Mini - project work on the topic "History of the development of trigonometry"

11 "a" student of the MBOU "Kilemar secondary school" of the Kilemar municipal district of the Republic of Mari El Ivantsov Vasily

Teacher: I.P. Konyushkova

Goals and objectives:

1. Find information about the development of trigonometry
2. Study the literature on the topic

Plan:

6. Development of modern trigonometry

In my work, I consider the history of the development of trigonometry.

1. The emergence of trigonometry as a science

Trigonometry arose and developed in antiquity as one of the branches of astronomy, as its computing apparatus. Some trigonometric information was known to the ancient Babylonians and Egyptians, but the foundations of this science are 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 first trigonometric tables were compiled by Hipparchus of Nicaea (180-125 BC). Hipparchus was the first to tabulate the corresponding magnitudes of arcs and chords for a series of angles.

More complete information on trigonometry is contained in Ptolemy's Almagest. Ptolemy divided the circumference into 360 degrees and the diameter into 120 parts. He considered the radius as 60 parts and used the sexagesimal number system. For a right triangle with a hypotenuse equal to the diameter of a circle, he wrote on the basis of the Pythagorean theorem: (chord α)² + (chord /180-α /)² = (diameter)², which corresponds to the modern formula sin²α + cos²α = 1. Ptolemy's table, which has survived to our time, is equivalent to a table of sines with five correct decimal places.

2. Development of trigonometry in India

In the 4th century, the center of development of mathematics moved to India. Indian mathematicians were well acquainted with the works of Greek astronomers and geometers. Their contribution to applied astronomy and computational aspects of trigonometry is very significant. First of all, the Indians changed some of the concepts of trigonometry, bringing them closer to modern ones. In India, trigonometry was laid as a general doctrine of the ratios in a triangle, although, unlike the Greek chords, the Indian approach was limited only to the functions of an acute angle. The Indians defined the sine somewhat differently than in modern mathematics, but they were the first to introduce the cosine into use.

3. Further development of trigonometry in the countries of the Middle and Near East

Trigonometry was further developed in the 9th-15th centuries. in the countries of the Middle and Near East. The earliest surviving works belong to al-Khwarizmi and al-Marvazi (IX century), who considered, along with the sine and cosine known to the Indians, new trigonometric functions: tangent, cotangent, secant and cosecant. Khorezmi (al-Khwarizmi) Muhammad bin Musa compiled tables of sines and cotangents. He is the author of a number of astronomical works: works on sundial, astrolabe; compiled a number of mathematical and astronomical tables. His manuscript "The Image of the Earth" (published in 1878), devoted to geography, has also been preserved. However, the fame of the scientist was brought primarily by his work in the field of mathematics. Great results in the development of trigonometry were achieved by Abu-l-Wafa in the second half of the 10th century, who for the first time used a circle of unit radius to determine trigonometric functions, as is done in modern mathematics.

One of the most important tasks of science at that time was the compilation of trigonometric tables with the smallest possible step. In the 9th century, al-Khwarizmi compiled tables of sines with a step of 1 °, his contemporary al-Marvazi added to them the first tables of tangents, cotangents and cosecants with the same step. At the beginning of the 10th century, al-Battani published tables with a step of 30", at the end of the same century Ibn Yunis compiled tables with a step of 1". When compiling tables, the key was to calculate the value. Skillful methods for calculating this value were invented along with Ibn Yunis and Abu-l-Wafa also by al-Biruni. The first specialized treatise on trigonometry was his work "The Book of the Keys of the Science of Astronomy" (995-996). Al-Kashi achieved the greatest success in the 15th century; in one of his works he calculated that(all characters are correct). His 1′ step trigonometric tables were unsurpassed for 250 years. At-Tusi, Nasir ad-Din (1201-1274) in his "Treatise on the complete quadrilateral" for the first time presented trigonometric information as an independent department of mathematics, and not an appendage to astronomy.

4. Continued development of trigonometry in Europe

After the Arabic treatises were translated into Latin in the XII-XIII centuries, many ideas of Indian and Persian mathematicians became the property of European science. In Europe, the development of trigonometry continued. Initially, information about trigonometry was given in writings on astronomy, but in Fibonacci's work "The Practice of Geometry", written around 1220, trigonometry is described as part of geometry. The first European work entirely devoted to trigonometry is often called the Four Treatises on Direct and Reversed Chords by the English astronomer Richard of Wallingford (circa 1320).

The most prominent European representative of this era was Regiomontanus. His work, set forth in the mathematical work Five Books on Triangles of All Kinds, was of great importance in 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. The expansion of the concept of 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. Leonhard Euler introduced both the very concept of a function and the symbolism accepted today. He gave all trigonometry its modern look. In the treatise Introduction to the Analysis of Infinites (1748), Euler gave a definition of trigonometric functions equivalent to the modern one and defined inverse functions. Euler's approach has since become generally accepted and entered the textbooks.

5. Development of trigonometry in Russia

In Russia, the first information about trigonometry was published in the collection "Tables of logarithms, sines and tangents for the study of wise caretakers", published with the participation of L.F. Magnitsky in 1703. In 1714, the informative manual "Geometry of Practice" appeared, the first Russian textbook on trigonometry, focused on applied problems of artillery, navigation and geodesy. The fundamental textbook of Academician M. E. Golovin (a student of Euler) “Plane and spherical trigonometry with algebraic proofs” (1789) can be considered the completion of the period of mastering trigonometric knowledge in Russia.

At the end of the 18th century, an authoritative trigonometric school arose in St. Petersburg, which made a great contribution to plane and spherical trigonometry.

Further development of the theory of trigonometry was continued in the 19th century by N. I. Lobachevsky and other scientists.

At the beginning of the 19th century, N. I. Lobachevsky added a third section to plane and spherical trigonometry - hyperbolic. In the 19th-20th centuries, the theory of trigonometric series and related areas of mathematics developed rapidly: for example, coding of audio and video information and others.

Nowadays, the most important part of trigonometry - the doctrine of trigonometric functions is considered in mathematical analysis, and the solution of triangles is part of geometry

While working on this topic, I studied a number of sources and found information about the development of trigonometry.

Literature: 1. Glazer G.I. History of mathematics at school: IX-X cells. A guide for teachers. - M .: Education, 1983.

2. Internet resources

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 analyzing the 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 section 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 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 obtained series were not the 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 deciphering 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".

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Introduction

Relevance: acquaintance with a new subject - trigonometry.

Purpose: To expand knowledge on the history of the development of trigonometry.

1. What brought to life the science of trigonometry

2. Application of trigonometry in astronomy, physics, biology and medicine.

Object: trigonometry, the history of the origin and development of trigonometry.

Hypothesis: many physical phenomena of nature can be described using trigonometry.

Novelty: familiarity with trigonometry.

Research methodology. The study of literature on this topic, information from Internet resources. Generalization of the found material.

Output: Booklet "History of Trigonometry" (Appendix 2).

Practical significance: given material can be used in the lessons of geometry and trigonometry for additional education. Any student can develop an interest in the science of trigonometry through this material.

The emergence of trigonometry

Historically, trigonometry has developed from problems for solving flat and spherical triangles.

Like any other science, trigonometry arose as a result of human practice in the process of solving specific practical problems.

The emergence of trigonometry is closely connected with the development of one of the oldest sciences - astronomy. The main role belongs to her in the formation and development of spherical trigonometry. From the time of ancient Babylon to the time of Euler and Laplace, astronomy has been the guiding and inspiring force behind the most remarkable mathematical discoveries.

The development of astronomy is caused, first of all, by the need to compile the correct calendar, which was important for the agricultural economy of antiquity. The farmer needed to know the change of seasons in order to carry out the necessary agricultural work in a timely manner. The calendar was also necessary for clergy, performing religious rites, to determine the days of the holiday and for many other people.

The development of trade, associated with the need for movement, both by land and by water, had a great influence on the development of astronomy: it was necessary to be able to correctly determine the course of a ship on the high seas.

A significant role in the development of astronomy and related trigonometry was undoubtedly played by the need to draw up accurate geographical maps, this required the correct determination of large distances on the earth's surface.

The level of development of mathematics among the ancient peoples of Mesopotamia was higher than that of other eastern peoples. The ancient peoples of Mesopotamia had especially developed astronomical observations. Consequently, they possessed some of the simplest information from trigonometry. Already 2-3 thousand years BC, the ancient Egyptians practically used astronomical observations when working on agriculture. The floods of the Nile were an important factor in the development of agriculture.

In the classic Chinese treatise "Mathematics in Nine Books", compiled in the 2nd-1st centuries AD according to earlier sources, book IX of the treatise contains a number of tasks on the use of right-angled triangles, where there are tasks for determining the distance to inaccessible objects. The ancient Maya achieved great success in astronomy; they created a fairly accurate calendar (calendar-chronological system).

Trigonometry in Ancient Greece

Much later, trigonometry entered the next stage of its development in ancient Greece, as part of astronomy. In connection with the needs of astronomy and geodesy, the computational problems of spherical trigonometry were of paramount importance. Thales of Miletus (640 - 548 BC - ancient Greek mathematician and astronomer (Appendix 1); in the first half of the 3rd century BC, the ancient Greek astronomer and mathematician Aristarchus of Samos (310 - 230 BC); Archimedes (Appendix 1), expressed a bold hypothesis that the Earth moves in a circle around the Sun (for this he was accused of godlessness and expelled from Athens).

Already in the middle of the 1st millennium BC. ancient Greek scientists knew that the Earth has the shape of a ball, in particular the length of its circumference. Several methods have been developed to solve this problem. The first measurement of the meridian arc and the radius of the Earth belongs to Eratosthenes of Cyrene (c. 276 - 194 BC) - an ancient Greek mathematician, geographer, historian, philosopher, poet (Appendix 1).

But the works of the ancient Greek scientist Hipparchus (c. 180 - 125 BC) (Appendix 1) - the founder of scientific astronomy - were of fundamental importance for the development of trigonometry in the era of its inception.

Hipparchus compiled a star catalog so that future astronomers could follow the appearance of new stars and the disappearance of old ones. The position in the sky of more than 1 thousand stars was entered in the catalog, subdivided by him by brightness into 6 magnitudes and determined by him by brightness by 6 magnitudes and determined very accurately for that time. Hipparchus was the founder of mathematical geography. He introduced the definition of points on the earth's surface using geographical coordinates- latitude and longitude.

It is important to note that neither Hipparchus nor other scientists of antiquity had trigonometry as a science in the modern sense of the word. But they, using the provisions of elementary geometry known to them, solved those problems that now relate to trigonometry. The basis of all trigonometric calculations among the Greeks was Ptolemy's theorem known to Hipparchus, which can be formulated as follows: "The product of the diagonals of a quadrangle inscribed in a circle is equal to the sum of the products of opposite sides."

Trigonometry in India

The next step in the development of trigonometry is associated with the development of the mathematical culture of the peoples of India from the 4th to the 12th centuries. Along with the "sine", the Indians introduced the "cosine" into trigonometry, more precisely, they began to use the cosine line in their calculations. The term "cosine" itself appeared much later in the works of European scientists, the Austrian mathematician Peirbach or Purbach (1423 - 1461) and the German mathematician Regiomontanus (1436 - 1476).) (Appendix 1).

The Indians also knew the ratio sin 2 a + cos 2 a \u003d r 2, as well as the formulas for the sine of a half angle and the sine of the sum and difference of two angles. Thus, the Indians laid the foundation for trigonometry as the doctrine of trigonometric quantities, although they paid little attention to the solution of triangles. For measuring heights and distances, several rules were developed based on changing the shadow of a vertical pole - the gnomon, and on the similarity of triangles. All this anticipated the introduction of tangent and cotangent.

Trigonometry in the countries of the Arab Caliphate

The next stage in the development of trigonometry is associated with the flourishing of the culture of the countries of the Arab Caliphate. This was the name of the association of various countries and peoples conquered by the Arabs in the 7th - 8th centuries. it included Tajiks, Uzbeks, Persians, Azerbaijanis, Egyptians, Syrians and other peoples. Many of these peoples stood at a higher level of social and cultural development than the Arabs themselves. The necessary information on astronomy, together with trigonometry, algebra and arithmetic, was first borrowed from India. And although Indian mathematics gave rise to the development of Arabic mathematics, Greek geometry and astronomy occupied a dominant position in the emerging science of science among the Arabs, thanks to the translation of all the works of Euclid, Apollonius, Archimedes, Ptolemy and their later commentators. The contribution made by Arabic-speaking peoples to mathematics is especially great. This is primarily a decimal number system, borrowed by the Arabs from the Indians and later, thanks to the works of Arabic-speaking scientists, which became widespread in Europe. Advances in mathematics, in particular in trigonometry, created the basis for achievements in astronomy and in some other sciences.

Trigonometry here also developed in close connection with astronomy and geography and had a pronounced "computational" character.

In Baghdad in different time engaged in scientific work such scientists as al - Khorezmi (783 - 830), al - Khabash (764 - 874), Ibn Kora (836 - 901), Ibn Iraq (965 - 1035), al - Biruni (973 - 1050) ( Annex 1) .)

Al - Khorezmi made a great contribution to the development of mathematics, astronomy and mathematical geography. His works had a strong influence on scientists in East and West for several centuries and served as a model for writing mathematics textbooks for a long time. Two of his treatises on arithmetic and algebra played an important role in the development of mathematics.

Trigonometry in Europe

In the 12th century, urban culture arose in Europe, and commodity-money relations developed within the feudal economic system. This was also facilitated by trade travels and crusades, which made it possible to partially get acquainted not only with the movements of Eastern culture, but also with the culture of ancient Greece. The independent work of European scientists begins. They had to rediscover much of what had been discovered long before them. Their first achievements relate specifically to trigonometry. This science spilled mainly on the basis of the achievements of the ancient Greeks. Translations of some "Arabic" works on trigonometry appeared. On the basis of these works in England, works on trigonometry were written by R. Walligrford (c. 1292 - 1335) and his contemporary D. Modyukt. English scientist Thomas Bradwardine (c. 1290 - 1349) (Appendix 1). For the first time in Europe, he proposed a unit radius of a trigonometric circle, introduced the cotangent under the designation of the “direct shadow” and the tangent under the name of the “reverse shadow” into trigonometric calculations. During this period, tables of sines are compiled.

Regiomontanus, independently of the Arabs (who were 400 years ahead of him) and T. Brodwardin, introduced the tangent function into Jewish science, compiled a table of sines through 1 'and a table of tangents through 1 o. he also compiled a table for calculating the leg of a right-angled triangle (spherical) according to the angle A lying opposite it and along the hypotenuse C according to the formula sina - sinCsinA, calling it a "double-entry" table. This work of Regiomontanus (Appendix 1) played a very important role in the further development of trigonometry.

An important contribution to the development of trigonometry was made by the Polish astronomer Nicolaus Copernicus (1473 - 1543) (Appendix 1), the creator of the heliocentric system of the world, a reformer of astronomy. Not familiar with the works of Regiomontanus, Copernicus independently substantiated some of the basic provisions of spherical trigonometry; for the first time he reduces the whole matter to a trihedron projecting a triangle from the center. Copernicus himself was engaged in compiling trigonometric tables. The German mathematician Peter Krüger (1480 - 1532) was the first European mathematician to compile separate tables of logarithms of trigonometric functions and tables of logarithms of numbers. Danish mathematician Thomas Fink (1561 - 1656) (Appendix 1) in his work "Geometry of the Round" (1583) for the first time introduces the terms "sine", "tangent" and "secant".

The English mathematician Abraham Moivre (1667 - 1754) (Appendix 1), a Frenchman by origin, finds a rule for exponentiating a complex number given in trigonometric form, which is widely used in trigonometry and algebra when solving two-term equations and is now known as the “Moivre formula ".

At present, trigonometry has ceased to exist as an independent science, having split into two parts. One of these parts is the doctrine of trigonometric functions, and the other is the calculation of the elements of trigonometric figures.

The first part, as we said above, is part of mathematical analysis, which has general methods for studying functions, and the second part relates to geometry and plays an auxiliary role in it.

The "geometric" part of trigonometry, in turn, is divided into two sections - "rectilinear trigonometry" and "spherical trigonometry". The main content of the first section is the calculation of the elements of flat triangles, and the second section is the calculation of the elements of a spherical triangle.

Application of trigonometry

Continuing the topic of trigonometry, it is important to note that trigonometric calculations are used in almost all areas of human life: astronomy, physics, nature, music, medicine, biology and many others.

2.1. Trigonometry in astronomy

Thus, in astronomy, the need arose for the "solution of triangles."

The tables of positions of the Sun and Moon compiled by Hipparchus made it possible to predict the moments of the onset of eclipses (with an error of 1-2 hours). Hipparchus was the first to use the methods of spherical trigonometry in astronomy.

2.2. Trigonometry in physics

In the world around us, we have to deal with periodic processes that repeat at regular intervals. These processes are called oscillatory. Oscillatory phenomena of different physical nature obey common laws and are described by the same equations. Exist different types oscillatory phenomena.

Mechanical vibrations. Mechanical vibrations are the movements of bodies that repeat exactly at regular intervals. The graphic representation of this function gives a visual representation of the course of the oscillatory process in time. Examples of simple mechanical oscillatory systems are a weight on a spring or a mathematical pendulum.

2.3 Trigonometry in nature

The rainbow theory was first given in 1637 by René Descartes. He explained the rainbow as a phenomenon associated with the reflection and refraction of light in raindrops.

Aurora borealis Penetration into the upper atmosphere of the planets of charged particles of the solar wind is determined by the interaction magnetic field planets with solar wind.

The force acting on a charged particle moving in a magnetic field is called the Lorentz force. It is proportional to the charge of the particle and the vector product of the field and the velocity of the particle.

2.4. Trigonometry in medicine

Scientists say that the brain estimates the distance to objects by measuring the angle between the ground plane and the plane of vision.

In addition, biology uses such a concept as sleepy sinus, carotid sinus and venous or cavernous sinus.

2.5. Trigonometry and trigonometric functions in medicine and biology, music

One of the fundamental properties of living nature is the cyclicity of most of the processes occurring in it. Biological rhythms, biorhythms are more or less regular changes in the nature and intensity of biological processes. The main terrestrial rhythm is daily. The model of biorhythms can be built using trigonometric functions.

Trigonometry plays an important role in medicine. With its help, Iranian scientists discovered the formula of the heart - a complex algebraic-trigonometric equality, consisting of 8 expressions, 32 coefficients and 33 main parameters, including several additional ones for calculations in cases of arrhythmia.

Biological rhythms, biorhythms are associated with trigonometry. A model of biorhythms can be built using graphs of trigonometric functions. To do this, you must enter the person's date of birth (day, month, year) and the duration of the forecast.

The movement of fish in water occurs according to the law of sine or cosine, if you fix a point on the tail, and then consider the trajectory of movement.

During the flight of a bird, the trajectory of the flap of the wings forms a sinusoid.

Frequencies corresponding to the same note in the first, second, etc. octaves are related as 1:2:4:8…

diatonic scale 2:3:5

Conclusion

During research work knowledge of trigonometry expanded, materials on the history of trigonometry were studied, and it was concluded that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.

We found out that trigonometry is a historically established science. It was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.

We were convinced that trigonometry ceased to exist as an independent science, having split into two parts.

We think that trigonometry has not only found its application in human life, that its scope will expand.

List of used sources and literature

Internet sources

https://ru.wikipedia.org/wiki

https://www.ucheba.ru/

http://www.math.ru/ library

https://sites.google.com/site/trigonometry history of trigonometry

http://fb.ru history of trigonometry

Literature

Voloshinov. Mathematics and Art // Moscow, 1992. Newspaper

History of mathematics from ancient times to the beginning of the 19th century in 3 volumes / / ed. A.P. Yushkevich. Moscow, 1970 - Volume 1-3 E. T. Bell Creators of Mathematics.

Maslova T.N. "Schoolchildren's Handbook in Mathematics" M .: Oniks Publishing House LLC: Mir and Education Publishing House LLC, 2008. - 672 p.

Mathematics. Supplement to the newspaper dated 1.09.98.

Predecessors of Modern Mathematics// ed. S. N. Niro. Moscow, 1983 A. N. Tikhonov, D. P. Kostomarov.

Stories about applied mathematics//Moscow, 1979. A.V.

Annex 1

Scientists who contributed to the development of trigonometry

Thales of Miletus

Aristarchus of Samos

Eratosthenes of Cyrene

Regiomontana

Thomas Bradwardine

al-Biruni

Nicholas Copernicus

Thomas Fincke

Abraham Moivre

Annex 2

The first measurement of the meridian arc and the radius of the Earth belongs to Eratosthenes of Cyrene (c. 276 - 194 BC) - an ancient Greek mathematician, geographer, historian, philosopher, poet.	Regiomontanus, independently of the Arabs (who were 400 years ahead of him) and T. Brodwardin, introduced the tangent function into Jewish science, compiled a table of sines through 1 'and a table of tangents through 1 o. Booklet prepared by: student 9 "A" class MOU secondary school No. 105 Pavlova Polina Alexandrovna		MOU secondary school №105 History of trigonometry Volgograd, 2018
Trigonometry is a branch of mathematics that studies trigonometric functions and their applications to geometry. The word trigonometry is made up of two Greek words: "trigwnon" - "triangle" and "metrew" - "to measure", means - "measurement of triangles". It is this task - "measuring triangles" or "solving triangles", determining all the elements of a triangle according to three data, from ancient times that formed the basis of practical applications of trigonometry.		The emergence of trigonometry is closely connected with the development of one of the oldest sciences - astronomy. The main role belongs to her in the formation and development of spherical trigonometry Hipparchus was the founder of mathematical geography. He introduced the definition of points on the earth's surface using geographical coordinates - latitude and longitude.		Doctors needed astronomy, algebra and trigonometry for astrological calculations, in order to draw up the horoscope of the patient and, by the location of the planets in the constellations, determine whether the patient would recover or not. These and other aspects of human activity already in ancient times came up against the need to familiarize themselves with the position and apparent movement of the heavenly bodies (the Sun, the Moon, the stars).

Sine, cosine, tangent - when pronouncing these words in the presence of high school students, you can be sure that two-thirds of them will lose interest in further conversation. The reason lies in the fact that the basics of trigonometry at school are taught in complete isolation from reality, and therefore students do not see the point in studying formulas and theorems.

In fact, this field of knowledge, upon closer examination, turns out to be very interesting, as well as applied - trigonometry is used in astronomy, construction, physics, music and many other areas.

Let's get acquainted with the basic concepts and name several reasons to study this branch of mathematical science.

Story

It is not known at what point in time humanity began to create future trigonometry from scratch. However, it is documented that already in the second millennium BC, the Egyptians were familiar with the basics of this science: archaeologists found a papyrus with a task in which it is required to find the angle of inclination of the pyramid on two known sides.

Scientists of Ancient Babylon achieved more serious successes. Being engaged in astronomy for centuries, they mastered a number of theorems, introduced special methods for measuring angles, which, by the way, we use today: degrees, minutes and seconds were borrowed by European science in Greco-Roman culture, in which these units came from the Babylonians.

It is assumed that the famous Pythagorean theorem, relating to the basics of trigonometry, was known to the Babylonians almost four thousand years ago.

Name

Literally, the term "trigonometry" can be translated as "measurement of triangles." The main object of study within this section of science for many centuries has been a right-angled triangle, or rather, the relationship between the magnitudes of the angles and the lengths of its sides (today, the study of trigonometry begins from this section from scratch). In life, situations are not uncommon when it is impossible to practically measure all the required parameters of an object (or the distance to the object), and then it becomes necessary to obtain the missing data through calculations.

For example, in the past, a person could not measure the distance to space objects, but attempts to calculate these distances occur long before our era. critical role trigonometry also played in navigation: with some knowledge, the captain could always navigate at night by the stars and correct the course.

Basic concepts

To master trigonometry from scratch, you need to understand and remember a few basic terms.

The sine of an angle is the ratio of the opposite leg to the hypotenuse. Let us clarify that the opposite leg is the side lying opposite the angle we are considering. Thus, if the angle is 30 degrees, the sine of this angle will always, for any size of the triangle, be equal to ½. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.

Tangent is the ratio of the opposite leg to the adjacent one (or, equivalently, the ratio of sine to cosine). The cotangent is the unit divided by the tangent.

It is worth mentioning the famous number Pi (3.14 ...), which is half the length of a circle with a radius of one unit.

Popular Mistakes

People who learn trigonometry from scratch make a number of mistakes - mostly due to inattention.

First, when solving problems in geometry, it must be remembered that the use of sines and cosines is possible only in a right triangle. It happens that the student “on the machine” takes the longest side of the triangle as the hypotenuse and receives incorrect calculation results.

Secondly, at first it is easy to confuse the values ​​of sine and cosine for the chosen angle: recall that the sine of 30 degrees is numerically equal to the cosine of 60, and vice versa. If you substitute the wrong number, all further calculations will be wrong.

Thirdly, until the problem is completely solved, it is not worth rounding off any values, extracting roots, writing an ordinary fraction as a decimal. Often, students strive to get a “beautiful” number in a trigonometry problem and immediately extract the root of three, although after exactly one action this root can be reduced.

Etymology of the word "sine"

The history of the word "sine" is truly unusual. The fact is that the literal translation of this word from Latin means "hollow". This is because the correct understanding of the word was lost when translating from one language to another.

The names of the basic trigonometric functions originated from India, where the concept of sine was denoted by the word "string" in Sanskrit - the fact is that the segment, together with the arc of a circle on which it rested, looked like a bow. During the heyday of the Arab civilization, Indian achievements in the field of trigonometry were borrowed, and the term passed into Arabic in the form of transcription. It so happened that this language already had a similar word for a depression, and if the Arabs understood the phonetic difference between a native and a borrowed word, then the Europeans, translating scientific treatises into Latin, by mistake literally translated the Arabic word, which had nothing to do with the concept of sine . We use them to this day.

Tables of values

There are tables that contain numerical values ​​​​for sines, cosines and tangents of all possible angles. Below we present data for angles of 0, 30, 45, 60 and 90 degrees, which must be learned as a mandatory section of trigonometry for "dummies", since it is quite easy to remember them.

If it so happened that the numerical value of the sine or cosine of the angle "flew out of my head", there is a way to derive it yourself.

Geometric representation

Let's draw a circle, draw the abscissa and ordinate axes through its center. The abscissa axis is horizontal, the ordinate axis is vertical. They are usually signed as "X" and "Y" respectively. Now we draw a straight line from the center of the circle in such a way that we get the angle we need between it and the X axis. Finally, from the point where the straight line intersects the circle, we lower the perpendicular to the X axis. The length of the resulting segment will be equal to the numerical value of the sine of our angle.

This method is very relevant if you forgot the desired value, for example, in an exam, and there is no trigonometry textbook at hand. You won’t get the exact figure this way, but you will definitely see the difference between ½ and 1.73 / 2 (sine and cosine of an angle of 30 degrees).

Application

One of the first specialists to use trigonometry were sailors who had no other reference point on the high seas than the sky above their heads. Today, captains of ships (aircraft and other modes of transport) do not look for the shortest path through the stars, but actively resort to the help of GPS navigation, which would be impossible without the use of trigonometry.

In almost every section of physics, you will find calculations using sines and cosines: whether it is the application of force in mechanics, calculations of the path of objects in kinematics, vibrations, wave propagation, light refraction - you simply cannot do without basic trigonometry in formulas.

Another profession that is unthinkable without trigonometry is a surveyor. Using a theodolite and a level, or a more sophisticated device - a tachometer, these people measure the difference in height between different points on the earth's surface.

Repeatability

Trigonometry deals not only with the angles and sides of a triangle, although this is where it began its existence. In all areas where cyclicity is present (biology, medicine, physics, music, etc.), you will encounter a graph whose name is probably familiar to you - this is a sinusoid.

Such a graph is a circle unfolded along the time axis and looks like a wave. If you've ever worked with an oscilloscope in a physics class, you know what I'm talking about. Both the music equalizer and the heart rate monitor use trigonometry formulas in their work.

Finally

When thinking about how to learn trigonometry, most middle and high school students begin to consider it a difficult and impractical science, because they get acquainted only with boring textbook information.

As for impracticality, we have already seen that, to one degree or another, the ability to handle sines and tangents is required in almost any field of activity. As for the complexity… Think about it: if people used this knowledge more than two thousand years ago, when an adult had less knowledge than today's high school student, is it really possible for you personally to study this field of science at a basic level? A few hours of thoughtful practice with problem solving - and you will achieve your goal by studying the basic course, the so-called trigonometry for "dummies".

Lecture #6

Topic: The origin and development of trigonometry

1.1. The origin and development of trigonometry.

1.2. Spherical trigonometry.

1.3. Trigonometry in Europe before Euler.

1.4. Euler's contribution to the development of trigonometry.

1.5. Euler's followers in the development of trigonometry.

1.1. THE ORIGIN AND DEVELOPMENT OF TRIGONOMETRY.

Trigonometry arose and developed in antiquity as one of the branches of astronomy, as its computing apparatus that meets the practical needs of man.

Trigonometry the word is Greek 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, 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 as early as 2000 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 in III century 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 ( I century AD), although they did not acquire a special name. Modern sine , for example, was studied as a half-chord, on which the central angle is based , or as a chord of a doubled arc.

The word cosine is much younger. Cosine is an abbreviation of the Latin expression completely sinus , i.e. “additional sine” (or otherwise “sine of an additional arc”; cos  \u003d 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 in X century by the Arab mathematician Abu-l-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 XIV 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" comes from the Latin tanger (to touch), appeared in 1583 Tangents translated as "touching" (line of tangents 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, that is, the facts that we now formulate in terms of trigonometric functions were formulated and proved using 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.

Since the XVII 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 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 mathematician XVIII century 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.

1.2. SPHERICAL TRIGONOMETRY.

Spherical trigonometry is a section of trigonometry that studies the relationship between angles and side lengths of spherical triangles. It is used to solve various geodetic and astronomical problems.

The foundations of spherical trigonometry were laid by the Greek mathematician and astronomer Hipparchus in the 2nd century BC. e. An important contribution to its development was made by such ancient scientists as Menelaus of Alexandria and Claudius Ptolemy. The spherical trigonometry of the ancient Greeks relied on the application of the theorem of Menelaus to a complete quadrilateral on a sphere. Ancient Greek mathematicians stated the condition of the Menelaus theorem not in the language of the ratios of sines, but in the language of the ratios of chords. To perform the required calculations, tables of chords were used, similar to the subsequent tables of sines.

As an independent discipline, spherical trigonometry was formed in the works of medieval mathematicians of Islamic countries. The greatest contribution to its development in this era was made by such scientists as Sabit ibn Korra, Ibn Iraq, Kushyar ibn Labban, Abu-l-Wafa, al-Biruni, Jabir ibn Aflah, al-Jayani, Nasir ad-Din at-Tusi. In their works, the basic trigonometric functions were introduced, the spherical sine theorem and a number of other theorems used in astronomical and geodetic calculations were formulated and proved, the concept of a polar triangle was introduced, which made it possible to calculate the sides of a spherical triangle from its three given angles.

The history of spherical trigonometry in Europe is connected with the works of such scientists as Regiomontanus, Nicolaus Copernicus, Francesco Mavrolico.

W The change of chords by sines was the main achievement of Medieval India. This replacement made it possible to introduce various functions related to the sides and angles of a right triangle. Thus, in India, the beginning of trigonometry as a doctrine of trigonometric quantities was laid.

Indian scientists used various trigonometric ratios. Trigonometry is necessary for astronomical calculations, which are drawn up in the form of tables. The first table of sines is found in the Surya Siddhanta and Aryabhata. Later, scientists compiled more detailed tables: for example, Bhaskara gives a table of sines through 1 °.

South Indian mathematicians in the 16th century made great progress in the field of summation of infinite number series. Apparently, they did this research when they were looking for ways to calculate more accurate values ​​of the number π. Nikalanta verbally gives the rules for expanding the arc tangent into an infinite power series. And in the anonymous treatise "Karanapaddhati" ("Computation Technique"), the rules for expanding the sine and cosine into infinite power series are given. It must be said that in Europe such results were approached only in the 17th-18th centuries. So, the series for sine and cosine was derived by Isaac Newton around 1666, and the arctangent series was found by J. Gregory in 1671 and G. W. Leibniz in 1673.

In the 8th c. scientists from the countries of the Near and Middle East got acquainted with the works of Indian mathematicians and astronomers and translated them into Arabic. In the middle of the 9th century, the Central Asian scholar al-Khwarizmi wrote the essay “On the Indian Account”. After the Arabic treatises were translated into Latin, many ideas of Indian mathematicians became the property of European and then world science.

1.3. trigonometry in Europe before Euler

The year 1637, which turned out to be so important for the development of the doctrine of geometric places, did not have the same significance for elementary geometry. In particular, Descartes himself did nothing for trigonometry as such. Both in his "Geometry" and in the comments to it, the trigonometric functions of the angles of a figure were expressed in terms of the ratios of the sides of the right triangles associated with it.

Nevertheless, the thirties of the 17th century completed, to a certain extent, the construction of trigonometry: we mean the creation of logarithmic methods of calculation. Note that in 1633 BriggsGellibrand's "British Trigonometry" was published, containing in the second part the most common methods for solving plane and spherical triangles, taking into account, first of all, formulas of a logarithmic nature. Along with Napier's analogies, Gellibrand applied formulas that determine half an angle on three sides for both a flat (Rhetik, earlier 1576, published in 1596) and a spherical triangle (Napier in Constructs, 1619). The case of three given angles in a spherical triangle Gellibrand, however, with the help of the polar triangle led to the case of three given sides, without giving any formulas. In the tables that made up the first part of "British Trigonometry", the authors introduced the decimal division of the degree. However, in the work "Artificial Trigonometry" published at the same time, Flaccus again used the old sexagesimal division and thus formed the basis of all later tables.

Let us also point to Krueger's book "The Use of Logarithmic Trigonometry" (Danzig, 1634), which also belongs to the period we are analyzing, since new editions of it appeared in 1648 and 1654. Kruger applied Napier analogies in the cases of spherical triangles with given a, b, f and a, p, c quite modern. For a flat triangle with three given sides a^>b^>c, he gave a long verbal indication, which in our notation says that first of all it is necessary to determine the auxiliary quantity x from the equation

and then from the equation

find the angle β and from the equation

angle γ . Here x denotes the difference p q of the projections of the sides b, c onto BC. This detour of the cosine theorem was characteristic of all English trigonometry of that time (for example, Outtred and J. Newton had the corresponding drawings. He ascended to Napier's Descriptio and was based, in essence, on the theorem that

which was known in various forms to the ancient Greeks and Arabs.

Both in algebra and in trigonometry of modern times, the form was first of all improved. Here and there, a gradual transition was made from the ancient presentation with the help of proportions and often long computational recipes to algebraic calculus and equations. Much of this progress has been made by the same scientists. P. Erigon extended his symbolism to trigonometry. As an example, we give only the form in which he presented the cosine theorem for the plane. At the same time, it should be noted that on the drawings of Erigon at the vertices of the corners there were large letters, which often themselves denoted these angles, and in the text small ones, and that D is the base of the height drawn from A. Erigon's theorem had the following form:

In England, he greatly contributed to a more symbolic exposition of trigonometry.

In Outred's work, the first (geometrical) proofs of both Napier's analogies were given. Napier, as well as Briguet and Gellibrand, published only the formulas themselves, moreover, in logarithmic form.

Analytical derivation of non-Perean analogies was first given by John Newton in his great work "British Trigonometry" (London, 1658). However, this conclusion, like other proofs of a computational nature given by Newton, was still very cumbersome. The work of J. Newton was a significantly improved and supplemented new edition of the work of the same name by Briggs Gellibrand. In it, as in the somewhat earlier "British Astronomy" by the same author (London, 1656), the terms "cosine" and "cotangent" were first regularly used. For "sine", "tangent" and "secant" this was done by T. Fink in 1583, so that now the name of the trigonometric functions has already been established, although it has not yet become commonly used.

We mention here that the formulas expressing the functions of a double angle in terms of functions of a single angle) were at the same time supplemented by a formula written in our notation in the form:

This theorem was given by John Pell in his book "Disputes about the true measurement of the circle" (Amsterdam, 1647); there it was proved in various ways by Roberval and other mathematicians. A more general rule for tg( + β) and, moreover, for sc ( + β) was first derived (geometrically) by J. Hermann in Acta Erud., 1706. Looking ahead, we add that the formula

appeared only in Euler's "Introduction" (1748) and that important formulas for rationally expressing sin 2a and cos 2a in terms of tg , were established even later by I. Lambert in the first book of his Essays on the Use of Mathematics, 1765.

The problem of determining two angles (β and γ ) of a triangle by the third angle a and the logarithms of the sides b and c was first solved according to Thomas Street (Astronomia Carolina, London, 1661) by Robert Anderson, who introduced an auxiliary angle for this purpose. He, translated into our notation, put tg = b / c , computed  and then determined the angleβ-γ :

The theorem was transmitted verbally; the proof, which assumed the well-known theorem of tangents, had a complex geometric character. From here, this technique has passed into most textbooks.

Of the encyclopedic expositions of trigonometry, trigonometry deserves mention due to its great clarity "Trigonometry" (in the book II Astronomia Britannica ”, London, 1669) by W. Wing. Wing relied on Napier, Norwood. For each case, he gave an example calculated using logarithms, but sometimes omitted proofs. The case of a spherical triangle with three given sides was dealt with by Wing on the basis of the formula

as it was often done at that time after Napier. Wing used the abbreviations s., cs., t., ct . In "Compendium of Mathematics" ( A mathematical compendium , London, 1674), compiled on the basis of the notes of J. Moore, N. Stephenson wrote S., Cos., T., Cot., but sometimes abbreviated in a different way (si:, si. co., cos, etc d.). J. Wallis, in his Treatise on Angular Sections, used the letters S, £, T, and m to designate the sine, cosine, tangent, and cotangent, respectively. Using these notations, he was the first to give the basic formulas of goniometry the form of equations

We are alien only to the "whole sine" (R), which continued to be used for a long time, as well as the use of the functions of secant, cosecant, sine-versus (1 cos) and sine-versus of an additional angle.

While with Wallis himself these important innovations played a rather subordinate role (for in print he applied them only in one small article), John Keswell, relying on them, compiled "Plane and Spherical Trigonometry", which also appeared in the appendix to " Algebra" by Wallis and in which we see the first, more saturated with formulas, presentation of trigonometry. In this work, for example, the formulas for half angles on three sides of a flat triangle were first derived using an algebraic transformation of the cosine theorem. The great progress in Keswell's methods becomes especially clear if we compare with his derivation the cumbersome geometric proofs of the same formulas that he gave in his "Mathematical Etudes" (Book. V) F. Scouten.

The cosine theorem of spherical trigonometry was derived by Keswell from a trihedron by superimposing a side face on the base. Apparently, this was the first case of the derivation of trigonometric formulas using the methods of descriptive geometry. Then Keswell algebraically transformed the theorem, and received the formulas of spherical trigonometry for half angles.

Denoting the base of the triangle, like Outred, through B, the sides through m, n , semiperimeter through 1 ), he, for example, found for the tangent of half the angle at the vertex the proportion

which easily transforms into a modern form

The Napier analogies were first proved by Keswell artificially geometrically, but then he gave four theorems from the heritage of the priest Thomas Becker, which he analytically derived from the usual theorems for a right triangle and from which he then obtained Napier analogies with the help of calculations.

The first graphically depicted was the sine function. Roberval drew its graph in connection with the determination of the area of ​​the cycloid. Namely, he drew inside the cycloid and on its base, with the help of the forming circle fixed at the left end of the base, the line he called Trochoidis comes (or also socia "companion of the trochoid"), identical with the complete revolution of the sinusoid. The name "line of sines" is first encountered by the French Jesuit Honoré Fabry in his work "Geometric work on the line of sines and the cycloid", published by him under the pseudonym A. Farbius and generally related to the prehistory of infinitesimal calculus. Wallis in part II In his Mechanics (1670), he correctly analyzed the question of the signs of the sine in all four quadrants and drew two complete turns of the sinusoid, noting that there are countless of them. In another place, he pointed out that the line of sines serves as the boundary of the surface of the “cylindrical hoof” turned onto a plane. Somewhat later, he drew the branches of the secant line, but did not notice that at the vertices they should be directed horizontally, and connected both halves at a right angle. J. Gregory in his "Geometric Studies" (1668) presented a part of the tangentoid lying in the first quadrant. The secant, tangent, and cosine curves for the first quadrant were shown in one drawing in Geometric Lectures (London, 1670, 2nd ed., 1674) by I. Barrow. However, the line of secants is drawn there at least very inaccurately, and in other drawings the line of tangents is even simply incorrect. The signs of the tangent in various quadrants were first correctly established by Lanyi, without giving either a drawing or proof. Cotes, in "Various Works" appended to the "Harmony of Measures" (published in 1722), gave the correct plots of tangent and secant for two revolutions. But even F. Mayer, who generally had very serious merits in the field of trigonometry, considered the sine and tangent of an obtuse angle to be positive, and the cosine and cotangent to be negative.

The great successes achieved by this time in England were only gradually consolidated on the European mainland. In Germany in XVII century, not a single significant book on trigonometry appeared, and only summaries, intended for land surveyors and reporting theorems without proof; the authors of some did not even use logarithmic calculations. Of these books, we will name only the very common "Tables of Universal Mathematics" (Wittenberg, 1664) by Egidius Strauch and "Pandora of Mathematical Tables" (Frankfurt, 1684 and 1688) by Grüneberger, in the introduction to which an overview of trigonometric theorems was given. A the author of "Clear Mathematics" (Nuremberg, 1689, 1695 and 1711), Johann Sturm, who spoke briefly about trigonometry in it, skipped secants as lines that can be easily dispensed with.

We find somewhat more among the French. First of all, we must point to the great encyclopedic works already known to us by Deschals and Ozanam. "Course or World of Mathematics" 1674, three volumes; 1690, four volumes) Deschal concluded in the first volume full review plane and spherical trigonometry, although still in the traditional form, with geometric proofs and without abbreviated notation. The problem of calculating two anglesβ and γ according to their sum and the ratio of their sines, he considered it somewhat differently than Street. He believed

sin β : sin γ = tg  : tg ψ ,

took  arbitrary and from here with the help of logarithms determinedψ . Deriving (of course, in the old form) from the above proportion a new one:

The introduction of infinite series prompted us to revisit the formulas for sin n  and cos n  already known for individual integer values ​​of n, and extend them to the case of arbitrary n. The basis for these studies was laid mainly by Vieta, who derived such formulas up n =10. He, like Bürgi (in one unpublished manuscript), had already noticed the law of the formation of coefficients. Although Outred did not essentially go further than Vieta's results, the expressions placed at the end of the first edition of his "Key to Mathematics" in 1631) up to n = 5, already significantly surpassed Vieta's exposition in algebraic form.

Outred's last result here was:

B sin a) in 5 B sin a K - f 5-2 sin a \u003d 2 sin 5a.

Outred here pointed out that the radius can generally be set equal to 1; however, only Euler accomplished this. In the second edition in 1648, Outred gave the corresponding formula for sin 7  . He compiled an even more extensive treatise on "sections of an angle" (published in Opuscula math, hactenus inedita , Oxford, 1677). There, the formula corresponding to the one we have given read:

In his works, Lanyi; which were devoted to "goniometry", introduced by him as a new science of measuring all angles. The first was devoted almost entirely to a purely geometrical method, which consisted in the fact that the relation between the arc corresponding to the angle and the semicircle was represented in the form of a continued fraction by superimposing the arc on the semicircle, the remainder on the arc, etc., continuing until the remainder is not will become inconspicuous. Another, purely analytical device was given in more detail mainly in the second article. In essence, it consisted in the use of a series for the arc tangent. But in order to refine the calculations, Lanyi introduced remarkable improvements almost everywhere. So, for example, he determined the smaller acute angle of a right triangle with sides 3, 4, 5 with an accuracy of 1/60 10 degrees. In the third article, he gave an expression for the common term of the series, which is obtained from the arc tangent series when adding each of its two consecutive terms, and determined the margins of error if the series terminates at some specific term. In the fourth article, Lanyi pointed out how long one should continue calculations when determining, using his method, one of the acute angles of a right triangle on the sides.

Important advances were made even before Euler in the 18th century in solving triangles. First of all, getting closer and closer to modern shape, although sometimes the proportions were expressed verbally. An example was given here by I. Newton in his "Universal Arithmetic" in 1707. He deals with the problem of solving a triangle by base; the sum of the sides and the angle at the top. Newton draws the bisector of an angle establishes a proportion

c: (a + b) \u003d sin 1/2 γ: sin 

In problems XI and XII, Newton specifies three sides, introduces lowercase letters himself, assuming, however, AB = a, AC = b and BC = c, and verbally establishes the proportion. The formulas he derived further for the half angle and Heron's formula for the area of ​​a triangle were, although not new, but they appeared in his usual form.

In Germany at this time Wolff's writings became widespread. However, the achievements of the British have not yet been used in them either in content or in relation to the form of theorems. It should only be noted the form that Wolf gave to Napier's rule for a right-angled spherical triangle. In France, after the writings of Deschal and Ozanam, a rather voluminous book by Deparcier "New Trigonometry Courses" (1741) appeared, which contained good seven-digit tables of sines, tangents and secants, the values ​​\u200b\u200bof which were given with an interval of one minute, and eight-digit tables of logarithms of sines and tangents with an interval 10 minutes.

The most prominent representative of trigonometry before Euler was, however, F. von Oppel. And he belonged to Mayer's followers. In his book Analysis of Triangles (1746), Oppel set out to analytically develop all plane and spherical trigonometry from a few geometrically deduced propositions. He really carried out his intention, although letter designation Mayer makes reading his book very difficult. Most importantly, both of the so-called Mollweide equations have been given here in their modern form. Oppel derived them by calculation from the geometrically proven tangent theorem.

Oppel was the first to systematically base spherical trigonometry on the consideration of a trihedron dissected at the upper edge, both side faces of which were superimposed on the plane of the base. With the help of such a drawing, he derived the theorems of sines and cosines, noticed that all other formulas can be derived from these two theorems, and showed how, using an additional triangle, one can obtain a reciprocal for each formula with it. He deduced a great many such formulas, without being interested, however, in their reduction to a logarithmic form, so that, for example, Napier's analogies were absent from him.

1.4. Euler's contribution to the development of trigonometry

It is clear that such a pronounced analytical genius as Euler, once engaged in computational trigonometry, should have significantly advanced it. The reason to turn to trigonometry presented itself to him in the already repeatedly mentioned "Introduction to Analysis" (1748). In the eighth chapter of his first volume, Euler first introduced angular functions into analysis as numerical quantities with which calculations can be performed, like with any others, so that henceforth they no longer affect the dimension of expressions. And although Euler did not explicitly define trigonometric functions anywhere as ratios of the sides of a right-angled triangle, he always considered them in this way. Apart from minor trifles, Euler's exposition and symbolism were quite modern. Already in one work in 1729 (1735) he wrote down the cosine theorem of spherical trigonometry in the form

cos: BC = cos: AB cos: AC + cos A sAB sAC;

the whole sine, which was still used by most of the previous authors, was already taken equal to 1 here. The designations of trigonometric functions in the "Introduction" were as follows: sin. A. z or sin. z (A = arcus ), cos . A. z or cos . z , tang . z, cot. z etc.

At the beginning of this chapter, formulas were systematically established for the first time for sin (z +), sin (z +  ), etc. By writing:

Euler opened the brackets and in this way obtained a formula for cosnz ; similarly, he found a formula for sinnz . Taking n infinitely large, a z infinitesimal, so cosz = l and sinz = z , he derived from these formulas infinite series for sine and cosine. From here he obtained series for the sine, cosine, tangent and cotangent, partly published by him already in Comm. Ac. Petr ., 1739 (1750). He then showed exhaustively how these series could be used to calculate trigonometric tables. Later in Nov. Comm. Ac. Petr ., 1754/55 (1760) he deduced further series for sin n  , cos n  , sin m  , cos n  , following the functions of angles that are multiples of . Euler came across the connection between exponential and trigonometric functions already in one paper on series, placed in Comm. Ac. Petr ., 1740 (1750). He gave the corresponding defining formula for the sine in Misc. Berol ., 1743, but the formulas for sine and cosine were proved only in the "Introduction". Euler obviously knew nothing about the results. Formulas

cos x \u003d (e ix + e - ix) and sin x \u003d (e ix e - ix)

he got in the "Introduction" from the expressions

assuming n =  . To this he added the formula

Definition of sin (x + iy) and cos (x + iy ) he first gave in 1749.

The summation of the series of sines and cosines, the arguments of which grow in arithmetic progression, was already done by Euler in 1748. In the "Introduction" he again returned to this issue from a more general point of view. Later (Petersburg, 1783) he took up similar series, the arguments of which form a geometric progression. Euler began to deal with the representation of trigonometric functions in the form of products already in 1734-35 (1740), where he decomposed the sine into an infinite product. He did the same for sine and cosine in 1740 (1750). All this, together with some additions, was included in the "Introduction", in the 14th chapter of which he also dealt in detail with the question of the multiplication and division of angles, that is, the trigonometric functions of multiple angles. We pointed out in the first part that in these various investigations Euler acted more creatively than critically. This was so deeply rooted in his nature that he disregarded the objections made to him chiefly by Nicholas I Bernoulli already in 1742 and 1743. Euler continued to perform calculations on any infinite series, extended the theorems on finite polynomials to infinite ones, and assigned any values ​​to the index n, which at the beginning of the proof was assumed to be integer. Despite this, his results were usually correct, although in some cases he came to erroneous conclusions, as, for example, in the article cited in Nov. Comm. Ac. Petr., 1754-55 (1760).

In the second volume of the Introduction (Chapter 22), Euler applied to the solution of transcendental equations such as s = cos s or s = sin 2 s etc., the rule of false position. As he himself reports, he came up with similar problems in order to see if it was possible to approach the squaring of the circle in this way. Later, when Lambert had already proved the irrationality , Euler again took up such considerations, emphasizing that Lambert's work had by no means proved the impossibility of squaring the circle.

Before turning to Euler's merits in spherical trigonometry, let us mention two more trigonometric expansions, which lie somewhat aside. Euler found them by developing the method proposed by Descartes and then repeatedly rediscovered for constructing a circle of a given length (Descartes, Amsterdam, 1701). It's an endless line

tg  + tg + tg +...= - 2 ctg 2 

and endless work

cos  coscos ... = ,

which Euler derived in another way already in 1737 (1744).

Euler dealt with spherical trigonometry in two large papers, approaching it from different points of view. In the first, placed in mem. Ac. Berl ., 1753 (1755) he quite generally constructed spherical trigonometry as the geometry of triangles composed on the surface of a sphere by lines of the shortest distance. Euler proceeded from a right-angled triangle, denoting the leg AP through x, the leg RM through y, the hypotenuse AM through s [rice. 4]. If O is the pole of the great circle (equator) on which AP lies, and Op is the meridian infinitely close to OP, then

Mm = ds, mn = dy, Pp = dx

and the line Mn, lying on a parallel circle of latitude y, is equal to dxcosy so

ds =  .

Further, Euler looked for conditions under which the integral of this element of the arc would have a minimum value, and thus obtained 10 equations arising from Napier's rule. Here for the first time appeared the designations which we are now inclined to take for granted, and whose absence has often given such an inconvenient appearance to earlier works. We mean the designation of three sides by the letters a, b , c, and the opposite vertices and corners of the triangle with the letters A, B, C. The fact that we designate the latter for the most part with the letters a, ,  is, of course, less significant. Greek letters were introduced only in XIX century, although sometimes ,  , were already used by A. Kestner in his Fundamentals of Arithmetic, Geometry and Trigonometry (Göttingen, 1759; 6th ed. 1800). The new notation allowed Euler to write down his ten equations in a completely modern form. He then derived from them six different basic equations for a right triangle. Euler did the same in the case of a general spherical triangle. Having determined the minimum of one of the sides, he first of all found five fundamental equations, from which he then derived the sine theorem, both cosine theorems and the so-called cotangent rule (first encountered in Vieta); the latter appeared in his form

sin a tg C sin B tg c \u003d cos a cos B tg C tgc,

passing into the one used today when dividing by tgCtgc . Euler wrote each theorem in three forms, which are obtained from each other by a cyclic permutation, although Euler himself did not use it. Euler did not mention the polar triangle, and in general, from the point of view of completeness, there were several minor gaps in the article. On the other hand, the applications and transformations of the fundamental theorems were extremely rich.

Among other things, there were all the formulas for half angles, although without the abbreviations for half-sums of sides and angles, then Napier's and Briggs' four analogies, the use of an auxiliary angle in the cosine theorem, the latter being given in a new form:

cos a =

a formula that was polar with the reduced formula was also reported.

Let us add that following this article, Euler in the same volume mem. Ac. Berl . placed a work detailing trigonometry on the surface of a spheroid, with particular regard to issues related to the measurement of the earth. Similar studies were carried out later by du Sejour.

In the second article on spherical trigonometry, Euler adopted an elementary basis for constructing a system of its formulas. He proceeded here from a trihedron, which he intersected with the corresponding planes, in order to then apply the theorems of plane trigonometry (like Copernicus). He thus deduced the sine theorem, the cosine theorem for sides, and a new formula relating the five elements:

cos A sin s = cos a sin b sin a cos b cos WITH ,

noting that these three formulas contain all of spherical trigonometry. The third equation obtained here was subjected to repeated transformations by Euler. He derived from it the so-called cotangent formula, the cosine theorem for angles, and, with the help of the sine theorem, a polar formula with it. Only after that did he introduce the polar triangle and explain its method of application, led, partially deriving them in a new way, logarithmic formulas and rightly declared that his article gives a complete exposition of the system of spherical trigonometry.

1.5. Euler's followers on the development of trigonometry

Euler's followers included a number of eminent mathematicians. Academician A. I. Leksel of St. Petersburg did a lot in the field of spherical trigonometry. He showed that the locus of vertices of all triangles with a common base and equal area is a small circle and that the products of the sine of a side (or, respectively, of an angle) and the corresponding height have a constant value d. And with the help of these equalities, which are generalizations of Heron's formula, he deduced elegant expressions for the tangents of the spherical radii of the circle inscribed in a triangle and the circumscribed circle, established formulas for cos ½ (A ± B ± C ), proved a theorem corresponding to Ptolemy's theorem in a spherical quadrilateral inscribed in a circle, extended some other proposals of planimetry to the ball.

A number of articles on spherical trigonometry were published by two

other Petersburg academicians N. Fus and Schubert. I. T. Mayer Jr. gave a logarithmic interpretation of the plane cosine theorem (Capital Course in Practical Geometry, vol. 1, Göttingen, 1777). Mention should also be made of Kestner with his "Geometric Articles" (1790/91), although he still used the whole sine. In terms of form, the construction of spherical trigonometry, given in 1783 (1786) by J. P. de Gua, was also unsuccessful.

Lagrange contributed to the success of trigonometry by applying series to the direct solution of equations, which was especially important for the practical purposes of astronomy and geodesy. O n solved the goniometric equation using imaginary quantities tg x = m tg y regarding x. He connected this equation with formulas for a right triangle and with Napier's analogies. Similar expansions into series were encountered, however, even with Lambert.

An excellent book, which had almost the same significance in Romance countries as Klugel's textbook in Germanic countries, was Cagnoli's Plane and Spherical Trigonometry (Italian and French edition, Paris) first published in 1786. Cagnoli still made calculations on trigonometric lines, but he took the radius of the circle equal to unity, and if he did not adopt the designations of the elements of the triangle introduced by Euler, then he generally joined him completely and built all his work on an analytical basis. Despite the fact that a lot has already been done in this direction, he still managed to make some improvements in a number of points. It is worth mentioning its reduction to a logarithmic form using the auxiliary angle of the plane cosine theorem (according to Mayer's method), as well as the spherical cosine theorem. B the derivation of new formulas for spherical right-angled triangles, which had the purpose of greater accuracy of calculations, and further the establishment of elegant formulas connecting the elements of a spherical triangle with the elements of the corresponding triangle composed of chords. Regardless of Kestner, Cagnoli solved the equation a cos A + b sin A = n using substitutions a = t cos B and b = t sin B, he added an important relationship between the six elements of a spherical triangle: sin with sin a + cos with cos a cos B \u003d sin A sin C cos A cos C cos b, as well as the derivation of the sum of the trigonometric series, and extended the summation to the case n x powers of such functions.

S. Lacroix in his "Elementary course of rectilinear and spherical trigonometry, etc." (Paris, 1798/99), in contrast to Cagnoli, he introduced the symbolism of Euler, but almost everywhere he retained in the formulas R.

To an account of the immediate successes of trigonometry, we will add a brief survey of the form that the system of trigonometry gradually took on in textbooks. We have repeatedly noted that, with the exception of Klugel, all other authors still introduced the radius r as an integer sine, partly, however, only in order to satisfy the still-remaining need for homogeneity of expressions. Only to simplify the formulas, r was often assumed to be equal to unity. The definition of the signs of segments has by no means been assimilated by the majority of mathematicians. Therefore, the signs of the functions of the angles, b o greater than 90 °, were clearly established only with the help of goniometric formulas, and for the tangent the proportion was especially important

tg  : r = sin  : cos 

This form of proportion, as well as the use of the radius r, often continued to be preserved in other cases, even in XIX century. However, it has become almost universal. modern designation functions given by Euler. In great books like Legendre's Principia Geometrie (1794), Theoretical Mathematics (Rostock, 1760), and Carsten's System of Mathematics, the functions of negative arguments were dealt with quite correctly, beginning with Euler's Introduction to Analysis.

Goniometric formulas were still often derived, each by itself, and moreover, geometrically. True, Legendre and Cagnoli derived the basic formulas from the addition theorem, but only Klugel clearly expressed the fundamental meaning of this theorem. Justify the addition theorem itself for angles, b o greater than 90°, only Legendre found it necessary. "Higher trigonometry" included in his work, in addition to Klugel, also Cagnoli. Carsten and A.R. Maudui did the same in their "Principles of spherical astronomy, etc." (Paris, 1765), and in a particularly wide volume L. Bertrand in the second volume of his work, published under the title "A New Exposition of the Elementary Part of Mathematics" (Geneva, 1778).

The designation of angles proposed by Euler was adopted only by Klugel and Kestner. In other authors, due to the lack of uniform symbolism, the formulas were unnecessarily complicated. Cagnoli had the most complete system of formulas for a plane triangle, including the so-called Mollweide equations, deduced in a completely modern way. However, even the calculation of flat right-angled triangles received its modern simple form only in the 19th century, when the whole sine was abandoned.

Spherical trigonometry was presented in textbooks almost entirely analytically. The exception was "Trigonometrie spherique" (Paris, 1757) by S. Valletta, who joined Boskovic's "Construction of Spherical Trigonometry", rediscovered everything that was included in Ptolemy's "Analemma", and graphically developed the entire sphere. In all other books, and after the publication of Euler's second work, all formulas for a right triangle were first established, and then an oblique triangle was considered separately, which for this was divided into two right triangles using height. The application of the theorems to constituent right-angled triangles gave, after eliminating the height, a system of five equations, which was already in the "Principles of Arithmetic and Geometry" (1739) by Segner and passed into everything.

In conclusion, we will briefly consider attempts to build all trigonometry on the simplest possible basis. Already Kestner, in later editions of the Foundations (for example, 3rd ed., 1774), derived the main formulas of trigonometry from the sine theorem and the fact that A + B + C \u003d 180 °, and Oppel showed even earlier how you can get all the formulas spherical trigonometry from the sine and cosine theorems. Even earlier, F.K. Mayer pointed out the possibility of achieving this with the help of the cosine theorem alone. Realized this possibility only de Gua.

Lagrange, who began with the same as de Gua, thanks to a much more skillful handling of formulas, first gave the derivation of all other trigonometric equalities a modern look (1798/99). First of all, he proved the Euler relation. Just as simply, he deduced the cotangent rule. Lagrange emphasized that with the help of these four formulas one can solve any problem, but he also deduced, constantly using the polar triangle, all the other formulas we already know. Lagrange's article was a fitting crowning achievement of trigonometry at the turn of the nineteenth century.

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