Trigonometry
Trigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle' and μέτρον (métron) 'measure')[1] is a branch of mathematics concerned with relationships between angles and side lengths of triangles.
The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.
[9] 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.
[11] Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today.
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, Islamic, and, later, Western European worlds.
The modern definition of the sine is first attested in the Surya Siddhanta, and its properties were further documented in the 5th century (AD) by Indian mathematician and astronomer Aryabhata.
[14][15] By the 10th century AD, in the work of Persian mathematician Abū al-Wafā' al-Būzjānī, all six trigonometric functions were used.
[23] 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.
[24] 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.
[26] 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.
The reciprocals of these ratios 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-".
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:[34] One way to remember the letters is to sound them out phonetically (i.e. /ˌsoʊkəˈtoʊə/ SOH-kə-TOH-ə, similar to Krakatoa).
[37] 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
For instance, sine and cosine have the following representations:[44] With these definitions the trigonometric functions can be defined for complex numbers.
[52] The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,[56] predicting eclipses, and describing the orbits of the planets.
[57] In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,[58] as well as in satellite navigation systems.
[19] Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.
[59] Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.
[68] Other fields that use trigonometry or trigonometric functions include music theory,[69] geodesy, audio synthesis,[70] architecture,[71] electronics,[69] biology,[72] medical imaging (CT scans and ultrasound),[73] chemistry,[74] number theory (and hence cryptology),[75] seismology,[67] meteorology,[76] oceanography,[77] image compression,[78] phonetics,[79] economics,[80] electrical engineering, mechanical engineering, civil engineering,[69] computer graphics,[81] cartography,[69] crystallography[82] and game development.
