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 mathematic althe expression of these ratios 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 gave impetus to 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 system of calculus, which gave rise to the modern decimal system used in many applied sciences.
It is assumed that trigonometry 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 ages
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 triangles (spherical). His writings date back to the 2nd century BC.
Also, one of the most important achievements of those times is the determination of the ratio of 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 of the world, which dominatedto 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. These are, in particular, the technology of astronomical instruments, the theory of cartographic projections, the celestial coordinate system, etc.
Middle Ages: research by Indian scientists
Indian medieval astronomers achieved significant success. The death of ancient science in the 4th century caused the center of mathematics to move to India.
The history of trigonometry as a separate section of mathematical teaching began in the Middle Ages. It was then that scientists replaced chords with sines. This discovery made it possible to introduce functions relating to the study of the sides and angles of a right triangle. That is, it was then that trigonometry began to separate from astronomy, turning into a branch of mathematics.
The first tables of sines were in Aryabhata, they were drawn through 3o, 4o, 5o . Later, detailed versions of the tables appeared: in particular, Bhaskara gave a table of sines through1o.
The first specialized treatise on trigonometry appeared in the X-XI century. Its author was the Central Asian scientist Al-Biruni. And in his main work "Canon Mas'ud" (book III), the medieval author goes even deeper into trigonometry, giving a table of sines (with a step of 15 ') and a table of tangents (with a step of 1 °).
History of the development of trigonometry in Europe
After the translation of Arabic treatises into Latin (XII-XIII c), 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 mention trigonometric functions in their writings.
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, Johannes Kepler, Francois Vieta. Copernicus devoted several chapters to trigonometry in his treatise On the Revolutions of the Celestial Spheres (1543). A little later, in the 60sXVI century, Retik - a student of Copernicus - gives fifteen-digit trigonometric tables in his work "The Optical Part of Astronomy".
François Viète in the "Mathematical Canon" (1579) gives a thorough and systematic, albeit unproven, characterization of plane and spherical trigonometry. And Albrecht Dürer was the one who gave birth to the sinusoid.
Merit of Leonhard Euler
Giving trigonometry a modern content and appearance 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 the inverse functions. But that's not all.
Determination of trigonometric functions on the entire number line became possible thanks to Euler's studies of not only permissible negative angles, but also angles greater than 360°. It was he who first proved in his works that the cosine and tangent of a right angle are negative. The expansion of integer powers of cosine and sine also became the merit of this scientist. The general theory of trigonometric series and the study of the convergence of the resulting series were not 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.
Fields of applicationtrigonometry
Trigonometry is not an applied science, 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 examinations, ultrasound and computed tomography), pharmaceuticals, chemistry, theory numbers, 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 studied and to this day. Perhaps in the future there will be even more areas of its application.
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. Workssuch 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 closely related sine (i.e. "bend").
The word "cosine" appeared much later. This term is a shortened version of the Latin phrase "additional sine".
The emergence of tangents is connected 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.”
