The history of trigonometry and of trigonometric functions may span about 4000 years. Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ... Sine redirects here. ...

Etymology

Our modern word sine is derived from the Latin word sinus, which means "bay" or "fold", from a mistranslation (via Arabic) of the Sanskrit word jiva, alternatively called jya.^[1] Aryabhata used the term ardha-jiva ("half-chord"), which was shortened to jiva and then transliterated by the Arabs as jiba (جب). European translators like Robert of Chester and Gherardo of Cremona in 12th-century Toledo confused jiba for jaib (جب), meaning "bay", probably because jiba (جب) and jaib (جب) are written the same in the Arabic script (this writing system, in one of its forms, does not provide the reader with complete information about the vowels). The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae.^[2] For other uses, see Latins and Latin (disambiguation). ... Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ... Sanskrit ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... For other uses, see Aryabhata (disambiguation). ... For other uses, see Arab (disambiguation). ... Robert of Chester (Robertus Castrensis) was an English arabist who flourished around 1150. ... For other uses, see Toledo (disambiguation). ... The Arabic alphabet is the script used for writing the Arabic language, which is the language of the Quran, the holy book of Islam. ... Writing systems of the world today. ...

Development

Trigonometry is not the work of any one man or nation. Its history spans thousands of years and has touched every major civilization. It should be noted that that from the time of Hipparchus until modern times there was no such thing as a trigonometric ratio. Instead, the Greeks and after them the Hindus and the Muslims used trigonometric lines. These lines first took the form of chords and later half chords, or sines. These chord and sine lines would then be associated with numerical values, possibly approximations, and listed in trigonometric tables.^[2] A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ...

Early trigonometry

The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But pre-Hellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry".^[3] Babylonian clay tablet YBC 7289 with annotations. ...

Based on one interpretation of the Plimpton 322 cuneiform tablet (circa 1900 BC), some have even asserted that the ancient Babylonians had a table of secants.^[4] There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table. Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ... Cuneiform redirects here. ... Babylonian clay tablet YBC 7289 with annotations. ...

Greek mathematics

The chord of an angle subtends the arc of the angle.

Ancient Greek and Hellenistic mathematicians made use of the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of the bisected angle, that is, , and consequently the sine function is also known as the "half chord". Due to this relationship, many of the trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form.^[5] Image File history File links TrigonometricChord. ... Image File history File links TrigonometricChord. ... Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ... The term Hellenistic (derived from Héllēn, the Greeks traditional self-described ethnic name) was established by the German historian Johann Gustav Droysen to refer to the spreading of Greek culture over the non-Greek people that were conquered by Alexander the Great. ...

Although there is no trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas.^[3] For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosine for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of the law of sines. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles.^[3] To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the well known theorem that, in modern notation, sin α/ sin β < α/β < tan α/ tan β whenever 0° < β < α < 90°, among other theorems.^[6] For other uses, see Euclid (disambiguation). ... For other uses, see Archimedes (disambiguation). ... Fig. ... In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ... For other uses of this name, including the grammarian Aristarchus of Samothrace, see Aristarchus Statue of Aristarchus at Aristotle University in Thessalonica, Greece Aristarchus (Greek: Ἀρίσταρχος; 310 BC - ca. ...

The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 - 125 BC), who is now consequently known as "the father of trigonometry."^[7] Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.^[7][1] For the Athenian tyrant, see Hipparchus (son of Pisistratus). ... Iznik ceramic pitcher with flower decoration from ca. ...

A medieval artist's rendition of Claudius Ptolemy

Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon (ca. 260 B.C.), since he measured an angle in terms of a fraction of a quadrant.^[6] It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy.^[8] In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts.^[2] It is due to the Babylonian sexagesimal number system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.^[2] Image File history File links Ptolemaeus. ... Image File history File links Ptolemaeus. ... This article is about the geographer, mathematician and astronomer Ptolemy. ... For other uses of this name, including the grammarian Aristarchus of Samothrace, see Aristarchus Statue of Aristarchus at Aristotle University in Thessalonica, Greece Aristarchus (Greek: Ἀρίσταρχος; 310 BC - ca. ... This article is about Hypsicles of Alexandria. ...

Menelaus of Alexandria (ca. 100 A.D.) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.^[5] He establishes a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles.^[5] Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.^[5] Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".^[5] He further gave his famous "rule of six quantities".^[9] Menelaus of Alexandria (c. ...

Later, Claudius Ptolemy (ca. 90 - ca. 168 A.D.) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity.^[10] A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Ptolemy further derived the equivalent of the half-angle formula . Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.^[10] This article is about the geographer, mathematician and astronomer Ptolemy. ... Almagest is the Latin form of the Arabic name (al-kitabu-l-mijisti, i. ...

Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.^[11]

Medieval Indian mathematics

Statue of Aryabhata

The next significant developments of trigonometry were in India. The mathematician-astronomer Aryabhata (476–550 A.D.), in his work Aryabhata-Siddhanta, first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine. His works also contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. He used the words jya for sine, kojya for cosine, ukramajya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation. Image File history File links Download high-resolution version (951x1361, 193 KB) [edit] Summary This image of a public statue in IUCAA Pune was photographed in May 2006 by myself, and I release all rights. ... Image File history File links Download high-resolution version (951x1361, 193 KB) [edit] Summary This image of a public statue in IUCAA Pune was photographed in May 2006 by myself, and I release all rights. ... For other uses, see Aryabhata (disambiguation). ... This article is under construction. ... For other uses, see Aryabhata (disambiguation). ... The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the coversed...

Other Indian mathematicians later expanded Aryabhata's works on trigonometry. In the 6th century A.D., Varahamihira used the formulas Varahamihira (505 – 587) was an Indian astronomer, mathematician, and astrologer born in Ujjain. ...

In the 7th century A.D., Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(x), which had a relative error of less than 1.9%: Bhāskara, or Bhāskara I, (c. ...

Later in the 7th century, Brahmagupta developed the formula as well as the Brahmagupta interpolation formula for computing sine values.
Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) was an Indian mathematician and astronomer. ... In trigonometry, the Brahmagupta interpolation formula is a special case of the Newton-Stirling interpolation formula, which calculates the values of sine at different intervals. ...

Islamic mathematics

Al-Khwārizmī depicted on a Soviet stamp

The Indian works were later translated and expanded in the Islamic world by Arab and Persian Muslim mathematicians. In the 9th century, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer of spherical trigonometry. Image File history File links Download high resolution version (735x985, 159 KB)Abu Abdullah Muhammad bin Musa al-Khwarizmi. ... Image File history File links Download high resolution version (735x985, 159 KB)Abu Abdullah Muhammad bin Musa al-Khwarizmi. ... al-KhwārizmÄ« redirects here. ... CCCP redirects here. ... This 1998 stamp of the Faroe Islands marks the 50th anniversary of the Universal Declaration of Human Rights. ... During the Islamic Golden Age, usually dated from the 8th century to the 13th century,[1] engineers, scholars and traders of the Islamic world contributed enormously to the arts, agriculture, economics, industry, literature, navigation, philosophy, sciences, and technology, both by preserving and building upon earlier traditions and by adding many... A 9th century picture of Arab scientists working in Baghdad, Iraq. ... This article is about Iranian scientists of the classical era. ... In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... al-KhwārizmÄ« redirects here. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...

By the 10th century, in the work of Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six trigonometric functions, after discovering the secant, cotangent and cosecant functions. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also developed the following trigonometric formula: (940 – 997/8) was a Persian mathematician and astronomer. ... In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... Secant is a term in mathematics. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ...

Also in the 10th century, Al-Battani was responsible for establishing a number of important trigometrical relationships such as: Al Battani (c. ...

Al-Jayyani (989–1079) of al-Andalus wrote a treatise on spherical trigonometry, entitled The book of unknown arcs of a sphere, which "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.^[12] Abu Abd Allah Muhammad ibn Muadh Al-Jayyani (Al-Jayyani; 989, Cordoba, Spain - 1079, Jaen, Spain) was an Arabic mathematician from present-day Spain. ... Al-Andalus is the Arabic name given the Iberian Peninsula by its Muslim conquerors; it refers to both the Caliphate proper and the general period of Muslim rule (711–1492). ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... Two types of special right triangles appear commonly in geometry, the angle based and the side based triangles. ... In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ... Right spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... For other uses, see Triangle (disambiguation). ... This article is about the mathematical concept. ... Johannes Müller von Königsberg (June 6, 1436 – July 6, 1476), known by his Latin pseudonym Regiomontanus, was an important German mathematician, astronomer and astrologer. ...

In the 11th century, Omar Khayyám (1048–1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ...

Other authors on trigonometric subjects include Bhaskara II and Nasir al-Din al-Tusi in the 13th century. Nasir al-Din al-Tusi stated the law of sines and provided a proof for it, and also listed the six distinct cases of a right angled triangle in spherical trigonometry. In the 14th century, Ghiyath al-Kashi gave trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg (14th century) also gives accurate tables of sines and tangents correct to 8 decimal places. Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... Tusi couple from Vat. ... In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ... Kashani, dubbed, the Second Ptolemy, was an outstanding Persian mathematician of the middle ages. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... Ulugh Beg, here depicted on a Soviet stamp, was one of Islams greatest astronomers during the Middle Ages. ...

The method of triangulation was first developed by Muslim mathematicians, who applied it to practical uses such as surveying.^[13] Triangulation can be used to find the distance from the shore to the ship. ... Surveyor at work with a leveling instrument. ...

Medieval China

Guo Shoujing (1231–1316)

In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty.^[14] Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the earlier Greek and then Indian and Islamic worlds.^[15] Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.^[14] However, this embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations.^[14] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.^[14] Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc of a circle s given the diameter d, sagita v, and length of the chord c subtending the arc, the length of which he approximated as s = c + 2v²/d.^[16] Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).^[17] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.^[18][14] Along with a later 17th century Chinese illustration of Guo's mathematical proofs, Needham states that: Guo Shoujing (Chinese: 郭守敬; Hanyu Pinyin: ; Wade-Giles: Kuo Shou-ching) (1231 – 1316) was a Chinese astronomer, engineer, and mathematician. ... The Treatise on Astrology of the Kaiyuan Era (Chinese: ; pinyin: ), also known as the Great Tang Treatise on Astrology of the Kaiyuan Era in the full title, is a Chinese astrology encyclopedia compiled by the lead editor Gautama Siddha and a numerous of scholars from 714 to 724 during the... For the band, see Tang Dynasty (band). ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... For other uses, see Liu Song Dynasty. ... Leonardo da Vinci is regarded in many Western cultures as the archetypal Renaissance Man. A polymath (Greek polymathÄ“s, πολυμαθής, having learned much)[1][2] is a person with encyclopedic, broad, or varied knowledge or learning. ... This is a Chinese name; the family name is Shen Shen Kuo or Shen Kua (Chinese: ; pinyin: ) (1031–1095) was a polymathic Chinese scientist and statesman of the Song Dynasty (960–1279). ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... Guo Shoujing (Chinese: 郭守敬; Hanyu Pinyin: ; Wade-Giles: Kuo Shou-ching) (1231 – 1316) was a Chinese astronomer, engineer, and mathematician. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... The Chinese calendar is a lunisolar calendar, incorporating elements of a lunar calendar with those of a solar calendar. ... The Dunhuang map from the Tang Dynasty (North Polar region). ...

Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).^[19] Illumination of Earth by the sun on the northern hemisphere summer solstice The summer solstice is an astronomical term regarding the position of the sun in relation to the celestial equator. ...

Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).^[20] The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... Xu Guangqi (Simplified Chinese: 徐光启; Traditional Chinese: 徐光啟; Pinyin: Xú Guāngqǐ) (1562–1633) was a Chinese agricultural scientist and mathematician born in Shanghai. ... Matteo Ricci. ...

Renaissance Europe

Isaac Newton in a 1702 portrait by Godfrey Kneller.

Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,^[21] in his De triangulis omnimodus written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed. Image File history File linksMetadata Download high resolution version (562x700, 107 KB) Description: National Portrait Gallery London Source: http://www. ... Image File history File linksMetadata Download high resolution version (562x700, 107 KB) Description: National Portrait Gallery London Source: http://www. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Sir Godfrey Kneller (August 8, 1646 -October 19, 1723) was an artist, court painter to several British monarchs. ... Johannes Müller von Königsberg (June 6, 1436 – July 6, 1476), known by his Latin pseudonym Regiomontanus, was an important German mathematician, astronomer and astrologer. ...

The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596. Georg Joachim von Lauchen, also known as Rheticus (February 16, 1514 – December 4, 1574), was a mathematician, cartographer, navigational and other instrument maker, medical practitioner, and teacher. ... Nicolaus Copernicus (in Latin; Polish Mikołaj Kopernik, German Nikolaus Kopernikus - February 19, 1473 – May 24, 1543) was a Polish astronomer, mathematician and economist who developed a heliocentric (Sun-centered) theory of the solar system in a form detailed enough to make it scientifically useful. ...

In the 17th century, Isaac Newton and James Stirling developed the general Newton-Stirling interpolation formula for trigonometric functions. Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... James Stirling (April 22, 1692–December 5, 1770) was a Scottish mathematician. ...

Trigonometric analysis

Madhava (c. 1400) made early strides in the analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the trigonometric series expansions of sine, cosine, tangent and arctangent. Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the θ, radius, diameter and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century.^[22][23] Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematician-astronomer from Kerala, India. ... Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... Series expansion redirects here. ... In mathematics, a Fourier series of a periodic function, named in honor of Joseph Fourier (1768-1830), represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... Look up Θ, θ in Wiktionary, the free dictionary. ... This article is about an authentication, authorization, and accounting protocol. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... The circumference is the distance around a closed curve. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...

Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, defining them as infinite series and presenting "Euler's formula" e^ix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec. Euler redirects here. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory and Colin Maclaurin were also very influential in the development of trigonometric series. Brook Taylor (August 18, 1685 – November 30, 1731) was an English mathematician. ... James Gregory For other people with the same name, see James Gregory. ... Colin Maclaurin Colin Maclaurin (February, 1698 - June 14, 1746) was a Scottish mathematician. ...

See also

• Greek mathematics
• History of mathematics
• Trigonometric functions
• Trigonometry

Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... For a timeline of events in mathematics, see timeline of mathematics. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ...

Citations and footnotes

Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... Donald Routledge Hill (1922–1994) was an engineer and historian of science. ...

References

• Boyer, Carl B. (1991). A History of Mathematics, Second Edition, John Wiley & Sons, Inc.. ISBN 0471543977. 
• Gauchet, L. (1917). Note Sur La Trigonométrie Sphérique de Kouo Cheou-King.
• Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
• Katz, Victor J. (2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton: Princeton University Press. ISBN 0691114854.
• Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
• O'Connor, J.J., and E.F. Robertson, "Trigonometric functions", MacTutor History of Mathematics Archive. (1996).
• O'Connor, J.J., and E.F. Robertson, "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2000).
• Pearce, Ian G., "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2002).
• Restivo, Sal. (1992). Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. ISBN 1402000391.