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Outline Of Trigonometry
Trigonometry is a branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. Basics * Geometry – mathematics concerned with questions of shape, size, the relative position of figures, and the properties of space. Geometry is used extensively in trigonometry. * Angle – the angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. * Ratio – a ratio indicates how many times one number contains another Content of trigonometry * Trigonometry * Trigonometric functions * Trigonometric identities * Euler's formula Scholars * Archimedes * Aristarchus * Aryabhata * Bhaskara I * Claudius Ptolemy * Euclid * Hipp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. 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. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus (mathematician), Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the philosophers Proclus and Pappus of Alexandria many centuries later. Until the early Renaissance he was often mistaken f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Marteloio
300px, The ''tondo e quadro'' (circle and square) from Andrea Bianco's 1436 atlas The rule of marteloio is a medieval technique of navigational computation that uses compass direction, distance and a simple trigonometric table known as the ''toleta de marteloio''. The rule told mariners how to plot the traverse between two different navigation courses by means of resolving triangles with the help of the ''Toleta'' and basic arithmetic. Those uncomfortable with manipulating numbers could resort to the visual ''tondo e quadro'' (circle-and-square) and achieve their answer with dividers. The rule of marteloio was commonly used by Mediterranean navigators during the 14th and 15th centuries, before the development of astronomical navigation. Etymology The etymology comes from the Venetian language. In his 1436 atlas, Venetian captain and cartographer Andrea Bianco introduced a table of numbers which he called the ''toleta de marteloio'' ("table of marteloio"), and the method of usi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ptolemy's Table Of Chords
The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's ''Almagest'', a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy (an earlier table of chords by Hipparchus gave chords only for arcs that were multiples of ). Centuries passed before more extensive trigonometric tables were created. One such table is the '' Canon Sinuum'' created at the end of the 16th century. The chord function and the table A chord of a circle is a line segment whose endpoints are on the circle. Ptolemy used a circle whose diameter is 120 parts. He tabulated the length of a chord whose endpoints are separated by an arc of ''n'' degrees, for ''n'' ranging from to 180 by increments o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Madhava's Sine Table
Madhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ..., and 90.00° (angles that are integral multiples of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is encoded in the letters of Devanagari using the Katapayadi system. This gives the entries in the table an appearance of the verses of a poem in Sanskrit. Madhava's original work containing the sine table has not yet been traced. The table is seen reproduced in the ''Aryabhatiyabhashya'' of Nilakantha Somayaji''The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part1-Gaṇitapāda,'' Edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.101. p. 55. https://archive.org/details/Trivandrum_Sanskrit_Series_TSS http://www.sanskritebooks.org/2013/02/t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jyā, Koti-jyā And Utkrama-jyā
Jyā, koṭi-jyā and utkrama-jyā are three trigonometric functions introduced by Indian mathematicians and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and koti-jyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of "sine" and "cosine" have been traced back to the Sanskrit words jyā and koti-jyā. Definition Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then: * ''jyā'' of arc AB = BM * ''koti-jyā'' of arc AB = OM * ''utkrama-jyā'' of arc AB = MA If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indian Astronomy
Astronomy has long history in Indian subcontinent stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley civilisation or earlier. Astronomy later developed as a discipline of Vedanga, or one of the "auxiliary disciplines" associated with the study of the Vedas,Sarma (2008), ''Astronomy in India'' dating 1500 BCE or older. The oldest known text is the ''Vedanga Jyotisha'', dated to 1400–1200 BCE (with the extant form possibly from 700 to 600 BCE). Indian astronomy was influenced by Greek astronomy beginning in the 4th century BCEHighlights of Astronomy, Volume 11B: As presented at the XXIIIrd General Assembly of the IAU, 1997. Johannes Andersen Springer, 31 January 1999 – Science – 616 pages. page 72/ref>Babylon to Voyager and Beyond: A History of Planetary Astronomy. David Leverington. Cambridge University Press, 29 May 2010 – Science – 568 pages. page 4/ref>The History and Practice of Anci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Astronomy
Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the Ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the language of scholarship throughout the Hellenistic world following the conquests of Alexander. This phase of Greek astronomy is also known as Hellenistic astronomy, while the pre-Hellenistic phase is known as Classical Greek astronomy. During the Hellenistic and Roman periods, much of the Greek and non-Greek astronomers working in the Greek tradition studied at the Museum and the Library of Alexandria in Ptolemaic Egypt. The development of astronomy by the Greek and notably Hellenistic astronomers is considered to be a major phase in the history of astronomy. Greek astronomy is characterized by seeking a geometrical model for celestial phenomena. Most of the names of the stars, planets, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bhaskara I's Sine Approximation Formula
In mathematics, Bhaskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhaskara I (c. 600 – c. 680), a seventh-century Indian mathematician. This formula is given in his treatise titled ''Mahabhaskariya''. It is not known how Bhaskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhaskara might have used to arrive at his formula. The formula is elegant, simple and enables one to compute reasonably accurate values of trigonometric sines without using any geometry whatsoever. The approximation formula The formula is given in verses 17 – 19, Chapter VII, Mahabhaskariya of Bhaskara I. A translation of the verses is given below: *(Now) I briefly state the rule (for finding the ''bhujaphala'' and the ''kotiphala'', etc.) without making use of the Rsine-differences 225, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aristarchus's Inequality
Aristarchus's inequality (after the Greek astronomer and mathematician Aristarchus of Samos; c. 310 – c. 230 BCE) is a law of trigonometry which states that if ''α'' and ''β'' are acute angles (i.e. between 0 and a right angle) and ''β'' < ''α'' then : used the first of these inequalities while constructing his table of chords. Proof The proof is a consequence of the more widely known inequalities :, : and :.Proof of the first inequality Using these inequalities we can first ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regiomontanus
Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrumental in the development of Copernican heliocentrism in the decades following his death. Regiomontanus wrote under the Latinized name of ''Ioannes de Monteregio'' (or ''Monte Regio''; ''Regio Monte''); the toponym ''Regiomontanus'' was first used by Philipp Melanchthon in 1534. He is named after Königsberg, Bavaria, Königsberg in Lower Franconia, not the larger Königsberg (modern Kaliningrad) in Prussia. Life Although little is known of Regiomontanus' early life, it is believed that at eleven years of age, he became a student at the University of Leipzig, Electorate of Saxony, Saxony. In 1451 he continued his studies at University of Vienna, Alma Mater Rudolfina, the university in Vienna, Duchy of Austria, Austria. There he became a pupil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, the Western philosophy, West in general. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Crotone, Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, asceticism, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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