Trigonometry

trigonometrictrigonometricaltrigonometricallytriangle identitiestrigonometristtrigonometric functionsplanar trigonometrytrigonometric stationHistory of trigonometryplane
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure" ) is a branch of mathematics that studies relationships between side lengths and angles of triangles.wikipedia
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Surveying

surveyorsurveyland surveyor
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.
Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages, and the law.

List of trigonometric identities

trigonometric identitiestrigonometric identityhalf-angle formula
Trigonometry is known for its many identities, which are equations used for rewriting trigonometrical expressions to solve equations, to find a more useful expression, or to discover new relationships.
They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

Mathematics

mathematicalmathmathematician
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure" ) is a branch of mathematics that studies relationships between side lengths and angles of triangles.
Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hipparchus of Nicaea (2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD).

Sine

sine functionsinnatural sines
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.
In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.

Hipparchus

HipparchosHipparchus of NicaeaHipparchus of Nicea
In 140 BC, Hipparchus (from Nicaea, 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.
He is considered the founder of trigonometry but is most famous for his incidental discovery of precession of the equinoxes.

Indian mathematics

Indian mathematicianmathematicianmathematics
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.
In addition, trigonometry

Geodesy

geodeticgeodesistgeodetic survey
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.
In plane geometry (valid for small areas on Earth's surface), the solutions to both problems reduce to simple trigonometry.

Nasir al-Din al-Tusi

Nasīr al-Dīn al-TūsīNasir al-Din Tusial-Tusi
The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.
He is often considered the creator of trigonometry as a mathematical discipline in its own right.

Law of tangents

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.
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 opposing sides.

Geometry

geometricgeometricalgeometries
The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.
The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.

Mathematics in medieval Islam

mathematicianmathematicsIslamic mathematics
These Greek and Indian works were translated and expanded by medieval Islamic mathematicians.
Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for The Compendious Book on Calculation by Completion and Balancing by scholar Al-Khwarizmi), and advances in geometry and trigonometry.

Similarity (geometry)

similarsimilaritysimilar triangles
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.
Similar triangles also provide the foundations for right triangle trigonometry.

Chord (geometry)

chordchords chord
In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes 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.
Chords were used extensively in the early development of trigonometry.

Aryabhata

AryabhattaĀryabhaṭaĀryabhaṭa I
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.
The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry.

Triangulation

triangulatetriangulatedtriangulating
Gemma Frisius described for the first time the method of triangulation still used today in surveying.
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to it from known points.

Bartholomaeus Pitiscus

Bartholemaeus PitiscusPitiscusPitiscus, Bartholomaeus
Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.
Bartholomaeus Pitiscus (also Barthélemy, Bartholomeo, August 24, 1561 – July 2, 1613) was a 16th-century German trigonometrist, astronomer and theologian who first coined the word trigonometry.

James Gregory (mathematician)

James GregoryGregory, JamesJames Gregorie
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.
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.

Law of sines

sine lawHyperbolic law of sinessine rule
With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles.

Law of cosines

cosine lawcosinescosine rule
With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem ) relates the lengths of the sides of a triangle to the cosine of one of its angles.

Spherical geometry

sphericalspherecurved surface
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.
Thus, in spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a triangle exceeds 180 degrees.

Spherical trigonometry

spherical triangleGirard's theoremspherical excess
In 140 BC, Hipparchus (from Nicaea, 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.
These identities approximate the cosine rule of plane trigonometry if the sides are much smaller than the radius of the sphere.

De revolutionibus orbium coelestium

De revolutionibusOn the Revolutions of the Heavenly SpheresDe revolutionibus orbium cœlestium
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.
Due to its friendly reception, Copernicus finally agreed to publication of more of his main work—in 1542, a treatise on trigonometry, which was taken from the second book of the still unpublished De revolutionibus.

Unit circle

circleBase circle (mathematics)base-circle
Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane.
Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane.

Slide rule

slide rulescircular slide ruleslide-rule
Slide rules had special scales for trigonometric functions.
The slide rule is used primarily for multiplication and division, and also for functions such as exponents, roots, logarithms, and trigonometry, but typically not for addition or subtraction.

Leonhard Euler

EulerLeonard EulerEuler, Leonhard
It was Leonhard Euler who fully incorporated complex numbers into trigonometry.
Euler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics.