History of trigonometry. Trigonometry in nature

History of trigonometry

Trigonometry is a Greek word and literally means the measurement of triangles ( - triangle, and  - I measure).

In this case, the measurement of triangles should be understood as the solution of triangles, i.e. determination of the sides, angles and other elements of a triangle, if some of them are given. A large number of practical problems, as well as problems of planimetry, stereometry, astronomy, and others, are reduced to the problem of solving triangles.

The emergence of trigonometry is associated with land surveying, astronomy and construction.

Although the name of science arose relatively recently, many of the concepts and facts now related to trigonometry were known two thousand years ago.

For the first time, methods for solving triangles based on dependencies between the sides and angles of a triangle were found by the ancient Greek astronomers Hipparchus (2nd century BC) and Claudius Ptolemy (2nd century AD). Later, the relationships between the ratios of the sides of a triangle and its angles began to be called trigonometric functions.

A significant contribution to the development of trigonometry was made by the Arab scientists Al-Batani (850-929) and Abu-l-Wafa, Mohamed-bin Mohamed (940-998), who compiled tables of sines and tangents in 10’ accurate to 1/60 4 . The sine theorem was already known by the Indian scientist Bhaskara (b. 1114, year of death unknown) and the Azerbaijani astronomer and mathematician Nasireddin Tusi Mukhamed (1201-1274). In addition, Nasireddin Tusi in his work "Treatise on the Complete Quadrilateral" outlined plane and spherical trigonometry as an independent discipline.

The concept of sine has a long history. In fact, various ratios of the segments of a triangle and a circle (and, in essence, trigonometric functions) are already found inIIIcentury BC in the works of the great mathematicians of ancient Greece - Euclid, Archimedes, Apollonius of Perga. In the Roman period, these relations were studied quite systematically by Menelaus (Icentury AD), although they did not acquire a special name. The modern sine , for example, was studied as a half-chord on which the central angle of magnitude  rests, or as a chord of a doubled arc.

 M

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Rice. one

IN IV- VFor centuries, a special term has already appeared in the works on astronomy of the great Indian scientist Aryabhata, after whom the first Indian satellite of the Earth is named. Segment AM (Fig. 1) he called ardhajiva (ardha - half, jiva - bowstring, which resembles a chord). Later, the shorter name jiva appeared. Arab mathematicians inIXcentury, this word was replaced by the Arabic word jaib (bulge). When translating Arabic mathematical texts in the century, it was replaced by the Latin sine (sinus- bend, curvature).

The word cosine is much younger. Cosine is an abbreviation of the Latin expressioncompletelysinus, i.e. “additional sine” (or otherwise “sine of an additional arc”;cos = sin(90  -  )).

Tangents arose in connection with the solution of the problem of determining the length of the shadow. The tangent (and also the cotangent) is introduced inXcentury by the Arab mathematician Abul-Wafa, who also compiled the first tables for finding tangents and cotangents. However, these discoveries remained unknown to European scientists for a long time, and tangents were rediscovered only inXIVcentury by the German mathematician, astronomer Regimontan (1467). He proved the tangent theorem. Regiomontanus also compiled detailed trigonometric tables; Thanks to his work, plane and spherical trigonometry became an independent discipline in Europe as well.

The name "tangent" comes from the Latintanger(to touch), appeared in 1583Tangentstranslated as “touching” (the line of tangents is tangent to the unit circle).

Trigonometry was further developed in the works of the outstanding astronomers Nicolaus Copernicus (1473-1543) - the creator of the heliocentric system of the world, Tycho Brahe (1546-1601) and Johannes Kepler (1571-1630), as well as in the works of the mathematician Francois Vieta (1540-1603), who completely solved the problem of determining all the elements of a flat or spherical triangle according to three data.

For a long time, trigonometry was purely geometric in nature, i.e., the facts that we now formulate in terms of trigonometric functions were formulated and proved with the help of geometric concepts and statements. It was like this even in the Middle Ages, although analytical methods were sometimes used in it, especially after the appearance of logarithms. Perhaps the greatest incentives for the development of trigonometry arose in connection with the solution of astronomical problems, which was of great practical interest (for example, for solving problems of determining the location of a ship, predicting blackout, etc.). Astronomers were interested in the relationship between the sides and angles of spherical triangles. And it should be noted that the mathematicians of antiquity successfully coped with the tasks set.

Beginning with XVIIcentury, trigonometric functions began to be applied to solving equations, problems of mechanics, optics, electricity, radio engineering, to describe oscillatory processes, wave propagation, the movement of various mechanisms, to study alternating electric current, etc. Therefore, trigonometric functions were comprehensively and deeply studied, and have become important for the whole of mathematics.

The analytic theory of trigonometric functions was mainly created by the eminent mathematicianXVIIIcentury Leonard Euler (1707-1783) a member of the St. Petersburg Academy of Sciences. Euler's vast scientific legacy includes brilliant results relating to calculus, geometry, number theory, mechanics, and other applications of mathematics. It was Euler who first introduced the well-known definitions of trigonometric functions, began to consider functions of an arbitrary angle, and obtained reduction formulas. After Euler, trigonometry took on the form of calculus: various facts began to be proved by the formal application of trigonometry formulas, proofs became much more compact, simpler,

Thus, trigonometry, which arose as the science of solving triangles, eventually developed into the science of trigonometric functions.

Later, the part of trigonometry that studies the properties of trigonometric functions and the relationships between them began to be called goniometry (in translation - the science of measuring angles, from the Greek  - angle,  - I measure). The term goniometry has not been used much in recent years.

The history of trigonometry is inextricably linked with astronomy, because it was to solve the problems of this science that ancient scientists began to study the ratios of various quantities in a triangle.

Today, trigonometry is a microsection of mathematics that studies the relationship between the values of the angles and lengths of the sides of triangles, as well as the analysis of algebraic identities of trigonometric functions.

The term "trigonometry"

The term itself, which gave its name to this branch of mathematics, was first discovered in the title of a book by the German mathematician Pitiscus in 1505. The word "trigonometry" is of Greek origin and means "I measure a triangle." To be more precise, we are not talking about the literal measurement of this figure, but about its solution, that is, determining the values of its unknown elements using the known ones.

General information about trigonometry

The history of trigonometry began more than two millennia ago. Initially, its occurrence was associated with the need to clarify the ratio of the angles and sides of the triangle. In the process of research, it turned out that the mathematical expression of these relationships requires the introduction of special trigonometric functions, which were originally drawn up as numerical tables.

For many sciences related to mathematics, it was the history of trigonometry that became the impetus for development. The origin of the units of measurement of angles (degrees), associated with the research of the scientists of Ancient Babylon, is based on the sexagesimal calculus, which gave rise to the modern decimal, used in many applied sciences.

Trigonometry is supposed to have originally existed as part of astronomy. Then it began to be used in architecture. And over time, the expediency of applying this science in various fields of human activity arose. These are, in particular, astronomy, sea and air navigation, acoustics, optics, electronics, architecture and others.

Trigonometry in the early centuries

Guided by data on surviving scientific relics, the researchers concluded that the history of the emergence of trigonometry is associated with the work of the Greek astronomer Hipparchus, who first thought about finding ways to solve (spherical) triangles. His writings date back to the 2nd century BC.

Also, one of the most important achievements of those times is the definition of the ratio of the legs and hypotenuse in right triangles, which later became known as the Pythagorean theorem.

The history of the development of trigonometry in Ancient Greece is associated with the name of the astronomer Ptolemy, the author of the geocentric system that prevailed before Copernicus.

Greek astronomers did not know sines, cosines and tangents. They used tables to find the value of the chord of a circle using a subtractive arc. The units for measuring the chord were degrees, minutes and seconds. One degree was equal to one sixtieth of the radius.

Also, the studies of the ancient Greeks advanced the development of spherical trigonometry. In particular, Euclid in his "Principles" gives a theorem on the regularities of the ratios of the volumes of balls of different diameters. His works in this area have become a kind of impetus in the development of related fields of knowledge. This, in particular, is the technology of astronomical instruments, the theory of cartographic projections, the system of celestial coordinates, etc.

Middle Ages: Studies of Indian Scholars

Medieval Indian astronomers achieved significant success. The death of ancient science in the 4th century led to the transfer of the center of development of mathematics to India.

The history of the emergence of trigonometry as a separate section of mathematical doctrine began in the Middle Ages. It was then that scientists replaced chords with sines. This discovery made it possible to introduce functions related to the study of sides and angles. That is, it was then that trigonometry began to separate from astronomy, turning into a branch of mathematics.

The first tables of sines were in Aryabhata, they were drawn through 3 o, 4 o, 5 o. Later, detailed versions of the tables appeared: in particular, Bhaskara gave a table of sines through 1 o.

The first specialized treatise on trigonometry appeared in the 10th-11th century. Its author was the Central Asian scientist Al-Biruni. And in his main work “Canon Masud” (book III), the medieval author goes even deeper into trigonometry, giving a table of sines (in increments of 15 ") and a table of tangents (in increments of 1 °).

History of the development of trigonometry in Europe

After the translation of Arabic treatises into Latin (XII-XIII century), most of the ideas of Indian and Persian scientists were borrowed by European science. The first mention of trigonometry in Europe dates back to the 12th century.

According to researchers, the history of trigonometry in Europe is associated with the name of the Englishman Richard of Wallingford, who became the author of the work "Four treatises on direct and reversed chords." It was his work that became the first work that is entirely devoted to trigonometry. By the 15th century, many authors in their writings mention trigonometric functions.

History of trigonometry: Modern times

In modern times, most scientists began to realize the extreme importance of trigonometry, not only in astronomy and astrology, but also in other areas of life. This is, first of all, artillery, optics and navigation in long-distance sea voyages. Therefore, in the second half of the 16th century, this topic interested many prominent people of that time, including Nicolaus Copernicus, Francois Vieta. Copernicus devoted several chapters to trigonometry in his treatise On the Revolutions of the Celestial Spheres (1543). A little later, in the 60s of the 16th century, Retik, a student of Copernicus, cites fifteen-digit trigonometric tables in his work “The Optical Part of Astronomy”.

In the "Mathematical Canon" (1579) he gives a detailed and systematic, albeit unproven, characterization of plane and spherical trigonometry. And Albrecht Dürer became the one thanks to whom the sinusoid was born.

Merits of Leonhard Euler

Giving trigonometry a modern content and form was the merit of Leonhard Euler. His treatise Introduction to the Analysis of Infinites (1748) contains a definition of the term "trigonometric functions" which is equivalent to the modern one. Thus, this scientist was able to determine But and that's not all.

The definition of trigonometric functions on the entire number line became possible thanks to Euler's studies not only of permissible negative angles, but also of angles greater than 360 °. It was he who first proved in his works that the cosine and tangent of a right angle are negative. The expansion of integer powers of cosine and sine also became the merit of this scientist. The general theory of trigonometric series and the study of the convergence of the resulting series were not objects of Euler's research. However, while working on solving related problems, he made many discoveries in this area. It was thanks to his work that the history of trigonometry continued. Briefly in his writings, he also touched on the issues of spherical trigonometry.

Applications of trigonometry

Trigonometry does not apply to applied sciences; in real everyday life, its problems are rarely used. However, this fact does not diminish its significance. Very important, for example, is the technique of triangulation, which allows astronomers to accurately measure the distance to nearby stars and control satellite navigation systems.

Trigonometry is also used in navigation, music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (for example, in decoding ultrasound, ultrasound and computed tomography), pharmaceuticals, chemistry, number theory, seismology, meteorology , oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography, etc. The history of trigonometry and its role in the study of natural and mathematical sciences are being studied to this day. Perhaps in the future there will be even more areas of its application.

The history of the origin of basic concepts

The history of the emergence and development of trigonometry has more than one century. The introduction of the concepts that form the basis of this section of mathematical science was also not instantaneous.

So, the concept of "sine" has a very long history. Mentions of various ratios of segments of triangles and circles are found in scientific works dating back to the 3rd century BC. The works of such great ancient scientists as Euclid, Archimedes, Apollonius of Perga already contain the first studies of these relationships. New discoveries required certain terminological clarifications. So, the Indian scientist Aryabhata gives the chord the name "jiva", meaning "bowstring". When Arabic mathematical texts were translated into Latin, the term was replaced by a sine (i.e., “bend”) that was close in meaning.

The word "cosine" appeared much later. This term is an abbreviated version of the Latin phrase "additional sine".

The emergence of tangents is associated with the decoding of the problem of determining the length of the shadow. The term "tangent" was introduced in the 10th century by the Arab mathematician Abul-Wafa, who compiled the first tables for determining tangents and cotangents. But European scientists did not know about these achievements. The German mathematician and astronomer Regimontan rediscovers these concepts in 1467. The proof of the tangent theorem is his merit. And this term is translated as "concerning".

Trigonometry arose and developed in antiquity as one of the branches of astronomy, as its computing apparatus; meeting the practical needs of the individual. It was astronomy that determined the fact that spherical trigonometry arose earlier than flat trigonometry.

Some trigonometric information was known to the ancient Babylonians and Egyptians, but the foundations of this science were laid in Ancient Greece. Ancient Greek astronomers successfully solved certain issues from trigonometry related to astronomy. However, they did not consider lines of sine, cosine, etc., but chords. The role of the line of sines of its angle in them was performed by a chord, contracting an arc equal to 2a.

Greek astronomer Hipparchus in the 2nd century. BC e. compiled a table of numerical values of chords, depending on the magnitude of the arcs contracted by them. More complete information from trigonometry is contained in the famous "Almagest" of Ptolemy.

Ptolemy divided the circumference into 360 degrees and the diameter into 120 parts. He considered the radius equal to 60 parts (60H). He divided each of the parts by 60", and each minute by 60", second by 60 thirds (60 ""), etc. In other words, he used the sexagesimal number system, in all likelihood, borrowed by him from Babylonians. Applying the indicated division, Ptolemy expressed the side of a regular inscribed hexagon or a chord subtending an arc of 60 ° in the form of 60 parts of a radius (60 H), and he equated the side of an inscribed square or a chord of 90 ° with the number 84 × 5110 ". A chord of 120 ° - the side of an inscribed equilateral triangle - he expressed the number 103 × 55 "23", etc.

Applying theorems known from geometry, the scientist found dependencies that are equivalent to the following modern formulas, provided:

Using these ratios and the values of the chords 60 ° and 72 ° expressed in parts of the radius, he calculated the chord that subtends the arc at 6 °, then 3 °; 1.5 ° and finally -0.75 °. (The value of the chord in He expressed approximately.)

The calculations made allowed Ptolemy to compile a table that contained chords from 0 to 180 °, calculated with an accuracy of 1 "radius.

This table, which has survived to our time, is equivalent to a table of sines from 0 to 90 ° in steps of 0.25 ° with five correct decimal places.

The names of the sine and cosine lines were first introduced by Indian scientists. They also compiled the first tables of sines, although less accurate than the Ptolemaic ones. In India, the doctrine of trigonometric quantities essentially begins, later called goniometry (from "gonia" - angle and "mehrio" - I measure).

The doctrine of trigonometric quantities was further developed in the IX-XV centuries. in the countries of the Middle and Near East in the works of a number of mathematicians who not only took advantage of the achievements in this field that existed at that time, but also made their significant contribution to science.

The famous Muhammad ibn Musa al-Khwarizmi (IX century) compiled tables of sines and cotangents. Al-Khabash or (Ahmed ibn Abdallah al-Marwazi) calculated tables for tangent, cotangent and cosecant.

The works of al-Battani (c. 850-929) and Abu-l-Vafa al-Buzjani (940-998) were of great importance in the development of trigonometry. The latter deduced the sine theorem of spherical trigonometry, calculated a table for sines with an interval of 15 ", the values in which are given with an accuracy of up to the 8th decimal place, found the segments corresponding to the secant and cosecant.

Abu Rayhan Muhammad ibn Ahmad-al-Beruni (according to another transcription of Biruni (973--1048)) summarized and at the same time specified the results achieved by his predecessors in the field of trigonometry. In the work "Canon Mas" ud "he outlined all the provisions of trigonometry known at that time and significantly supplemented them. Al-Beruni confirmed the important innovation undertaken by Abu-l-Vafa. Instead of dividing the radius into parts made by Ptolemy, they took unit radius Al-Beruni explained in detail the reason for this change, showing that all calculations with a unit radius are much simpler.

Nasir ad-Din Muhammad at-Tusi (1201--1274) in his "Treatise on the Complete Quadrilateral" for the first time presented trigonometric information as an independent branch of mathematics, and not an appendage to astronomy. His treatise later had a great influence on the work of Regiomontanus (1436--1476).

In the first half of the XV century. Jamshid ibn Masud al-Kashi calculated trigonometric tables with a step of c with great accuracy. G, which for 250 years remained unsurpassed.

In Europe of the XII-XV centuries, after some classical mathematical and astronomical works were translated from Arabic and Greek into Latin, the development of trigonometry continued. When solving flat triangles, the sine theorem was widely used, rediscovered by Leo Gersonides (1288-1344), who lived in southern France, whose trigonometry was translated into Latin in 1342. The most prominent European representative of this era in the field of trigonometry was Regiomontanus. His extensive tables of sines through Г with an accuracy of up to the 7th significant figure and his masterfully presented trigonometric work Five Books on Triangles of All Kinds were of great importance for the further development of trigonometry in the 16th-17th centuries.

On the threshold of the 17th century in the development of trigonometry, a new direction is outlined - analytical. If before that the main goal of trigonometry was considered to be the solution of triangles, the calculation of the elements of geometric figures and the doctrine of trigonometric functions was built on a geometric basis, then in the XVII-XIX centuries. trigonometry gradually becomes one of the chapters of mathematical analysis. It finds wide application in mechanics, physics and technology, especially in the study of oscillatory motions and other periodic processes. Viet knew about the property of periodicity of trigonometric functions, the first mathematical studies of which were related to trigonometry. The Swiss mathematician Johann Bernoulli (1642--1727) already used the symbols of trigonometric functions. And if the development of algebraic symbolism, the introduction of negative numbers and directed segments contributed to the expansion of the concept of angle and arc, then the development of the doctrine of oscillatory movements, of sound, light and electromagnetic waves led to the fact that the study and description of oscillatory processes became the main content of trigonometry. It is known from physics that the equation of a harmonic oscillation (for example, oscillations of a pendulum, alternating electric current) has the form:

Graphs of harmonic oscillations are sinusoids, therefore, in physics and technology, harmonic oscillations themselves are often called sinusoidal oscillations.

In the first half of the XIX century. the French scientist J. Fourier proved that any periodic motion can be represented (with any degree of accuracy) as a sum of simple harmonic oscillations.

The expansion of ideas about trigonometric functions led to their substantiation on a new, analytical basis: trigonometric functions are defined independently of geometry using power series and other concepts of mathematical analysis.

I. Newton and L. Euler contributed to the development of the analytic theory of trigonometric functions. L. Euler should be considered the founder of this theory. He gave the whole trigonometry a modern look. Further development of the theory was continued in the 19th century. N.I. Lobachevsky and other scientists. Nowadays, trigonometry is no longer considered as an independent branch of mathematics. Its most important part, the doctrine of trigonometric functions, is part of a more general theory, constructed from a unified point of view, of the theory of functions studied in mathematical analysis; the other part - the solution of triangles - is considered as the head of geometry (flat and spherical).

Ptolemy divided the circumference into 360 degrees and the diameter into 120 parts. He considered the radius equal to 60 parts (60H). He divided each of the parts by 60 ", and every minute by 60", second - by 60 thirds (60 ""), etc. In other words, he used the sexagesimal number system, most likely borrowed by him from the Babylonians . Using the indicated division, Ptolemy expressed the side of a regular inscribed hexagon or a chord subtending an arc of 60 ° in the form of 60 parts of a radius (60 H), and equated the side of an inscribed square or a chord of 90 ° to the number 84 × 5110 ". A chord of 120 ° is a side an inscribed equilateral triangle - he expressed the number 103 × 55 "23", etc.

Applying theorems known from geometry, the scientist found dependencies that are equivalent to the following modern formulas, provided:

Using these ratios and the values of the chords 60 ° "and 72 ° expressed in parts of the radius, he calculated the chord that subtends the arc at 6 °, then 3 °; 1.5 ° and, finally, -0.75 °. (The value of the chord in Г he was approximate.)

The calculations made allowed Ptolemy to compile a table that contained chords from 0 to 180 °, calculated with an accuracy of 1 "radius.

This table, which has survived to our time, is equivalent to a table of sines from 0 to 90 ° in steps of 0.25 ° with five correct decimal places.

The doctrine of trigonometric quantities was further developed in the 9th-15th centuries. in the countries of the Middle and Near East in the works of a number of mathematicians who not only took advantage of the achievements in this field that existed at that time, but also made their significant contribution to science.

The famous Muhammad ibn Musa al-Khwarizmi (IX century) compiled tables of sines and cotangents. Al-Khabash or (Ahmed ibn Abdallah al-Marwazi) calculated tables for tangent, cotangent and cosecant.

Abu Rayhan Muhammad ibn Ahmad-al-Beruni (according to another transcription of Biruni (973-1048)) summarized and at the same time specified the results achieved by his predecessors in the field of trigonometry. In the work "Canon Mas" ud "he outlined all the provisions of trigonometry known at that time and significantly supplemented them. Al-Beruni confirmed the important innovation undertaken by Abu-l-Vafa. Instead of dividing the radius into parts made by Ptolemy, they took unit radius Al-Beruni explained in detail the reason for this change, showing that all calculations with a unit radius are much simpler.

Nasir ad-Din Muhammad at-Tusi (1201-1274) in his "Treatise on the Complete Quadrilateral" for the first time presented trigonometric information as an independent branch of mathematics, and not an appendage to astronomy. His treatise later had a great influence on the work of Regiomontanus (1436-1476).

In the first half of the XV century. Jamshid ibn Masud al-Kashi calculated trigonometric tables with a step of c with great accuracy. G, which for 250 years remained unsurpassed.

In Europe in the 12th-15th centuries, after some classical mathematical and astronomical works were translated from Arabic and Greek into Latin, the development of trigonometry continued. When solving plane triangles, the sine theorem was widely used, rediscovered by Leo Gersonides (1288-1344), who lived in southern France, whose trigonometry was translated into Latin in 1342. The most prominent European representative of this era in the field of trigonometry was Regiomontanus. His extensive tables of sines through Г up to the 7th significant figure and his masterfully presented trigonometric work Five Books on Triangles of All Kinds were of great importance for the further development of trigonometry in the 16th-17th centuries.

On the threshold of the 17th century in the development of trigonometry, a new direction is outlined - analytical. If before that the main goal of trigonometry was considered to be the solution of triangles, the calculation of the elements of geometric figures and the doctrine of trigonometric functions was built on a geometric basis, then in the XVII-XIX centuries. trigonometry gradually becomes one of the chapters of mathematical analysis. It finds wide application in mechanics, physics and technology, especially in the study of oscillatory motions and other periodic processes. Viet knew about the property of periodicity of trigonometric functions, the first mathematical studies of which were related to trigonometry. The Swiss mathematician Johann Bernoulli (1642-1727) already used the symbols for trigonometric functions. And if the development of algebraic symbolism, the introduction of negative numbers and directed segments contributed to the expansion of the concept of angle and arc, then the development of the doctrine of oscillatory movements, of sound, light and electromagnetic waves led to the fact that the study and description of oscillatory processes became the main content of trigonometry. It is known from physics that the equation of a harmonic oscillation (for example, oscillations of a pendulum, alternating electric current) has the form:

Graphs of harmonic oscillations are sinusoids, therefore, in physics and technology, harmonic oscillations themselves are often called sinusoidal oscillations.

In the first half of the XIX century. the French scientist J. Fourier proved that any periodic motion can be represented (with any degree of accuracy) as a sum of simple harmonic oscillations.

I. Newton and L. Euler contributed to the development of the analytic theory of trigonometric functions. L. Euler should be considered the founder of this theory. He gave the whole trigonometry a modern look. Further development of the theory was continued in the 19th century. N.I. Lobachevsky and other scientists. Nowadays, trigonometry is no longer considered as an independent branch of mathematics. Its most important part - the doctrine of trigonometric functions - is part of a more general theory, built from a unified point of view, of the theory of functions studied in mathematical analysis; the other part - the solution of triangles - is considered as the head of geometry (flat and spherical).

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Report on the history of the emergence of trigonometric functions

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Report on mathematics

On the topic: "The history of the development of trigonometry"

Completed by a student:

Udalova Evgeniya

Groups: 1-T-1

Moscow 2012

The word trigonometry first occurs in 1505 in the title of a book by the German mathematician Pitiscus.

Trigonometry is a Greek word and literally means the measurement of triangles (trigwnon - a triangle, and metrew - I measure).

In this case, the measurement of triangles should be understood as the solution of triangles, that is, the determination of the sides, angles and other elements of the triangle, if some of them are given. A large number of practical problems, as well as problems of planimetry, stereometry, astronomy, and others, are reduced to the problem of solving triangles.

The emergence of trigonometry is associated with land surveying, astronomy and construction.

Although the name of science arose relatively recently, many of the concepts and facts now related to trigonometry were known two thousand years ago.

For the first time, methods for solving triangles based on dependencies between the sides and angles of a triangle were found by the ancient Greek astronomers Hipparchus (2nd century BC) and Claudius Ptolemy (2nd century AD). Later, the relationship between the ratios of the sides of a triangle and its angles began to be called trigonometric functions.

A significant contribution to the development of trigonometry was made by the Arab scientists Al-Batani (850-929) and Abu-l-Wafa, Mohamed-bin Mohamed (940-998), who compiled tables of sines and tangents through 10 "with an accuracy of 1/604. Theorem sines were already known by the Indian scientist Bhaskara (b. 1114, the year of death is unknown) and the Azerbaijani astronomer and mathematician Nasireddin Tusi Mukhamed (1201-1274). discipline.

The concept of sine has a long history. In fact, various ratios of segments of a triangle and a circle (and, in essence, trigonometric functions) are found already in the 3rd century BC. e. in the works of the great mathematicians of Ancient Greece - Euclid, Archimedes, Apollonius of Perga. In the Roman period, these relations were studied quite systematically by Menelaus (1st century AD), although they did not acquire a special name. The modern sine a, for example, was studied as a half-chord supported by a central angle of magnitude a, or as a chord of a doubled arc.

In the 4th-5th centuries, a special term appeared in the works on astronomy of the great Indian scientist Aryabhata, after whom the first Indian satellite of the Earth was named. He called the segment AM ardhajiva (ardha - half, jiva - bowstring, which resembles a chord). Later, the shorter name jiva appeared. Arab mathematicians in the 9th century replaced this word with the Arabic word jayb (bulge). When translating Arabic mathematical texts in the century, it was replaced by the Latin sine (sinus - bend, curvature).

Tangents arose in connection with the solution of the problem of determining the length of the shadow. The tangent (and also the cotangent) was introduced in the 10th century by the Arab mathematician Abul-Wafa, who also compiled the first tables for finding tangents and cotangents. However, these discoveries remained unknown to European scientists for a long time, and tangents were rediscovered only in the 14th century by the German mathematician, astronomer Regimontan (1467). He proved the tangent theorem. Regiomontanus also compiled detailed trigonometric tables; Thanks to his work, plane and spherical trigonometry became an independent discipline in Europe as well.

The name "tangent", which comes from the Latin tanger (to touch), appeared in 1583. Tangens is translated as "touching" (the line of tangents is tangent to the unit circle).

Starting from the 17th century, trigonometric functions began to be applied to solving equations, problems of mechanics, optics, electricity, radio engineering, to describe oscillatory processes, wave propagation, the movement of various mechanisms, to study alternating electric current, etc. Therefore, trigonometric functions are comprehensive and deeply studied, and acquired an important significance for all of mathematics.

The analytical theory of trigonometric functions was mainly created by the outstanding mathematician of the 18th century, Leonhard Euler (1707-1783), a member of the St. Petersburg Academy of Sciences. Euler's vast scientific legacy includes brilliant results relating to calculus, geometry, number theory, mechanics, and other applications of mathematics. It was Euler who first introduced the well-known definitions of trigonometric functions, began to consider functions of an arbitrary angle, and obtained reduction formulas. After Euler, trigonometry took on the form of calculus: various facts began to be proved by the formal application of trigonometry formulas, the proofs became much more compact and simpler.