History of Trigonometry

Running Head: History of Trigonometry History of Trigonometry Rome Fiedler History of Mathematics 501 University of Akron April 29, 2012 History of Trigonometry: An Introduction Trigonometry is useful in our world. By exploring where these concepts come from provides an understanding in putting this mathematics to use. The term Trigonometry comes from the Greek word trigon, meaning triangle and the Greek word meatria meaning measurement. However it is not native to Greek in origin. The mathematics comes from multiple people over a p of thousands of years and has touched over every major civilization.

It is a combination of geometry, and astronomy and has many practical applications over history. Trigonometry is a branch of math first created by 2nd century BC by the Greek mathematician Hipparchus. The history of trigonometry and of trigonometric functions sticks to the general lines of the history of math. Early research of triangles could be found in the 2nd millennium BC, in Egyptian and Babylonian math. Methodical research of trigonometric functions started in Greek math, and it reached India as part of Greek astronomy.

In Indian astronomy, the research of trigonometric functions flourished in the Gupta dynasty, particularly as a result of Aryabhata. Throughout the Middle Ages, the research of trigonometry continued in Islamic math, while it was implemented as a discrete subject in the Latin West beginning in the Renaissance with Regiomontanus. The growth of contemporary trigonometry shifted in the western Age of Enlightenment, starting with 17th-century math and reaching its contemporary type with Leonhard Euler (1748) Etymology The word "trigonometry" originates from the Greek "trigonometria", implying "triangle measuring", from triangle + to measure.

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The name developed from the study of right triangles by applying the relation ships between the measures of its sides and angles to the study of similar triangles (Gullberg, 1996). The word was introduced by Barthoolomus ptiticus in the title of his work Trigonometria sice de solutione triangularumtractus brevis et perspicius… in 1595. The contemporary word "sine", 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 "chord".

Sanskrit jiva in learned used was a synonym of jya "chord", primarily the word for "bow-string". Sanskrit jiva was taken into Arabic as jiba (Boyer, 1991). This word was then changed into the real Arabic word jayb, implying "bosom, fold, bay", either by the Arabs or erroneously of the European translators such as Robert of Chester, who translated jayb into Latin as sinus. In particular Fibonacci's sinus rectus arcus was significant in creating the word sinus. Early Beginnings The origin of the subject has rich diversity. Trigonometry is not the work of one particular person or place but rather a development over time.

The primitive Egyptians and Babylonians had known of theorems on the ratios of the sides of analogous triangles for many 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 known as "trilaterometry"(Boyer, 1991). The Babylonian astronomers kept comprehensive records on the rising and setting of stars, the movement of the planets, and the solar and lunar eclipses, all of which needed knowledge with angular distances measured on the celestial sphere.

Founded on one explanation of the Plimpton 322 cuneiform tablet, some have even claimed that the primitive Babylonians had a table of secants. There was, on the other hand, much discussion as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table. The Egyptians, in contrast, applied an ancient kind of trigonometry for construction of pyramids and surveying the land in the 2nd millennium BC. The early beginnings of trigonometry ar thought to be the first numerical sequences correlating shadow lengths to time of day.

Shadow tables were simple sequences of numbers which applied the shadow of a vertical stick, called a gnomon, is long in the morning and shortens to a minimum at noon. Then becomes longer and longer as the afternoon progresses (Kennedy, 1969). The shadow tables would correlate a particular hour to a particular length and were used as early as 1500 BC by the Egyptians. Similar tables were developed by other civilizations such as the Indians and Greeks. Greek mathematics Shadow tables were the primary development in creation of trigonometry however the Greeks really developed Trigonometry into an ordered science.

The Greeks continued as the Babylonians astronomers did and studied the relation between angles and circles in lengths of chords to develop their theories on planetary 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 chord's perpendicular bisector traverses the center of the circle and bisects the angle. One half of the bisected chord is the sine of the bisected angle, that is, [pic] nd consequently the sine function is also known as the "half-chord". As a result of this relationship, several trigonometric identities and theorems that are known at present were also known to Greek mathematicians, however in their equivalent chord form. Though there is no trigonometry in the works of Euclid and Archimedes, there are theorems presented in a geometric method that are similar to particular trigonometric laws or rules. Theorems on the lengths of chords are applications of the law of sines. In addition Archimedes' theorem on broken 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 expansion of astronomical knowledge from the Babylonians, there was a gap in the improvement of trigonometry until the time of Hipparchus. Hipparchus The first trigonometric table was in fact compiled by Hipparchus of, who is known as an as "the father of trigonometry"(Boyer, 1991). Hipparchus was the first to put into a table the corresponding values of arc and chord for a series of angles. He did this by considering every triangle was inscribed in a circle of fixed radius. Each side of the triangle became a chord, a straight line drawn between two points on a circle.

To find the parts of the triangle he needed to find the length of the chord as a function of the central angle. [pic] For Example, in the diagram 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 formula for the? length of a chord sketched from one point on the circumference of? a circle to another (Motz, 1993). This could therefore be used to help understand the positioning of the planets on the sphere.

Though it is not known when the methodical use of the 360° circle came into math, it is known that the methodical introduction of the 360° circle introduced a little 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 systematic used of the 360° circle was mainly as a result of Hipparchus and his table of chords. Hipparchus might have taken the idea of that division from Hypsicles who had previously divided the day into 360 parts, a division of the day that might have been recommended by Babylonian astronomy.

In primeval astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A recurring cycle of approximately 360 days 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 degree was divided into 60 minutes and each minute was divided into 60 seconds. Though Hipparchus is attributed as the father of trigonometry all of his work is lost except one but we gain knowledge of his work through Ptolemy. [pic] http://www. ies. co. p/math/java/vector/menela/menela. html Menelaus Menelaus of Alexandria wrote in three books his Sphaerica. In Book I, he created a basis for spherical triangles analogous to the Euclidean basis for plane triangles. He established a theorem that is without Euclidean analogue, that two spherical triangles were similar if corresponding angles are equal, however he did not differentiate between congruent and symmetric spherical triangles. Another theorem that he established was that the sum of the angles of a spherical triangle is more than 180°. Book II of Sphaerica applied spherical geometry to astronomy.

In addition Book III contained the "theorem of Menelaus"(Boyer, 1991). He further gave his well-known "rule of six quantities"(Needham, 1986). This theorem came to paly a major role in spherical trigonometry and astronomy. It was also believed that Melaus mya have developed a second table of chords based 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 Mathematical Syntaxis. The Almagest was mainly a work on astronomy, and astronomy relied on trigonometry.

The 13 books of the Almagest were the most prominent and important trigonometric work of ancient times. This book was a composition 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 distinguished that Menelaus started by dividing a circle into 360o, and the diameter into 120 parts. He did this because 3 x 120 = 360, using the previous application of 3 for pi.

Then each part is divided into sixty 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 only system available at the time for handling 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 same 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’ construction _ crd · 2_ and said that the complement angle could be written 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_), using a similar argument as the one shown above (van Brummelen, 2009). From these two expressions, one of the greatest 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). [pic]http://nrich. maths. org/6853 [pic] http://en. ikipedia. org/wiki/Ptolemy's_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 Ptolemy's calculation of chords was what was still known at present as Ptolemy's theorem, that the sum of the products of the opposite sides of a recurring quadrilateral 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 Ptolemy’s discoveries, given a chord of? an arc in a circle, the chord of half an arc can be determined 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 made 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 surface 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 interest and application of astronomers. In fact, spherical trigonometry was the most prevalent branch of trigonometry until the 1450s, even though Ptolemy did introduce a basis for plane trigonometry in the Almagest in 150 A. D. India

The next major contribution to trigonometry came from India. The trigonometry of Ptolemy was based on the functional relationship between chords of a circle and central angles they subtend. The Siddhantas, a book thought to be written by Hindu scholars in late fourth century, early fifth century A. D. , changed Ptolemy’s trigonometry to the study of the relationship between half of a chord of a circle and half of the angle subtended at he center by the whole 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 transformation 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 cosine. 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 simply meant, “chord” (Smith 615). Sines were given in minutes, at intervals of 225 minutes. This measurement 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 total 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-Bhata’s calculations were inaccurate. 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. Islamic mathematics

The ancient works were translated and developed in the medieval Islamic world by Muslim mathematicians of mostly Persian and Arab descent, who explained a large number of theorems which freed the subject of trigonometry from reliance upon the complete 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 actually (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 compute the place of the moon and stars, although that method proved to be ungainly and complex.

It engaged creation of two intersecting 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 sun's 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. Abu’l-Wafa is believed to have helped introduced the concept of the tangent function. He also may have had something to do with the development 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 Abu’l-Wafa, even? hough it was first introduced by Ptolemy. This is in part? due to the fact that Abu’l-Wafa presented a? straightforward 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. Western Europe favored Arabic mathematics over 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 acquainted 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 than that, little development occurred from the time of the 1200’s to the 1500’s, aside for the developments of the Germans in the late 15th and early 16th century.

Germany was becoming a prosperous nation at the time and was engaged in much trade. Their interests 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 around 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 finally 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 determine 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 respect to arcs of circles were disregarded. Francois Viete who practiced law and spent his leisure time devoted to mathematics also . contributed trigonometry around 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 focus 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 emphasis to a functional approach. This formed the basis of 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 ordinary 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 infinite 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 are periodic functions as well. The use of trigonometric series was introduced to determine the positions of the planets and interpolation, which is a mathematical procedure 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 invention of the Scientific Calculator in 1968. In society 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 minimal information available on the history of Trigonometry it is still a vital part of mathematics. The History 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. By exploring the history of trigonometry we see the importance of it in our world. References Boyer, Carl B. (1991), A History of Mathematics (Second ed. ). John Wiley & Sons, Inc. 3 Bressoud, D. M. (2010). Historical Refelctions on Teaching Trigonometry. Mathematics Teacher, 104 (2), 106-112. Brummelen, G. V. (2009). The Mathematics of the Heavens and the Earth. Princeton, NJ: Princeton University Press. Gingerich, Owen (1986), "Islamic astronomy". Scientific American 254 (10): 74.

Gullberg, Jan. (1996)Mathematics from the Birth Of Numbers. New York:W. W. Norton and Company, Inc. Joyce, D. E. (n. d. ). History of Trigonometry Outline. Retrieved 3 21, 2012, from History of Trigonometry Outline: http://aleph0. clarku. edu/~djoyce/ma105/trighist. html Kennedy, E. S. (1969), "The History of Trigonometry". 31st Yearbook (National Council of Teachers of Mathematics, Washington DC) (cf. Haq, Syed Nomanul. The Indian and Persian background. pp. 60–3, in Seyyed Hossein Nasr, Oliver Leaman (1996). History of Islamic Philosophy. Routledge. pp. 52–70.

Kline, Morris. (1972) Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. Kluemper, A. (2010, 3 24). History of Trigonometry. Retrieved 3 5, 2012, from www. xtimeline. com: http://www. xtimeline. com/timeline/History-of-Trigonometry Mankiewicz, Richard. (2001)The Story of Mathematics. New Jersy:Princetion University Press. Maor, E. (1998). Trigonometric Delights. New Jersey: Princeton University Press. Miller, S. (2001). Understanding Transformations of Periodic Functions through Art. Mathematics Teacher , 94 (8), 632-635.

Moussa, Ali (2011), "Mathematical Methods in Abu al-Wafa's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy. Cambridge University Press. 21 (1): 1–56. Needham, Joseph (1986), Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. Rogers, L. (n. d. ). The History of Trigonometry- Part 1. Retrieved 3 1, 2012, from Enriching Mathematics: http://nrich. maths. org/6843/index Suzuki, J. (2009). Mathematics in Historical Context. Washington D. C. : The Mathematical Association of America.

Smith, D. E. (1958)History of Mathematics. New York:Dover Publications, Inc. Toomer, G. J. (1998), Ptolemy's Almagest, Princeton University Press. Weber, K. (2005). Students Understanding of Trigonometric Functions. Mathematics Education Research Journal , 17 (3), 91-112. www. cartage. org. (n. d. ). Trigonometry History. Retrieved 3 5, 2012, from Trigonometry History: http://www. cartage. org. lb/en/themes/sciences/Mathematics/Trigonometry/history/History%20. html van Brummelen, G. (2009)The Mathematics of the Heavens and Earth. Princeton University Press. Princeton and Oxford.

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