History of Trigonometry

Running Head bill of trig level of trig Rome Fiedler autobiography of math 501 University of Akron April 29, 2012 storey of trig An Introduction trig is useful in our world. By exploring where these concepts come from provides an rationality in putting this math to use. The term trig comes from the Hellenic phrase trigon, meaning triplicity and the Hellenic word meatria meaning eyeshadement. However it is non primal to Greek in bu sinss breeze. The maths comes from multiple plurality everyplace a span of thousands of forms and has affected over every major civilization.It is a faction of geometry, and astronomy and has m either(prenominal) an(prenominal) practical coverings over story. trigonometry is a branch of math startle created by warrant century BC by the Greek mathematician Hipp cuthus. The history of trig and of trigonometric subr byines sticks to the general lines of the history of math. Early query of tri weights could be found in the 2 nd millennium BC, in Egyptian and Babylonian math. Methodical rese waiverh of trigonometric comp adeptnts started in Greek math, and it r severallyed India as lay out of Greek astronomy.In Indian astronomy, the rese flickerh of trigonometric functions flourished in the Gupta dynasty, busyly as a result of Aryabhata. Throughout the Middle Ages, the research of trig continued in Moslem math, while it was enforced as a discrete clear in the Latin West beginning in the metempsychosis with Regiomontanus. The growth of coetaneous trig shifted in the occidental Age of Enlightenment, starting with 17th-century math and reaching its contemporary type with Leonhard Euler (1748) Etymology The word trig originates from the Greek trigonometria, implying triangle measuring, from triangle + to measure.The name develop from the guinea pig of honor of chastise triangles by applying the carnal fuckledge ships amidst the measures of its sides and angles to the field of identical tri angles (Gullberg, 1996). The word was go ind by Barthoolomus ptiticus in the epithet of his hunt d sustain Trigonometria sice de soluti adept triangularumtractus brevis et perspicius in 1595. The contemporary word hell, is originated from the Latin word sinus, which implied bay, bosom or fold, translation from Arabic word jayb. The Arabic word is in origin of version of Sanskrit jiva fit in.Sanskrit jiva in learned utilize was a alike word of jya harmonize, primarily the word for bow-string. Sanskrit jiva was interpreted into Arabic as jiba (Boyer, 1991). This word was past changed into the real Arabic word jayb, implying bosom, fold, bay, either by the Arabs or err building blockaryously of the European translators such as Robert of Chester, who translated jayb into Latin as sinus. In particular Fibonaccis sinus rectus arcus senilis was significant in creating the word sinus. Early Beginnings The origin of the subject has rich diversity. trig is not the train of one part icular person or space alone rather a development over season.The primitive Egyptians and Babylonians had cognise of theorems on the ratios of the sides of analogous triangles for m whatever an(prenominal) centuries. However pre-Greek societies were deficient of the concept of an angle measure and as a result, the sides of triangles were analyzed rather, a field that would be better cognise as trilaterometry(Boyer, 1991). The Babylonian astronomers kept comprehensive records on the rising and move of stars, the movement of the categoricts, and the solar and lunar eclipses, all of which postulate association with angular distances deliberate on the aerial sphere.Founded on one explanation of the Plimpton 322 cunei shape panelt, virtually take in even claimed that the primitive Babylonians had a set back of secants. There was, on the other hand, much intelligence as to whether it is a flurry of Pythagorean triples, a solution of quadratic equations, or a trigonometri c send back. The Egyptians, in contrast, applied an ancient kind of trig for rebuildion of pyramids and surveying the land in the 2nd millennium BC. The aboriginal beginnings of trigonometry ar plan to be the firstly numerical sequences correlating shadow lengths to age of mean solar day.Shadow remits were simple sequences of numbers which applied the shadow of a vertical stick, called a gnomon, is long in the forenoon and shortens to a minimum at noon. Then becomes lengthy and longer as the afternoon progresses (Kennedy, 1969). The shadow tables would correlate a particular hour to a particular length and were used as earlier as 1500 BC by the Egyptians. Similar tables were real by other civilizations such as the Indians and Greeks. Greek mathematics Shadow tables were the primary development in cosmos of trigonometry however the Greeks really certain trig into an arrayed science.The Greeks continued as the Babylonians astronomers did and studied the relation mingle d with angles and bandings in lengths of harmonises to develop their theories on global position and motion (Mankiewicz, 2001). pic The chord of an angle subtends the arc of the angle. Ancient Greek mathematicians used the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chords perpendicular bisector traverses the center of the circle and bisects the angle. wiz half of the bisected chord is the sin of the bisected angle, that is, pic nd accordingly the sinfulness function is as well cognize as the half-chord. As a result of this relationship, several trigonometric identities and theorems that ar know at bear were likewise known to Greek mathematicians, however in their equivalent chord form. Though there is no trigonometry in the works of Euclid and Archimedes, there atomic number 18 theorems presented in a geometric system that atomic number 18 similar to particular trigonometric righteousnesss or rules. Theorems on the lengths of chords are applications of the law of sines. In addition Archimedes theorem on mazed chords is similar to rules for sines of sums and differences of angles.From the primitive landmarks of shadow tables and the Greeks gain and stretch outing upon of astronomic knowledge from the Babylonians, there was a flutter in the improvement of trigonometry until the period of Hipparchus. Hipparchus The first trigonometric table was in fact compiled by Hipparchus of, who is known as an as the commence of trigonometry(Boyer, 1991). Hipparchus was the first to put into a table the constitute values of arc and chord for a serial publication of angles. He did this by considering every triangle was engrave in a circle of fixed radius. to each one side of the triangle became a chord, a neat line drawn between two points on a circle.To point out the parts of the triangle he wanted to find the length of the chord as a function of the underlying angle. pic For Example, in the plot triangle ACB is? inscribed in circle O. So the sides of the triangle become chord? AC, chord CB and chord AB. Hipparchus would have sought to? find the length of the chord, AC, as a function of the central? angle. He deduced a trigonometric ordinance for the? length of a chord sketched from one point on the circuit of? a circle to another (Motz, 1993). This could therefore be used to help understand the positioning of the run downts on the sphere.Though it is not known when the mannerical use of the 360 circle came into math, it is known that the methodical introduction of the 360 circle introduced a teentsy after Aristarchus of Samos comprised of On the Sizes and Distances of the Sun and Moon, since he measured an angle a part of a quadrant. It seemed that the magisterial used of the 360 circle was generally as a result of Hipparchus and his table of chords. Hipparchus might have taken the idea of that division from Hypsicles who had previously carve up the day into 36 0 parts, a division of the day that might have been recommended by Babylonian astronomy.In primeval astronomy, the zodiac had been shared into cardinal signs or thirty- half-dozen decans. A recurring unit of ammunition of approximately 360 long time could have corresponded to the signs and decans of the zodiac by dividing each sign into 30 parts and each decan into 10 parts. It was as a result of the Babylonian sexagesimal numeral system that each peak was divided into 60 minutes and each minute was divided into 60 seconds. Though Hipparchus is attri barelyed as the father of trigonometry all of his work is lost except one barely we gain knowledge of his work through and through Ptolemy. pic http//www. ies. co. p/math/java/vector/menela/menela. hypertext markup language Menelaus Menelaus of Alexandria wrote in three defys his Sphaerica. In Book I, he created a basis for globose triangles analogous to the euclidian basis for plane triangles. He established a theorem that is w ithout Euclidean analogue, that two orbiculate triangles were similar if alike angles are equal, however he did not specify between congruent and symmetric spherical triangles. some other theorem that he established was that the sum of the angles of a spherical triangle is more(prenominal) than than 180. Book II of Sphaerica applied spherical geometry to astronomy.In addition Book deuce-ace contained the theorem of Menelaus(Boyer, 1991). He but gave his well-known rule of sestet quantities(Needham, 1986). This theorem came to paly a major role in spherical trigonometry and astronomy. It was in any case believed that Melaus mya have essential a second table of chords ground on Hipparchus works, however these were lost (Smith, 1958). Ptolemy Afterwards, Claudius Ptolemy developed upon Hipparchus Chords in a Circle in his Almagest, or the numeral Syntaxis. The Almagest was of importly a work on astronomy, and astronomy relied on trigonometry.The 13 books of the Almagest w ere the most prominent and important trigonometric work of ancient times. This book was a typography of both astronomy and trigonometry and was derived from the work of Hipparchus and Menelaus. Almagest contains a table of lengths of chords in a circle and a detailed set of instructions on how to construct the table. These instructions contain some of the earliest derivtions of trigonometry. Ptolemy magisterial that Menelaus started by dividing a circle into 360o, and the diameter into cxx parts. He did this because 3 x 120 = 360, development the previous application of 3 for pi.Then each part is divided into cardinal parts, each of these again into sixty parts, and so on. This system of parts was based on the Babylonian sexagesimal or base 60-numeration system, which was the moreover system available at the time for treatment fractions (Maor, 1998). This system was based on 60 so that the number of degrees corresponding to the circumference of a circle would be the homogeneo us as the number of days in a year, which the Babylonians believed to be 360 days (Ball 1960). From Menlaus Ptolemy developed the concept that the sine is half of a chord.Ptolemy took Menelaus winding _ crd 2_ and said that the complement angle could be scripted as _ crd (180 o -2_), since 180o was half the circumference of the circle. Since today, cos_ = sin(90 o -_), it can be shown that cos_ = _ crd (180 o -2_), development a similar argument as the one shown above (van Brummelen, 2009). From these two expressions, one of the enormousest identities known today was created. That is, (_ crd 2_) 2 + _ crd (180 o -2_) 2 = 1 which is exactly sin2_ + cos2_ = 1 (van Brummelen, 2009). pichttp//nrich. maths. org/6853 pic http//en. ikipedia. org/wiki/Ptolemys_table_of_chords Using his table, Ptolemy believed that one could solve any planar triangle, if given at least one side of the triangle (Maor, 1998). A theorem that was fundamental to Ptolemys tally of chords was what was s till known at present as Ptolemys theorem, that the sum of the products of the opposite sides of a recurring many-sided was equivalent to the product of the diagonals. Ptolemy used these results to develop his trigonometric tables however whether these tables were originated from Hipparchus work could not be proved.Neither the tables of Hipparchus nor those of Ptolemy had survived to the present day, though descriptions by other ancient authors exhibits they existed. In his work, Ptolemy founded formulas for the chord of? difference and an equivalent for our modern day half-angle? formulas. Because of Ptolemys discoveries, given a chord of? an arc in a circle, the chord of half an arc can be assignd as? well. Ptolemy also discovered chords of sum and difference, chords of half an arc, and chords of half degree, from which he then built up his tables to the nearest second of chords of arcs from half degree.In the Almagest, a true distinction was make up between plane and spherical trigonometry. Plane trigonometry is the branch of trigonometry which applies its principles to plane triangles Spherical trigonometry, on the other hand, is the branch of trigonometry in which its principles are applied to spherical triangles, which are triangles on the move up of the sphere. Ptolemy began with spherical trigonometry, for he worked with spherical triangles in many of his theorems and proofs. However, when calculating the chords of arcs, he unintentionally developed a theory for plane trigonometry. Trigonometry was created for use in astronomy and because spherical trigonometry was for this purpose the more useful tool, it was the first to be developed. The use of plane trigonometry is foreign to Greek mathematicians (Kline, 1972). Spherical trigonometry was developed out of necessity for the divert and application of astronomers. In fact, spherical trigonometry was the most popular branch of trigonometry until the 1450s, even though Ptolemy did introduce a basis for plane trigonometry in the Almagest in 150 A. D. IndiaThe next major parcel to trigonometry came from India. The trigonometry of Ptolemy was based on the operational relationship between chords of a circle and central angles they subtend. The Siddhantas, a book theme to be write by Hindu scholars in late fourth century, early fifth century A. D. , changed Ptolemys trigonometry to the need of the relationship between half of a chord of a circle and half of the angle subtended at he center by the livelong chord (Kennedy, 1969). This came from the basis for the modern trigonometric function known as the sine.The Siddhantas introduction to the sine function is the chief contribution from India and marks a displacement in trigonometry. Indian mathematicians also contributed by creating their own sine table. Arya-Bhata, born in 476, was a great Indian mathematician and astronomer (Ball, 1960). He composed a book called Aryabhathiya, which contained most of the essential ideas we associate with sine and cos. His most outstanding contribution to the topic, which distinguishes him from the other mathematicians of this time, was his work on sine differences (van Brummelen, 2009).His definition of sine was literally half chord and was abbreviated jya or jiva, which exactly meant, chord (Smith 615). Sines were given in minutes, at intervals of 225 minutes. This criterion was not of the sines themselves, but instead, it was the measurement of the differences between the sines. His method of calculating them was as follows. The first sine was equal to 225. The second sine was defined as any particular sine being worked with in order to calculate the sine that directly follows (Clark 29).It was found using the following pattern (225 the previous sine) + (225 + the previous sine) 225 this come in was then subtracted from 225 to obtain the sine table. Second sine 225 225 = 0 225 / 225 = 1 0 + 1= 1 225 1 = 224 Third sine? 225 224 = 1 (225 + 224) / 225 ? 2 225 2 = 222 (van Brummelen, 2009). Arya-Bhata concluded that dividing a quarter of the circumference of a circle (essentially one quadrant of the unit circle) into as many equal parts, with the resulting triangles and quadrilaterals would have, on the radius, the same amount of sines of equal arcs.Doing this, he was able to form a table of natural sines corresponding to the angles in the first quadrant (van Brummelen, 2009). Although much of his work had the right idea, many of Arya-Bhatas calculations were in spotless. Later, in 1150AD, an Indian mathematician known as Bhaskara gave a more accurate method of constructing a table of sines, which considered sines in every degree (van Brummelen, 2009). Although the Indian mathematicians made attempts at creating a table to help with astronomy, their table of sines was not as accurate as that of the Greeks. Muslim mathematicsThe ancient works were translated and developed in the medieval Islamic world by Muslim mathematicians of mostly Iranian and Arab descent, who explained a medium-large number of theorems which freed the subject of trigonometry from reliance upon the come quadrilateral, as was the case in Greek mathematics as a result of the application of Menelaus theorem. In accordance with E. S. Kennedy, it was following that development in Islamic math that the first real trigonometry appeared, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles (Kennedy, 1969).E. S. Kennedy pointed out that whilst it was possible in pre-Islamic math to calculate the magnitudes of a spherical figure, in theory, by use of the table of chords and Menelaus theorem, the application of the theorem to spherical problems was very complex very (Kennedy, 1969). With the aim of observing holy days on the Islamic calendar in which timings were established by phases of the moon, astronomers at first used Menalaus method to lick the place of the moon and stars, although th at method proved to be ungainly and complex.It engaged creation of two cross right triangles by applying Menelaus theorem it was possible to solve one of the 6 sides, however only if the other 5 sides were known. To tell the time from the suns elevation, for example, repeated applications of Menelaus theorem were needed. For medieval Islamic astronomers, there was a clear challenge to find a simpler trigonometric rule (Gingerich, 1986). In the early 9th century, Muhammad ibn Musa al-Khwarizmi c a Persian Mathematician, was an early pioneer in spherical trigonometry and wrote a treatise on the subject creating accurate sine and cosine tables.By the 10th century, in the work of Abu al-Wafa al-Buzjani, another Persian Mathematician established the angle addition formulas, e. g. , sin(a + b), and discovered the sine formula for spherical trigonometry. Abul-Wafa is believed to have helped introduced the concept of the topaz function. He also may have had something to do with the develo pment of secant and cosecant. His trigonometry took on a more systematic form in which he proved theorems for double and half angle formulas. The law of sines, is also attributed to Abul-Wafa, even? hough it was first introduced by Ptolemy. This is in part? due to the fact that Abul-Wafa presented a? unreserved formulation of the law of sines for? spherical triangles, which states pic where A, B, and C are surface angles of the spherical? triangle and a, b, and c are the central angles of the? spherical triangle. In 830, Habash al-Hasib al-Marwazi created the first table of cotangents. Muhammad ibn Jabir al-Harrani al-Battani found the reciprocal functions of secant and cosecant, and created the first table of cosecants for each degree from 1 to 90.By 1151 AD, the ideas of the six trigonometric functions existed, they were just not named as we know them today. Europe It is from the Arabic influence that trigonometry reached Europe. horse opera Europe favored Arabic mathematics ove r Greek geometry. Arabic arithmetic and algebra were on a more elementary level than Greek geometry had been during the time of the Roman Empire. Romans did not display much interest in Greek trigonometry or any facets of Greek math. Therefore, Arabic math appealed to them since it was easier for them to comprehend.Leonardo Fibonacci was one mathematician who became present with trigonometry during his extensive travels in Arab countries. He then presented the knowledge he gained in Practica geometriae in 1220 AD (Gullberg, 1996). The first distinction of trigonometry as a science separate from astronomy is credited to the Persian, Nasir Eddin. He helped to differentiate plane trigonometry and spherical trigonometry. other(a) than that, little development occurred from the time of the 1200s to the 1500s, aside for the developments of the Germans in the late 15th and early sixteenth century.Germany was becoming a prosperous nation at the time and was engaged in much trade. Their in terests also developed in navigation, calendar formation, and astronomy. This interest in astronomy precipitated a general interest and need for trigonometry (Kline, 1972). Included in this movement slightly the time of 1464, the German astronomer and mathematician, Regiomontanus (also known as Iohannes Molitoris) formulated a work known as De Triangulis Omnimodis, a compilation of the trigonometry of that time.When it was in the long run printed in 1533, it became an important medium of spreading the knowledge of trigonometry throughout Europe (Gullberg, 1996). The first book began with fifty propositions on the solutions of triangles using the properties of right triangles. Although the word sine was derived from the Arabs, Regiomontanus read the term in an Arabic manuscript in Vienna and was the first to use it in Europe. The second book began with a proof of the law of sines and then included problems involving how to larn sides, angles, and areas of plane triangles.The third book contained theorems found on Greek spherics before the use of trigonometry, and the fourth was based on spherical trigonometry. In the sixteenth century, Nicholas Copernicus was a revolutionary astronomer who could also be deemed as a trigonometer. He studied law, medicine and astronomy. He completed a treatise, known as De revolutionibus orbium coelestium, the year he died in 1543. This work-contained information on trigonometry and it was similar to that of Regiomontanus, although it is not clear if they were connected or not.While this was a great achievement, Copernicus student, Rheticus, an Indian mathematician, who lived during the years 1514-1576, went further and combined the work of both these men and published a two-volume work, Opus palatinum de triangulus. Trigonometry really began to expand and formalize at this point as the functions with enjoy to arcs of circles were disregarded. Francois Viete who practiced law and spent his leisure time devoted to mathematics also . contributed trigonometry approximately this time. He came to be known as the father of the generalized analytic approach to trigonometry (Boyer, 1991).He thought of trigonometry as? an independent branch of mathematics, and he worked? without direct reference to chords in a circle. He made? tables for all six trigonometric functions for angles to the? nearest minute. Viete was also one of the first to use the? formula for the law of tangents, which states the following pic Viete was one of the first mathematicians to point on analytical trigonometry, the branch of trigonometry which focuses on the relations and properties of the trigonometric functions.This form of trigonometry became more prevalent around the time of 1635 with the work of Roberval and Torricelli. They developed the first sketch of half an arch of a sine curve. This important development assisted in the progression of trigonometry from a computational fury to a functional approach. This formed the basis o f the European contribution of trigonometry. From the influence of oriental scientists, the Europeans focused on the computation of tables and the discovery of functional relations between parts of triangles.Europe developed appropriate symbols, which replaced the verbal rules and public language in which the subject was usually presented. Previously, trigonometry was expressed in lengthy passages of confusing words, but the Europeans introduced such symbols as sin, cos, tan, etc. to simplify the subject and make it more concise. Prior to the analytic approach, the main usage of trigonometry was to measure geometric figures, but the transition of its influence from geometry to calculus began with the discovery of uncounted series representations for the trigonometric functions.Trigonometric series became useful in the theory of astronomy, around the time of the eighteenth century. Since astronomical phenomena are periodic, it was useful to have trigonometric series because they ar e periodic functions as well. The use of trigonometric series was introduced to determine the positions of the planets and interpolation, which is a mathematical functioning that estimates the values of a function at positions between given values (Kline, 1972). Many continued to make contributions to Trigonometry looking for more accurate tables to determine the six functions.These works continued up until the designing of the Scientific Calculator in 1968. In indian lodge today, trigonometry is used in physics to aide in the understanding of space, engineering and chemistry. Within mathematics it is typically seen in mainly in calculus, but also in linear algebra and statistics. Despite the nominal information available on the history of Trigonometry it is still a vital part of mathematics. The storey shows progression from astronomy and geometry and the movement from spherical to plane geometry.Today, Trigonometry is used to understand space, engineering, chemistry as well as mathematics. 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