Who invented trigonometry identities

Skip to content Home Essay Why is trigonometry important in real life? Ben Davis May 3, Why is trigonometry important in real life? What is the point of trigonometry? Where is trigonometry used in real life? What is the most important identity in trigonometry? Who invented trigonometry in India? Who is the father of trigonometry in India?

How do you learn trigonometry? Why was trigonometry invented? What is cos 2x identity? What are the six trigonometric functions?

What are the three trigonometry identities? The most important use was in navigation, for calculating the distance between two points on a sphere. The perpendicular distance from the mid point of a chord to a curve is still used as a measure of 'deviation from straightness', for example, by railway engineers.

It is used also in optics for measuring the curvature of lenses and mirrors, where he versine is sometimes called the sagitta from the Latin for arrow. The Hindu word jiya for the sine was adopted by the Arabs who called the sine jiba. Eventually jiba became jaib and this word actually meant a 'fold'. When Europeans translated the Arabic works into Latin they translated jaib into the word sinus meaning a fold in Latin.

In his Practica Geometriae Fibonacci uses the term sinus rectus arcus which soon encouraged the universal use of the word sine. Main menu Search. History of Trigonometry - Part 3. The Arabs collect knowledge from the known world The Arab civilisation traditionally marks its beginning from the year CE the date when Muhammad, threatened with assassination, fled from Mecca to Medina where Muhammad and his followers found safety and respect.

Over a century later, the Arabs had established themselves as a powerful unified force across large parts of the Middle East and The Caliph Abu Ja'far Al-Mansour moved from Damascus to establish the city of Baghdad during the years to Al-Mansour sent his emissaries to search for and collect knowledge. From China, they learnt how to produce paper, and using this new skill they started a programme of translation of texts on mathematics, astronomy, science and philosophy into Arabic.

The quest for knowledge became a lasting and significant part of Arab culture. Al-Mansour had founded a scientific academy that became called 'The House of Wisdom'. This academy attracted scholars from many different countries and religions to Baghdad to work together and establish the traditions of Arabic science that were to continue well into the Middle Ages. Some of this work was later translated into Latin by Mediaeval scholars and passed on into Europe.

The dominance of Baghdad and the influence of the Arab World was to last for the next years. The scholars in the House of Wisdom came from many cultures and translated the works of Egyptian, Babylonian, Greek, Indian and Chinese astronomers and mathematicians. The name ' Almagest ' has continued to this day and it is recognized as both the great synthesis and the culmination of mathematical astronomy of the ancient Greek world.

It was translated into Arabic at least five times and constituted the basis of the mathematical astronomy carried out in the Islamic world. Greek astronomy began to be known in India during the period CE. However, Indian astronomers had long been using planetary data and calculation methods from the Babylonians, and even though it was well after Ptolemy had written the Almagest, 4th century Indian astronomers did not entirely take over Greek planetary theory.

Ancient works like the Panca-siddhantica now lost that had been transmitted through the version by Vrahamihira [ See Part 1 section 3 ] and Aryabhata's Aryabhatiya CE demonstrated that Indian scholars had their own ways of dealing with astronomical problems and that they had great skill in calculation.

Even in the oldest Indian texts, the Chord [to remind yourself about Chords see the section on Claudius Ptolemy in the previous article ] is not used, and instead there appear some very early versions of trigonometric tables using Sines. So the relation between the jiya and our sine is:. By the 5th century, two other functions had been defined and used. This was sometimes called the sama meaning an 'arrow', or sagitta in Latin.

This became the standard for later works. Comparison with Varhamihira's Sines in sexagesimal numbers and Hipparchs' table in lengths of chords suggests a possible transmission of at least some of the Greek works to the Hindus. However, we have no way of knowing this for certain, and it is quite possible that the Hindus calculated their values independently.

He produced a remarkable method for approximating values for the Sines, by using the ratio of two quadratic functions. This was based entirely on comparing the results of his calculations with earlier values.

The introduction and development of trigonometry into an independent science in the Arab civilisation took, in all, some years. In the early s Indian astronomical works reached the C aliph Al-Mansur in Baghdad, and were translated as the Zij al-Sindhind , and this introduced Indian calculation methods into Islam.

Famous for his algebra book, Abu Ja'far Muhammad ibn Musa al-Khwarizmi see The Development of Algebra Part 1 had also written a book on Indian methods of calculation al-hisab al-hindi and he produced an improved version of the Zij al-Sindhind.

Al-Khwarizmi's version of Zij used Sines and Versines, and developed procedures for tangents and cotangents to solve astronomical problems. Al-Khwarizmi's Zij was copied many times and versions of it were used for a long time. Among these were the works of Euclid, Archimedes Apollonius and of course, Ptolemy.

The Arabs now had two competing versions of astronomy, and soon the Almagest prevailed. The Indian use of the sine and its related functions were much easier to apply in calculations, and the sexagesimal system from the Babylonians continued to be used, so apart from these two changes, the early Arabic versions of the Almagest remained faithful to Ptolemy. Abu al-Wafa al-Buzjani Abul Wafa made important contributions to both geometry and arithmetic and was the first to study trigonometric identities systematically.

The study of identities was important because by establishing relationships between sums and differences, and fractions and multiples of angles, more efficient astronomical calculations could be conducted and more accurate tables could be established.

The sine, versine and cosine had been developed in the context of astronomical problems, whereas the tangent and cotangent were developed from the study of shadows of the gnomon.

In his Almagest , Abul Wafa brought them together and established the relations between the six fundamental trigonometric functions for the first time. From these relations Abul Wafa was able to demonstrate a number of new identities using these new functions:. Greek astronomers had long since introduced a model of the universe with the stars on the inside of a vast sphere.

They had also worked with spherical triangles, but Abul Wafa was the first Arab astronomer to develop ways of measuring the distance between stars using his new system of trigonometric functions including the versine. By an ingenious application of Menelaos' Theorem [See History of Trigonometry Part 2 ] using special cases of great circles with two right angles, Abul Wafa showed how the theorem could be applied in spherical triangles.

This was a considerable advance in Spherical Trigonometry that enabled the calculation of the correct direction for prayer the quibla and was to have important applications in Navigation and Cartography. Abu al-Rayhan Muhammad ibn Ahmad Al-Biruni was an outstanding scholar reputed to have written over treatises on astronomy, science, mathematics, geography, history, geodesy and philosophy.

Only about twenty of these works now survive, and only about a dozen of these have been published. Al-Biruni's treatise entitled Maqalid 'ilm al-hay'a Keys to the Science of Astronomy ran to over one thousand pages and contained extensive developments in on trigonometry. Among many theorems, he produced a demonstration of the tangent formula, shown below.

While many new aspects of trigonometry were being discovered, the chord, sine, versine and cosine were developed in the investigation of astronomical problems, and conceived of as properties of angles at the centre of the heavenly sphere.

In contrast, tangent and cotangent properties were derived from the measurement of shadows of a gnomon and the problems of telling the time.

In his Demarcation of the Coordinates of Cities he used spherical triangles for finding the coordinates of cities and other places to establish local meridian the quibla and thereby finding the correct direction of Mecca, and in his Exhaustive Treatise on Shadows he showed how to use gnomons [See A Brief History of Time Measurement ] for finding the time of day.

Abu Muhammad Jabir ibn Aflah Jabir ibn Aflah c - c probably worked in Seville during the first part of the 12th century. His work is seen as significant in passing on knowledge to Europe. Jabir ibn Aflah was considered a vigorous critic of Ptolemy's astronomy.

His treatise helped to spread trigonometry in Europe in the 13th century, and his theorems were used by the astronomers who compiled the influential Libro del Cuadrante Sennero Book of the Sine Quadrant under the patronage of King Alfonso X the Wise of Castille A result of this project was the creation of much more accurate astronomical tables for calculating the position of the Sun, Moon and Planets, relative to the fixed stars, called the Alfonsine Tables made in Toledo somewhere between and These were the tables Columbus used to sail to the New World, and they remained the most accurate tables until the 16th century.

By the end of the 10th century trigonometry occupied an important place in astronomy texts with chapters on sines and chords, shadows tangents and cotangents and the formulae for spherical calculations. There was also considerable interest in the resolution of plane triangles. But a completely new type of work by Nasir al-Din al-Tusi Al-Tusi entitled Kashf al-qina 'an asrar shakl al-qatta Treatise on the Secrets of the Sector Figure , was the first treatment of trigonometry in its own right, as a complete subject apart from Astronomy.

The work contained a systematic discussion on the application of proportional reasoning to solving plane and spherical triangles, and a thorough treatment of the formulae for solving triangles and trigonometric identities. Al-Tusi originally wrote in Persian, but later wrote an Arabic version. The only surviving Persian version of his work is in the Bodleian Library in Oxford. This was a collection and major improvement on earlier knowledge. Natural sciences. Paleontology Physics.

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See also: Islamic mathematics and Science and technology in the Ottoman Empire. The Greeks, and after them the Hindus and the Arabs, used trigonometric lines. These at first took the form, as we have seen, of chords in a circle, and it became incumbent upon Ptolemy to associate numerical values or approximations with the chords.

Our common system of angle measure may stem from this correspondence. Moreover since the Babylonian position system for fractions was so obviously superior to the Egyptians unit fractions and the Greek common fractions, it was natural for Ptolemy to subdivide his degrees into sixty partes minutae primae , each of these latter into sixty partes minutae secundae , and so on. It is from the Latin phrases that translators used in this connection that our words "minute" and "second" have been derived.

It undoubtedly was the sexagesimal system that led Ptolemy to subdivide the diameter of his trigonometric circle into parts; each of these he further subdivided into sixty minutes and each minute of length sixty seconds. Episodes from the Early History of Astronomy. New York: Springer, A history of ancient mathematical astronomy. ISBN Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians.

The second book of the Sphaerica describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest. In this work Aristarchus made the observation that when the moon is just half-full, the angle between the lines of sight to the sun and the moon is less than a right angle by one thirtieth of a quadrant. In trigonometric language of today this would mean that the ratio of the distance of the moon to that of the sun the ration ME to SE in Fig.

It is possible that he took over from Hypsicles, who earlier had divided the day into parts, a subdivision that may have been suggested by Babylonian astronomy. We do not know when or where Euclid and Ptolemy were born. We know that Ptolemy made observations at Alexandria from A.

Suidas, a writer who lived in the tenth century, reported that Ptolemy was alive under Marcus Aurelius emperor from A. Ptolemy's Almagest is presumed to be heavily indebted for its methods to the Chords in a Circle of Hipparchus, but the extent of the indebtedness cannot be reliably assessed.

It is clear that in astronomy Ptolemy made use of the catalogue of star positions bequeathed by Hipparchus, but whether or not Ptolemy's trigonometric tables were derived in large part from his distinguioshed predecrssor cannot be determined. Similar reasoning leads to the formula [ It was the formula for sine of the difference - or, more accurately, chord of the difference - that Ptolemy found especially useful in building up his tables.

Another formula that served him effectively was the equivalent of our half-angle formula. Islamic Astronomy. Scientific American. Retrieved Lennart Princeton University Press. Katz, Robin J. The historical development of the calculus. Springer Study Edition Series 3 ed.