All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. The field emerged in the Hellenistic trigonometric table from 0 to 90 pdf during the 3rd century BC from applications of geometry to astronomical studies. The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology.

Trigonometry is also the foundation of surveying. Thus the majority of applications relate to right-angle triangles. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

Trigonometry basics are often taught in schools, either as a separate course or as a part of a precalculus course. Hipparchus, credited with compiling the first trigonometric table, is known as “the father of trigonometry”. A thick ring-like shell object found at the Indus Valley Civilization site of Lothal, with four slits each in two margins served as a compass to measure angles on plane surfaces or in the horizon in multiples of 40 degrees, up to 360 degrees. 12 whole sections of the horizon and sky, explaining the slits on the lower and upper margins.

12 fold division of horizon and sky, as well as an instrument for measuring angles and perhaps the position of stars, and for navigation. Sumerian 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 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. 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. 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, Islamic, and, later, Western European worlds. 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 Western Europe via Latin translations of Ptolemy’s Greek Almagest as well as the works of Persian and Arabic 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.