nilpotent
algebra, number theory, plane trigonometry, political arithmetic,
projective geometry, proper subalgebra, quaternian algebra,
reducible algebra, set theory, simple algebra, solid geometry,
speculative geometry, spherical trigonometry, statistics, subalgebra, systems analysis,
topology, trig, universal algebra, universal
geometry, vector algebra, zero algebra