Application of inverse trigonometric functions in life. History of trigonometry: origin and development

MUNICIPAL EDUCATIONAL INSTITUTION

"GYMNASIUM №1"

"TRIGONOMETRY IN REAL LIFE"

information project

Completed:

Krasnov Egor

9th grade student

Supervisor:

Borodkina Tatyana Ivanovna

Zheleznogorsk

Introduction………………………………………………………..……3

Relevance…………………………………………………….3

Purpose……………………………………………………………4

Tasks………………………………………………………….4

1.4 Methods………………………………………………………...4

2. Trigonometry and the history of its development………………………………..5

2.1. Trigonometry and stages of formation….………………….5

2.2. Trigonometry as a term. Feature……………….7

2.3. Occurrence of the sinus……………………….……………….7

2.4. The emergence of cosine…………………….……………….8

2.5. The emergence of tangent and cotangent……...……………….9

2.6 Further development of trigonometry……...………………..9

3. Trigonometry and real life……………………..……………...12

3.1.Navigation……………………………..…………………….....12

3.2 Algebra….……………………………..…………………….....14

3.3.Physics….……………………………..…………………….....14

3.4. Medicine, biology and biorhythms.…..…………………….....15

3.5.Music…………………………….…..……………………....19

3.6.Informatics..…………………….…..……………………....21

3.7. The sphere of construction and geodesy.…………………………....22

3.8 Trigonometry in art and architecture………………..…....22

Conclusion. ……………………………..…………………………..…..25

References.………………………….…………….……………27

Appendix 1 .…....………………………….…………….………………29

Introduction

In the modern world, much attention is paid to mathematics as one of the areas of scientific activity and study. As we know, one of the components of mathematics is trigonometry. Trigonometry is a branch of mathematics that studies trigonometric functions. I believe that this topic is, firstly, relevant from a practical point of view. We are graduating from school, and we understand that for many professions, knowledge of trigonometry is simply necessary, because. allows you to measure distances to nearby stars in astronomy, between landmarks in geography, control satellite navigation systems. The principles of trigonometry are also used in such areas as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound and computed tomography), pharmaceuticals, chemistry, number theory ( and, as a result, cryptography), seismology, meteorology, oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography.

Secondly, relevance The topic "Trigonometry in real life" is that knowledge of trigonometry will open up new ways to solve various problems in many areas of science and simplify the understanding of some aspects of various sciences.

It has long been established practice in which students are faced with trigonometry three times. So we can say that trigonometry has three parts. These parts are interconnected and depend on time. At the same time, they are absolutely different, they do not have similar features both in terms of the meaning that is laid down when explaining the basic concepts, and in terms of functions.

The first acquaintance occurs in the 8th grade. This is the period when schoolchildren study: "The ratios between the sides and angles of a right triangle." In the process of studying trigonometry, the concept of cosine, sine and tangent is given.

The next step is to continue acquaintance with trigonometry in grade 9. The level of complexity increases, the ways and methods of solving examples change. Now, in place of cosines and tangents comes the circle and its possibilities.

The last stage is the 10th grade, in which trigonometry becomes more complex, the ways of solving problems change. The concept of the radian measure of an angle is introduced. Graphs of trigonometric functions are introduced. At this stage, students begin to solve and learn trigonometric equations. But not like geometry. To fully understand trigonometry, you need to get acquainted with the history of its origin and development. After getting acquainted with the historical background and studying the activities of the works of great figures, mathematicians and scientists, we can understand how trigonometry affects our lives, how it helps to create new objects, make discoveries.

aim my project is to study the influence of trigonometry in human life and develop interest in it. After solving this goal, we will be able to understand what place trigonometry occupies in our world, what practical problems it solves.

To achieve this goal, we have identified the following tasks:

1. Get acquainted with the history of the formation and development of trigonometry;

2. Consider examples of the practical impact of trigonometry in various fields of activity;

3. Show with examples, the possibilities of trigonometry and its application in human life.

Methods: Search and collection of information.

1. Trigonometry and the history of its development

What is trigonometry? This term implies a section in mathematics that studies the relationship between different angles, studies the lengths of the sides of a triangle and the algebraic identities of trigonometric functions. It is difficult to imagine that this area of ​​mathematics occurs to us in everyday life.

1.1. Trigonometry and the stages of its formation

Let's turn to the history of its development, the stages of formation. Since ancient times, trigonometry has gained its beginnings, developed and showed the first results. We can see the very first information about the emergence and development of this area in the manuscripts that are in ancient Egypt, Babylon, and Ancient China. By examining the 56th problem from the Rhinda Papyrus (2nd millennium BC), one can see that it proposes to find the slope of the pyramid, whose height is 250 cubits high. The length of the side of the base of the pyramid is 360 cubits (Fig. 1). It is curious that the Egyptians in solving this problem simultaneously used two measurement systems - "elbows" and "palms". Today, when solving this problem, we would find the tangent of the angle: knowing half the base and apothem (Fig. 1).

The next step was the stage of development of science, which is associated with the astronomer Aristarchus of Samos, who lived in the III century BC. e. The treatise, which considers the magnitudes and distances of the Sun and the Moon, set itself a specific task. It was expressed in the need to determine the distance to each celestial body. In order to make such calculations, it was required to calculate the ratio of the sides of a right triangle with a known value of one of the angles. Aristarchus considered a right-angled triangle formed by the Sun, Moon and Earth during the quadrature. To calculate the value of the hypotenuse, which was the basis of the distance from the Earth to the Sun, using the leg, which is the basis of the distance from the Earth to the Moon, with a known value of the included angle (87 °), which is equivalent to calculating the value sin angle 3. According to Aristarchus, this value lies in the range from 1/20 to 1/18. This suggests that the distance from the Sun to the Earth is twenty times greater than from the moon to the Earth. However, we know that the Sun is 400 times further away than the location of the Moon. An erroneous judgment arose due to an inaccuracy in the measurement of the angle.

A few decades later, Claudius Ptolemy, in his Ethnogeography, Analemma, and Planisferium, provides a detailed exposition of trigonometric additions to cartography, astronomy, and mechanics. Among other things, a stereographic projection is shown, a number of factual issues are studied, for example: to set the height and angle of a celestial body according to its declination and hour angle. From the point of view of trigonometry, this means that it is necessary to find the side of the spherical triangle according to the other 2 faces and the opposite angle (Fig. 2)

Collectively, it can be noted that trigonometry was used to:

Clearly establishing the time of day;

Calculation of the upcoming location of celestial bodies, episodes of their rising and setting, eclipses of the Sun and Moon;

Finding the geographic coordinates of the current location;

Calculation of the distance between megacities with known geographical coordinates.

Gnomon is an ancient astronomical mechanism, a vertical object (stele, column, pole), which allows using the smallest length of its shadow at noon to determine the angular height of the sun (Fig. 3).

Thus, the cotangent was presented to us as the length of the shadow from a vertical gnomon 12 (sometimes 7) units high. Note that in the original version, these definitions were used to calculate sundial. The tangent was represented by a shadow falling from a horizontal gnomon. Cosecant and secant are understood as hypotenuses, which correspond to right triangles.

1.2. Trigonometry as a term. Characteristic

For the first time, the specific term "trigonometry" occurs in 1505. It was published and used in the book of the German theologian and mathematician Bartholomeus Pitiscus. While science was already used to solve astronomical, architectural problems.

The term trigonometry is characterized by Greek roots. And it consists of two parts: "triangle" and "measure". By studying translation, we can say that we have before us a science that studies the changes in triangles. The appearance of trigonometry is associated with land surveying, astronomy and the construction process. Although the name appeared relatively recently, many of the definitions and data currently attributed to trigonometry were known before the year 2000.

1.3. The occurrence of the sinus

The representation of the sine has a long history. In fact, various relationships between segments of a triangle and a circle (and, in essence, trigonometric functions) are found earlier in the 3rd century. BC. in the works of famous mathematicians of ancient Greece - Euclid, Archimedes, Apollonius of Perga. In the Roman period, these relationships were already quite regularly studied by Menelaus (1st century AD), although they did not receive a special name. The modern sine of the angle α, for example, is studied as a half-chord on which the central angle of magnitude α rests, or as a chord of a doubled arc.

In the subsequent period, mathematics was for a long time most rapidly formed by Indian and Arab scientists. In the 4th-5th centuries, in particular, a special term arose earlier in the works on astronomy of the famous Indian scientist Aryabhata (476-ca. 550), after whom the first Hindu satellite of the Earth is named. He called the segment ardhajiva (ardha-half, jiva-bowstring break, which resembles an axis). Later, a more abbreviated name jiva took root. Arab mathematicians in the IX century. the term jiva (or jiba) was replaced by the Arabic word jaib (concavity). During the transition of Arabic mathematical texts in the XII century. this word was replaced by the Latin sinus (sinus-bend) (Fig. 4).

1.4. The emergence of the cosine

The definition and emergence of the term "cosine" is of a more short-term and narrow-minded nature. By cosine is meant "additional sine" (or otherwise "sine of additional arc"; remember cosα= sin(90° - a)). An interesting fact is that the first ways to solve triangles, which are based on the relationship between the sides and angles of a triangle, were found by the astronomer Hipparchus from Ancient Greece in the second century BC. This study was also carried out by Claudius Ptolemy. Gradually, new facts appeared about the relationship between the ratios of the sides of a triangle and its angles, a new definition began to be applied - the trigonometric function.

A significant contribution to the formation of trigonometry was made by the Arab experts Al-Batani (850-929) and Abu-l-Wafa, Mohamed-bin Mohamed (940-998), who compiled tables of sines and tangents using 10 'with accuracy up to 1/604. The sine theorem was previously known by the Indian professor Bhaskara (b. 1114, the year of death is unknown) and the Azerbaijani astrologer and scientist Nasireddin Tusi Mukhamed (1201-1274). In addition, Nasireddin Tusi in his work “Work on the complete quadrilateral” described direct and spherical trigonometry as an independent discipline (Fig. 4).

1.5. The emergence of tangent and cotangent

Tangents arose in connection with the conclusion of the problem of establishing the length of the shadow. The tangent (and besides the cotangent) was established in the 10th century by the Arabian arithmetician Abul-Wafa, who also compiled the original tables for finding tangents and cotangents. But these discoveries remained unfamiliar to European scientists for a long time, and tangents were rediscovered only in the 14th century by the German arithmetic, astronomer Regimontan (1467). He argued 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 designation "tangent", which comes from the Latin tanger (to touch), arose in 1583. Tangens is translated as "affecting" (the line of tangents is tangent to the unit circle).
Trigonometry was further developed in the works of the outstanding astrologers Nicolaus Copernicus (1473-1543), Tycho Brahe (1546-1601) and Johannes Kepler (1571-1630), and also in the works of the mathematician Francois Vieta (1540-1603), who completely solved the problem in determining absolutely all components of a flat or spherical triangle according to three data (Fig. 4).

1.6 Further development of trigonometry

For a long time, trigonometry had an exclusively geometric form, that is, the data that we currently formulate in the definitions of trigonometric functions were formulated and argued with the support of geometric concepts and statements. It existed as such even in the Middle Ages, although analytical methods were sometimes used in it, especially after the appearance of logarithms. Perhaps, the maximum incentives for the formation of trigonometry appeared in conjunction with the solution of astronomical problems, which gave great positive interest (for example, in order to solve the issues of establishing the location of a ship, forecasting blackout, etc.). Astrologers were occupied with the relationship between the sides and angles of spherical triangles. And arithmetic of antiquity successfully coped with the questions posed.

Starting from the 17th century, trigonometric functions began to be applied to solving equations, questions of mechanics, optics, electricity, radio engineering, in order to display oscillatory actions, wave propagation, movement of various elements, to study alternating galvanic current, etc. For this reason, trigonometric functions have been comprehensively and deeply studied, and have become essential for the whole of mathematics.

The analytical theory of trigonometric functions was mainly created by the outstanding mathematician of the 18th 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.

Later, the part of trigonometry, which studies the properties of trigonometric functions and the relationships between them, began to be called goniometry (in translation - the science of measuring angles, from the Greek gwnia - angle, metrew - I measure). The term goniometry has not been used much in recent years.

2. Trigonometry and real life

Modern society is characterized by constant changes, discoveries, the creation of high-tech inventions that improve our lives. Trigonometry meets and interacts with physics, biology, mathematics, medicine, geophysics, navigation, computer science.

Let's get acquainted in order with the interaction in each industry.

2.1 Navigation

The first point explaining to us the use and benefits of trigonometry is its relationship with navigation. By navigation we mean the science whose purpose is to study and create the most convenient and useful ways of navigation. So, scientists are developing simple navigation, which is building a route from one point to another, evaluating it and choosing the best option from all those offered. These routes are necessary for seafarers who, during their journey, face many difficulties, obstacles, and questions on the course of movement. Navigation is also necessary: ​​pilots who fly complex high-tech aircraft orient themselves, sometimes in very extreme situations; cosmonauts, whose work is associated with a risk to life, with the complex construction of the route and its development. Let us study the following concepts and tasks in more detail. As a task, we can imagine the following condition: we know the geographical coordinates: latitude and longitude between points A and B on the earth's surface. It is necessary to find the shortest path between points A and B along the earth's surface (the radius of the Earth is considered known: R = 6371 km).

We can also present a solution to this problem, namely: first, we clarify that the latitude of the point M of the earth's surface is the value of the angle formed by the radius OM, where O is the center of the Earth, with the plane of the equator: ≤ , and north of the equator, the latitude is considered positive, and south is negative. For the longitude of the point M, we take the value of the dihedral angle passing in the planes COM and SON. By C we mean the North Pole of the Earth. As H, we understand the point corresponding to the Greenwich observatory: ≤ (to the east of the Greenwich meridian, longitude is considered positive, to the west - negative). As we already know, the shortest distance between points A and B on the earth's surface is represented by the length of the smallest of the arcs of the great circle that connects A and B. We can call this kind of arc an orthodrome. Translated from Greek, this term is understood as a right angle. Because of this, our task is to determine the length of the side AB of the spherical triangle ABC, where C is understood as the northern polis.

An interesting example is the following. When creating a route by sailors, precise and painstaking work is necessary. So, for laying the course of the ship on the map, which was made in the projection of Gerhard Mercator in 1569, there was an urgent need to determine the latitude. However, when going to sea, in locations until the 17th century, navigators did not indicate the latitude. For the first time, Edmond Gunther (1623) applied trigonometric calculations in navigation.

With its help of trigonometry, pilots could calculate wind errors for the most accurate and safe aircraft handling. In order to carry out these calculations, we turn to the triangle of speeds. This triangle expresses the formed airspeed (V), wind vector (W), ground speed vector (Vp). PU - track angle, SW - wind angle, KUV - heading wind angle (Fig. 5) .

To get acquainted with the type of dependence between the elements of the navigation triangle of speeds, you need to look below:

Vp \u003d V cos US + W cos SW; sin US = * sin SW, tg SW

To solve the navigation triangle of speeds, counting devices are used that use the navigation ruler and mental calculations.

2.2 Algebra

The next area of ​​interaction of trigonometry is algebra. It is thanks to trigonometric functions that very complex equations and tasks that require large calculations are solved.

As we know, in all cases where it is necessary to interact with periodic processes and oscillations, we come to the use of trigonometric functions. It does not matter what it is: acoustics, optics or pendulum swing.

2.3 Physics

In addition to navigation and algebra, trigonometry has a direct influence and impact in physics. When objects are immersed in water, they do not change their shape or volume in any way. The full secret is the visual effect that forces our vision to perceive the subject in a different way. Simple trigonometric formulas and the values ​​of the sine of the angle of incidence and refraction of the half-line provide the probability of calculating a constant refractive index as the light beam passes from sphere to sphere. For example, a rainbow appears due to the fact that sunlight is refracted in water droplets suspended in the air according to the law of refraction:

sinα / sinβ = n1 / n2

where: n1 is the refractive index of the first medium; n2 is the refractive index of the second medium; α-angle of incidence, β-angle of refraction of light.

Charged elements of the solar wind enter the upper layers of the atmosphere of the planets due to the interaction of the earth's magnetic field with the solar wind.

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

Revealing the practical aspects of the application of trigonometry in physics, we give an example. This problem should be solved using trigonometric formulas and solutions. Conditions of the problem: on an inclined plane, the angle of which is 24.5o, there is a body with a mass of 90 kg. It is necessary to find what force the body exerts pressure on the inclined plane (i.e. what pressure the body exerts on this plane) (Fig. 6).

Having designated the X and Y axes, we will begin to build projections of forces on the axes, first using this formula:

ma = N + mg, then look at the picture,

X: ma = 0 + mg sin24.50

Y: 0 = N - mg cos24.50

we substitute the mass, we find that the force is 819 N.

Answer: 819 N

2.4. Medicine, biology and biorhythms

The fourth area where trigonometry has a serious influence and help is two areas at once: medicine and biology.

One of the fundamental properties of living nature is the cyclicity of most of the processes occurring in it. There is a connection between the movement of celestial bodies and living organisms on Earth. Living organisms not only capture the light and heat of the Sun and Moon, but also have various mechanisms that accurately determine the position of the Sun, respond to the rhythm of the tides, the phases of the Moon and the movement of our planet.

Biological rhythms, biorhythms, are more or less regular changes in the nature and intensity of biological processes. The ability for such changes in vital activity is inherited and found in almost all living organisms. They can be observed in individual cells, tissues and organs, whole organisms and populations. Biorhythms are divided into physiological, having periods from fractions of a second to several minutes and environmental, in duration coinciding with any rhythm of the environment. These include daily, seasonal, annual, tidal and lunar rhythms. The main earthly rhythm is daily, due to the rotation of the Earth around its axis, therefore, almost all processes in a living organism have a daily periodicity.

Many environmental factors on our planet, primarily the light regime, temperature, air pressure and humidity, atmospheric and electromagnetic fields, sea tides, naturally change under the influence of this rotation.

We are seventy-five percent water, and if at the time of the full moon the waters of the world's oceans rise 19 meters above sea level and the tide begins, then the water in our body also rushes into the upper parts of our body. And people with high blood pressure often experience exacerbations of the disease during these periods, and naturalists who collect medicinal herbs know exactly in which phase of the moon to collect "tops - (fruits)", and in which - "roots".

Have you noticed that at certain periods your life makes inexplicable jumps? Suddenly, out of nowhere - emotions overflow. Sensitivity increases, which can suddenly be replaced by complete apathy. Creative and barren days, happy and unhappy moments, mood swings. It is noted that the capabilities of the human body change periodically. This knowledge underlies the "theory of three biorhythms".

Physical biorhythm - regulates physical activity. During the first half of the physical cycle, a person is energetic, and achieves better results in his activity (the second half - energy is inferior to laziness).

Emotional rhythm - during periods of its activity, sensitivity increases, mood improves. A person becomes excitable to various external cataclysms. If he is in a good mood, he builds castles in the air, dreams of falling in love and falls in love. With a decrease in the emotional biorhythm, a decline in mental strength occurs, desire and joyful mood disappear.

Intelligent biorhythm - he controls memory, the ability to learn, logical thinking. In the activity phase, there is an increase, and in the second phase, a decline in creative activity, there is no luck and success.

Theory of three rhythms:

· Physical cycle -23 days. Determines energy, strength, endurance, coordination of movement

Emotional cycle - 28 days. The state of the nervous system and mood

· Intellectual cycle - 33 days. Determines the creative ability of the individual

Trigonometry is also found in nature. 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. When swimming, the body of the fish takes the form of a curve that resembles the graph of the function y=tgx.

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

Trigonometry in medicine. As a result of a study conducted by a student at the Iranian University of Shiraz, Wahid-Reza Abbasi, physicians for the first time were able to streamline information related to the electrical activity of the heart, or, in other words, electrocardiography.

The formula, called Tehran, was presented to the general scientific community at the 14th Conference of Geographical Medicine and then at the 28th Conference on the Application of Computer Technology in Cardiology, held in the Netherlands.

This formula is a complex algebraic-trigonometric equation, consisting of 8 expressions, 32 coefficients and 33 main parameters, including several additional ones for calculations in cases of arrhythmia. According to doctors, this formula greatly facilitates the process of describing the main parameters of the activity of the heart, thereby speeding up the diagnosis and the start of the actual treatment.

Many people have to do an ECG of the heart, but few know that the ECG of the human heart is a sine or cosine plot.

Trigonometry helps our brain determine the distances to objects. American scientists claim that the brain estimates the distance to objects by measuring the angle between the ground plane and the plane of vision. This conclusion was made after a series of experiments in which participants were asked to look at the world around them through prisms that increase this angle.

This distortion caused experimental prism wearers to perceive distant objects as closer and could not cope with the simplest tests. Some of the participants in the experiments even leaned forward, trying to align their bodies perpendicular to the misrepresented ground surface. However, after 20 minutes, they got used to the distorted perception, and all problems disappeared. This circumstance indicates the flexibility of the mechanism by which the brain adapts the visual system to changing external conditions. It is interesting to note that after the prisms were removed, the opposite effect was observed for some time - an overestimation of the distance.

The results of the new study, as you might expect, will be of interest to engineers designing navigation systems for robots, as well as specialists who are working on creating the most realistic virtual models. Applications are also possible in the field of medicine, in the rehabilitation of patients with damage to certain areas of the brain.

2.5.Music

The musical field also interacts with trigonometry.

I present to your attention interesting information about a certain method that accurately provides a connection between trigonometry and music.

This method of analyzing musical works is called "geometric theory of music". With its help, the main musical structures and transformations are translated into the language of modern geometry.

Each note within the framework of the new theory is represented as the logarithm of the frequency of the corresponding sound (the note "to" the first octave, for example, corresponds to the number 60, the octave to the number 12). A chord is thus represented as a point with given coordinates in geometric space. Chords are grouped into different "families" that correspond to different types of geometric spaces.

When developing a new method, the authors used 5 known types of musical transformations that were not previously taken into account in music theory when classifying sound sequences - octave permutation (O), permutation (P), transposition (T), inversion (I) and cardinality change (C) . All these transformations, as the authors write, form the so-called OPTIC-symmetries in n-dimensional space and store musical information about the chord - in what octave its notes are, in what sequence they are played, how many times they are repeated, and so on. Using OPTIC symmetries, similar but not identical chords and their sequences are classified.

The authors of the article show that various combinations of these 5 symmetries form many different musical structures, some of which are already known in music theory (a sequence of chords, for example, will be expressed in new terms as OPC), and others are fundamentally new concepts that , perhaps, will be adopted by the composers of the future.

As an example, the authors give a geometric representation of various types of chords of four sounds - a tetrahedron. The spheres on the graph represent the types of chords, the colors of the spheres correspond to the size of the intervals between chord sounds: blue - small intervals, warmer tones - more "rarefied" chord sounds. The red sphere is the most harmonious chord with equal intervals between notes, which was popular with composers of the 19th century.

The "geometric" method of music analysis, according to the authors of the study, can lead to the creation of fundamentally new musical instruments and new ways of visualizing music, as well as to make changes in modern methods of teaching music and ways of studying various musical styles (classical, pop music, rock music). music, etc.). The new terminology will also help to compare the musical works of composers from different eras in more depth and present the results of research in a more convenient mathematical form. In other words, it is proposed to single out their mathematical essence from musical works.

Frequencies corresponding to the same note in the first, second, etc. octaves, relate as 1:2:4:8... According to the legends that have come down from antiquity, the first who tried to do this were Pythagoras and his students.

Diatonic scale 2:3:5 (Fig. 8).

2.6.Computer science

Trigonometry with its influence did not bypass computer science. So, its functions are applicable for accurate calculations. Thanks to this point, we can approximate any (in some sense "good") function by expanding it into a Fourier series:

a0 + a1 cos x + b1 sin x + a2 cos 2x + b2 sin 2x + a3 cos 3x + b3 sin 3x + ...

The process of selecting a number in the most appropriate way numbers a0, a1, b1, a2, b2, ..., can be represented in the form of such an (infinite) sum by almost any function in a computer with the required accuracy.

Trigonometry has a serious role and assistance in the development and in the process of working with graphic information. If you need to simulate a process, with a description in electronic form, with the rotation of a certain object around a certain axis. There is a rotation through a certain angle. To determine the coordinates of the points, you will have to multiply by sines and cosines.

So, you can cite Justin Windell, a programmer and designer working at the Google Grafika Lab, as an example. He posted a demo that shows an example of using trigonometric functions to create dynamic animations.

2.7. Sphere of construction and geodesy

An interesting branch that interacts with trigonometry is the field of construction and geodesy. The lengths of the sides and the angles of an arbitrary triangle on the plane are interconnected by certain relations, the most important of which are called the cosine and sine theorems. Formulas containing a, b, c imply that the letters are represented by the sides of the triangle, which lie respectively opposite the angles A, B, C. These formulas allow us to restore the remaining three elements from the three elements of the triangle - the lengths of the sides and the angles. They are used in solving practical problems, for example, in geodesy.

All "classical" geodesy is based on trigonometry. Since, in fact, since ancient times, surveyors have been fascinated by the fact that they "solve" triangles.

The process of erecting buildings, roads, bridges and other buildings begins with survey and design work. Without exception, all measurements at a construction site are carried out with the support of geodetic instruments, such as a total station and a trigonometric level. With trigonometric leveling, the height difference between several points on the earth's surface is established.

2.8 Trigonometry in art and architecture

From the time that man began to exist on earth, science has become the basis for improving everyday life and other areas of life. The foundations of everything that is created by man are various directions in the natural and mathematical sciences. One of them is geometry. Architecture is not the only field of science in which trigonometric formulas are used. Most of the compositional decisions and construction of drawings took place precisely with the help of geometry. But theoretical data mean little. Consider an example of the construction of one sculpture by the French master of the Golden Age of Art.

The proportional relationship in the construction of the statue was perfect. However, when the statue was raised to a high pedestal, it looked ugly. The sculptor did not take into account that many details are reduced in perspective towards the horizon, and when viewed from the bottom up, the impression of its ideality is no longer created. A lot of calculations were carried out so that the figure from a great height looked proportional. Basically, they were based on the method of sighting, that is, an approximate measurement, by eye. However, the coefficient of difference of certain proportions made it possible to make the figure closer to the ideal. Thus, knowing the approximate distance from the statue to the point of view, namely from the top of the statue to the eyes of a person and the height of the statue, we can calculate the sine of the angle of incidence of the gaze using the table, thereby finding the point of view (Fig. 9).

In Figure 10, the situation changes, since the statue is raised to the height AC and HC increase, we can calculate the cosine of angle C, using the table we find the angle of incidence of the gaze. In the process, you can calculate AH, as well as the sine of angle C, which will allow you to check the results using the basic trigonometric identity cos 2 a + sin 2 a = 1.

By comparing the measurements of AH in the first and second cases, one can find the proportionality coefficient. Subsequently, we will receive a drawing, and then a sculpture, when raised, the figure will be visually close to the ideal.

Iconic buildings around the world have been designed with mathematics that can be considered the genius of architecture. Some famous examples of such buildings are the Gaudí Children's School in Barcelona, ​​the Mary Ax in London, the Bodegas Isios Winery in Spain, and the Los Manantiales Restaurant in Argentina. The design of these buildings was not without trigonometry.

Conclusion

Having studied the theoretical and applied aspects of trigonometry, I realized that this branch is closely connected with many sciences. At the very beginning, trigonometry was essential for making and taking measurements between angles. However, later a simple measurement of angles grew into a full-fledged science that studies trigonometric functions. We can identify the following areas in which there is a close connection between trigonometry and the physics of architecture, nature, medicine, and biology.

So, thanks to trigonometric functions in medicine, the formula of the heart was discovered, which is a complex algebraic-trigonometric equality, which consists of 8 expressions, 32 coefficients and 33 main parameters, including the possibility of additional miscalculations in the event of arrhythmia. This discovery helps doctors to provide more qualified and high-quality medical care.

Let's also note. that all classical geodesy is based on trigonometry. Since, in fact, since ancient times, surveyors have been engaged in "solving" triangles. The process of building buildings, roads, bridges and other structures begins with survey and design work. All measurements at the construction site are carried out using surveying instruments such as theodolite and trigonometric level. With trigonometric leveling, the height difference between several points on the earth's surface is determined.

Getting acquainted with its influence in other areas, we can conclude that trigonometry actively affects human life. The connection of mathematics with the outside world allows you to "materialize" the knowledge of schoolchildren. Thanks to this, we can more adequately perceive and assimilate the knowledge and information that we are taught at school.

The goal of my project has been successfully completed. I studied the influence of trigonometry in life and the development of interest in it.

To achieve this goal, we completed the following tasks:

1. We got acquainted with the history of the formation and development of trigonometry;

2. Considered examples of the practical impact of trigonometry in various fields of activity;

3. Showed with examples the possibilities of trigonometry and its application in human life.

Studying the history of the emergence of this industry will help to arouse interest among schoolchildren, form the right worldview and improve the general culture of a high school student.

This work will be useful for high school students who have not yet seen the beauty of trigonometry and are not familiar with the areas of its application in the surrounding life.

Bibliography

Glazer G.I.

Glazer G.I.

Rybnikov K.A.

Bibliography

A.N. Kolmogorov, A.M. Abramov, Yu.P. Dudnitsin et al. "Algebra and the Beginnings of Analysis" Textbook for grades 10-11 of educational institutions, M., Education, 2013.

Glazer G.I. History of mathematics at school: VII-VIII class. - M.: Education, 2012.

Glazer G.I. History of mathematics at school: IX-X cells. - M.: Education, 2013.

Rybnikov K.A. History of Mathematics: Textbook. - M.: Publishing house of Moscow State University, 1994 and. - M.: Higher School, 2016. - 134 p.

Olechnik, S.N. Problems in algebra, trigonometry and elementary functions / S.N. Olekhnik. - M.: Higher school, 2013. - 645 p.

Potapov, M.K. Algebra, trigonometry and elementary functions / M.K. Potapov. - M.: Higher school, 2014. - 586 p.

Potapov, M.K. Algebra. Trigonometry and elementary functions / M.K. Potapov, V.V. Aleksandrov, P.I. Pasichenko. - M.: [not specified], 2015. - 762 p.

Appendix 1

Fig.1Image of a pyramid. Slope calculation b / h .

Goniometer Seked In general, the Egyptian formula for calculating the seked of the pyramid looks like So:.

Ancient Egyptian term seked” denoted the angle of inclination. It was across the height, divided into half the base. "The length of the pyramid on the east side is 360 (cubits), the height is 250 (cubits). You need to calculate the slope of the eastern side. To do this, take half of 360, i.e. 180. Divide 180 by 250. You get: 1 / 2 , 1 / 5 , 1 / 50 elbow. Note that one cubit is equal to 7 hand widths. Now multiply the resulting numbers by 7 as follows: " Fig.2Gnomon Fig.3 Determination of the angular height of the sun Fig.4 Basic formulas of trigonometry Fig.5 Navigation in trigonometry Fig.6 Physics in trigonometry Fig.7 Theory of three rhythms ( The physical cycle is 23 days. Determines energy, strength, endurance, coordination of movement; The emotional cycle is 28 days. The state of the nervous system and mood; Intellectual cycle - 33 days. Determines the creative ability of the individual) Rice. 8 Trigonometry in music Fig.9, 10 Trigonometry in architecture

MBOU Tselinnaya secondary school

Report Trigonometry in real life

Prepared and conducted

mathematic teacher

qualification category

Ilyina V.P.

Tselinny March 2014

Table of contents.

1. Introduction .

2. The history of the creation of trigonometry:

Early centuries.

Ancient Greece.

Middle Ages.

New time.

From the history of the development of spherical geometry.

3. Trigonometry and real life:

Application of trigonometry in navigation.

Trigonometry in algebra.

Trigonometry in physics.

Trigonometry in medicine and biology.

Trigonometry in music.

Trigonometry in computer science

Trigonometry in construction and geodesy.

4. Conclusion .

5. List of references.

Introduction

It has long been established in mathematics that in the systematic study of mathematics, we students have to meet with trigonometry three times. Accordingly, its content appears to consist of three parts. These parts during training are separated from each other in time and do not resemble each other both in terms of the meaning invested in the explanations of the basic concepts, and in terms of the developed apparatus and service functions (applications).

Indeed, for the first time we met trigonometric material in the 8th grade when studying the topic “Ratios between the sides and angles of a right triangle”. So we learned what sine, cosine and tangent are, learned how to solve flat triangles.

However, some time passed and in the 9th grade we returned to trigonometry again. But this trigonometry is not like the one studied before. Its ratios are now defined with the help of a circle (a unit semicircle), and not a right triangle. Although they are still defined as functions of angles, these angles are already arbitrarily large.

Having moved to the 10th grade, we again encountered trigonometry and saw that it had become even more difficult, the concept of the radian measure of an angle was introduced, and trigonometric identities, and the formulation of problems, and the interpretation of their solutions look different. Graphs of trigonometric functions are introduced. Finally, trigonometric equations appear. And all this material appeared before us already as part of algebra, and not as geometry. And it became very interesting for us to study the history of trigonometry, its application in everyday life, because the use of historical information by a mathematics teacher is not mandatory when presenting the material of the lesson. However, as K. A. Malygin points out, “... excursions into the historical past enliven the lesson, give relaxation to mental stress, raise interest in the material being studied and contribute to its lasting assimilation.” Moreover, the material on the history of mathematics is very extensive and interesting, since the development of mathematics is closely connected with the solution of urgent problems that have arisen in all periods of the existence of civilization.

Having learned about the historical reasons for the emergence of trigonometry, and having studied how the fruits of the activities of great scientists influenced the development of this area of ​​\u200b\u200bmathematics and the solution of specific problems, we, among schoolchildren, increase interest in the subject being studied, and we will see its practical significance.

Objective of the project - development of interest in the study of the topic "Trigonometry" in the course of algebra and the beginning of analysis through the prism of the applied value of the material being studied; expansion of graphic representations containing trigonometric functions; application of trigonometry in such sciences as physics, biology, etc.

The connection of trigonometry with the outside world, the importance of trigonometry in solving many practical problems, the graphical capabilities of trigonometric functions make it possible to "materialize" the knowledge of schoolchildren. This allows you to better understand the vital need for knowledge acquired in the study of trigonometry, increases interest in the study of this topic.

Research objectives:

1. Consider the history of the emergence and development of trigonometry.

2. Show practical applications of trigonometry in various sciences with concrete examples.

3.Explain on specific examples the possibilities of using trigonometric functions, which allow turning "little interesting" functions into functions whose graphs have a very original look.

"One thing remains clear, that the world is arranged menacingly and beautifully."

N. Rubtsov

Trigonometry - is a branch of mathematics that studies the relationship between the angles and the lengths of the sides of triangles, as well as the algebraic identities of trigonometric functions. It is hard to imagine, but we encounter this science not only in mathematics lessons, but also in our daily life. We might not be aware of this, but trigonometry is found in such sciences as physics, biology, it plays an important role in medicine, and, most interestingly, even music and architecture could not do without it. Problems with practical content play a significant role in developing the skills to apply theoretical knowledge gained in the study of mathematics in practice. Every student of mathematics is interested in how and where the acquired knowledge is applied. This work provides an answer to this question.

The history of the creation of trigonometry

Early centuries

The usual measurement of angles in degrees, minutes and seconds originates from Babylonian mathematics (the introduction of these units into ancient Greek mathematics is usually attributed to the 2nd century BC).

The main achievement of this period was the ratio of the legs and hypotenuse in a right triangle, which later received the name.

Ancient Greece

A general and logically coherent presentation of trigonometric relations appeared in ancient Greek geometry. Greek mathematicians did not yet single out trigonometry as a separate science, for them it was part of astronomy.
The main achievement of the ancient trigonometric theory was the general solution of the problem of “solving triangles”, that is, finding the unknown elements of a triangle, based on three given elements (of which at least one is a side).

Middle Ages

In the IV century, after the death of ancient science, the center of development of mathematics moved to India. They changed some of the concepts of trigonometry, bringing them closer to modern ones: for example, they were the first to introduce the cosine into use.
The first specialized treatise on trigonometry was the work of the Central Asian scientist (X-XI century) "The Book of the Keys of the Science of Astronomy" (995-996). The whole course of trigonometry contained the main work of Al-Biruni - "The Canon of Mas'ud" (Book III). In addition to the tables of sines (with a step of 15 "), Al-Biruni gave tables of tangents (with a step of 1 °).

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. Apparently, the first acquaintance of Europeans with trigonometry took place thanks to the zij, two translations of which were made in the 12th century.

The first European work devoted entirely to trigonometry is often called the Four Treatises on Direct and Reversed Chords by an English astronomer (circa 1320). Trigonometric tables, often translated from Arabic, but sometimes original, are contained in the works of a number of other authors of the 14th-15th centuries. Then trigonometry took its place among the university courses.

new time

The word "trigonometry" is first encountered (1505) in the title of a book by the German theologian and mathematician Pitiscus. The origin of this word is Greek: triangle, measure. In other words, trigonometry is the science of measuring triangles. Although the name arose relatively recently, many of the concepts and facts now related to trigonometry were already known two thousand years ago.

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 found already in the ӀӀӀ c. BC e in the works of the great mathematicians of Ancient Greece - Euclid, Archimedes, Apollonius of Perga. In the Roman period, these relations were already quite systematically studied by Menelaus (Ӏ century BC), although they did not acquire a special name. The modern minus of an angle, for example, was studied as a product of half-chords, on which the central angle is supported by a value, or as a chord of a doubled arc.

In the subsequent period, mathematics was most actively developed by Indian and Arab scientists for a long time. In ӀV- Vcenturies In particular, a special term appeared in the works on astronomy of the great Indian scientist Aryabhata (476-ca. 550), after whom the first Indian satellite of the Earth is named.

Later, a shorter name jiva was adopted. Arab mathematicians in ΙXin. the word jiva (or jiba) was replaced by the Arabic word jaib (bulge). When translating Arabic mathematical texts intoXΙΙin. this word was replaced by the Latin sine (sinus- bend, curvature)

The word cosine is much younger. Cosine is an abbreviation of the Latin expressioncomplementsinus, i.e. "additional sine" (or otherwise "sine of the additional arc"; remembercosa= sin(90°- a)).

Dealing with trigonometric functions, we essentially go beyond the scope of the task of "measuring triangles". Therefore, the famous mathematician F. Klein (1849-1925) proposed to call the theory of "trigonometric" functions otherwise - goniometry (angle). However, this name did not stick.

Tangents arose in connection with the solution of the problem of determining the length of the shadow. Tangent (as well as cotangent, secant and cosecant) is introduced inXin. 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 inXIVin. first by the English scientist T. Braverdin, and later by the German mathematician, astronomer Regiomontanus (1467). The name "tangent" comes from the Latintanger(to touch), appeared in 1583Tangentstranslated as "touching" (remember: the line of tangents is tangent to the unit circle)

Modern designationsarc sin and arctgappear in 1772 in the works of the Viennese mathematician Sherfer and the famous French scientist J.L. Lagrange, although J. Bernoulli had already considered them a little earlier, who used a different symbolism. But these symbols became generally accepted only at the endXVΙΙΙcenturies. The prefix "arc" comes from the Latinarcusx, for example -, this is an angle (or, one might say, an arc), the sine of which is equal tox.

For a long time, trigonometry developed as part of geometry, i.e. the facts that we now formulate in terms of trigonometric functions were formulated and proved with the help of geometric concepts and statements. Perhaps the greatest incentives for the development of trigonometry arose in connection with solving problems of astronomy, which was of great practical interest (for example, for solving problems of determining the location of a ship, predicting eclipses, etc.)

Astronomers were interested in the relationship between the sides and angles of spherical triangles made up of great circles lying on a sphere. And it should be noted that the mathematicians of antiquity successfully coped with problems that were much more difficult than problems on solving plane triangles.

In any case, in geometric form, many trigonometry formulas known to us were discovered and rediscovered by ancient Greek, Indian, Arab mathematicians (although the formulas for the difference of trigonometric functions became known only inXVΙӀ v. - they were brought out by the English mathematician Napier to simplify calculations with trigonometric functions. And the first drawing of a sinusoid appeared in 1634.)

Of fundamental importance was the compilation by K. Ptolemy of the first table of sines (for a long time it was called the table of chords): a practical tool appeared for solving a number of applied problems, and first of all, problems of astronomy.

When dealing with ready-made tables, or using a calculator, we often do not think about the fact that there was a time when tables had not yet been invented. In order to compile them, it was necessary to perform not only a large amount of calculations, but also to come up with a way to compile tables. Ptolemy's tables are accurate to five decimal places, inclusive.

The modern form of trigonometry was given by the largest mathematicianXVΙӀΙ century L. Euler (1707-1783), a Swiss by birth, worked for many years in Russia and was a member of the St. Petersburg Academy of Sciences. It was Euler who first introduced the well-known definitions of trigonometric functions, began to consider functions of an arbitrary angle, and received reduction formulas. All this is a small fraction of what Euler managed to do in mathematics over a long life: he left over 800 papers, proved many theorems that have become classical, related to the most diverse areas of mathematics. But if you are trying to operate with trigonometric functions in geometric form, that is, in the way that many generations of mathematicians did before Euler, then you will be able to appreciate Euler's merits in the systematization of trigonometry. After Euler, trigonometry acquired a new form of calculus: various facts began to be proved by the formal application of trigonometry formulas, the proofs became much more compact, simpler.

From the history of the development of spherical geometry .

It is widely known that Euclidean geometry is one of the most ancient sciences: already inIIIcentury BC Euclid's classic work "Beginnings" appeared. Less well known is that spherical geometry is only slightly younger. Her first systematic exposition refers toI- IIcenturies. In the book "Sphere", written by the Greek mathematician Menelaus (Ic.), the properties of spherical triangles were studied; it was proved, in particular, that the sum of the angles of a spherical triangle is greater than 180 degrees. Another Greek mathematician Claudius Ptolemy made a big step forward (IIin.). In essence, he was the first to compile tables of trigonometric functions and introduce the stereographic projection.

Just like the geometry of Euclid, spherical geometry arose when solving problems of a practical nature, and primarily problems of astronomy. These tasks were necessary, for example, for travelers and navigators who navigated by the stars. And since in astronomical observations it is convenient to assume that both the Sun and the Moon and the stars move along the depicted "celestial sphere", it is natural that knowledge of the geometry of the sphere was required to study their movement. It is no coincidence, therefore, that Ptolemy's most famous work was called "The Great Mathematical Construction of Astronomy in 13 Books".

The most important period in the history of spherical trigonometry is associated with the activities of scientists in the Middle East. Indian scientists successfully solved problems of spherical trigonometry. However, the method described by Ptolemy and based on the theorem of Menelaus of the complete quadrilateral was not used by them. And in spherical trigonometry, they used projective methods that corresponded to those in Ptolemy's Analemma. As a result, they obtained a set of specific computational rules that made it possible to solve almost any problem of spherical astronomy. With their help, such a problem was ultimately reduced to comparing similar flat right-angled triangles with each other. When solving, the theory of quadratic equations and the method of successive approximations were often used. An example of an astronomical problem that Indian scientists solved using the rules they developed is the problem considered in the work Panga Siddhantika by Varahamihira (V- VI). It consists in finding the height of the Sun, if the latitude of the place is known, the declination of the Sun and its hour angle. As a result of solving this problem, after a series of constructions, a relation is established that is equivalent to the modern cosine theorem for a spherical triangle. However, this relationship, and another equivalent to the sine theorem, have not been generalized as rules applicable to any spherical triangle.

Among the first Eastern scholars who turned to the discussion of the theorem of Menelaus, one should name the Banu Mussa brothers - Muhammad, Hasan and Ahmad, the sons of Musa ibn Shakir, who worked in Baghdad and studied mathematics, astronomy and mechanics. But the earliest surviving work on Menelaus' theorem is the "Treatise on the figure of secant" by their student Thabit ibn Korra (836-901)

The treatise of Thabit ibn Korra has come down to us in the Arabic original. And in Latin translationXIIin. This translation by Gerando of Cremona (1114-1187) was widely used in Medieval Europe.

The history of trigonometry, as a science of the relationships between the angles and sides of a triangle and other geometric figures, covers more than two millennia. Most of these relationships cannot be expressed using ordinary algebraic operations, and therefore it was necessary to introduce special trigonometric functions, originally presented in the form of numerical tables.
Historians believe that trigonometry was created by ancient astronomers, and a little later it began to be used in architecture. Over time, the scope of trigonometry has constantly expanded, today it includes almost all natural sciences, technology and a number of other areas of activity.

Applied trigonometric problems are very diverse - for example, measurable results of operations on the listed quantities (for example, the sum of angles or the ratio of side lengths) can be set.

In parallel with the development of plane trigonometry, the Greeks, under the influence of astronomy, advanced spherical trigonometry far. In Euclid's "Principles" on this topic, there is only a theorem on the ratio of the volumes of balls of different diameters, but the needs of astronomy and cartography caused the rapid development of spherical trigonometry and related areas - the celestial coordinate system, the theory of cartographic projections, and the technology of astronomical instruments.

courses.

Trigonometry and real life

Trigonometric functions have found application in mathematical analysis, physics, computer science, geodesy, medicine, music, geophysics, and navigation.

Application of trigonometry in navigation

Navigation (this word comes from the Latinnavigation- sailing on a ship) - one of the most ancient sciences. The simplest tasks of navigation, such as, for example, determining the shortest route, choosing the direction of movement, faced the very first navigators. At present, these and other tasks have to be solved not only by sailors, but also by pilots and astronauts. Let's consider some concepts and tasks of navigation in more detail.

Task. Geographic coordinates are known - the latitude and longitude of points A and B of the earth's surface:, and, . It is required to find the shortest distance between points A and B along the earth's surface (the radius of the Earth is considered known:R= 6371 km)

Decision. Recall first that the latitude of the point M of the earth's surface is the value of the angle formed by the radius OM, where O is the center of the Earth, with the plane of the equator: ≤ , and to the north of the equator, the latitude is considered positive, and to the south - negative

The longitude of the point M is the value of the dihedral angle between the planes COM and SON, where C is the North Pole of the Earth, and H is the point corresponding to the Greenwich observatory: ≤ (to the east of the Greenwich meridian, the longitude is considered positive, to the west - negative).

As already known, the shortest distance between points A and B on the earth's surface is the length of the smaller of the arcs of a large circle connecting A and B (such an arc is called the orthodrome - translated from Greek means "straight run"). Therefore, our task is reduced to determining the length of the side AB of the spherical triangle ABC (C is the north pole).

Applying the standard notation for the elements of the triangle ABC and the corresponding trihedral angle OABS, from the condition of the problem we find: α = = - , β = (Fig. 2).

Angle C is also not difficult to express in terms of the coordinates of points A and B. By definition, ≤ , therefore, either angle C = if ≤ , or - if. Knowing = using the cosine theorem: = + (-). Knowing and, therefore, the angle, we find the required distance: =.

Trigonometry in navigation 2.

To plot the ship's course on a map made in the projection of Gerhard Mercator (1569), it was necessary to determine the latitude. When sailing in the Mediterranean Sea in sailing directions up toXVIIin. latitude was not specified. For the first time, Edmond Gunther (1623) applied trigonometric calculations in navigation.

Trigonometry helps calculate the effect of wind on aircraft flight. The velocity triangle is the triangle formed by the airspeed vector (V), wind vector(W), ground velocity vector (V P ). PU - track angle, SW - wind angle, KUV - heading wind angle.

The relationship between the elements of the navigation velocity triangle has the form:

V P = V cos US + W cos UV; sin US = * sin UV, tg SW =

The navigation triangle of speeds is solved with the help of counting devices, on the navigation ruler and approximately in the mind.

Trigonometry in algebra.

Here is an example of solving a complex equation using trigonometric substitution.

Given the equation

Let be , we get

;

where: or

subject to restrictions, we get:

Trigonometry in physics

Wherever we have to deal with periodic processes and oscillations - be it acoustics, optics or the swing of a pendulum - we are dealing with trigonometric functions. Oscillation formulas:

where A- oscillation amplitude, - angular frequency of oscillation, - initial phase of oscillation

Oscillation phase.

When objects are immersed in water, they do not change their shape or size. The whole secret is the optical effect that makes our vision perceive the object in a different way. The simplest trigonometric formulas and the values ​​of the sine of the angle of incidence and refraction of the beam make it possible to calculate the constant refractive index during the transition of a light beam from medium to medium. For example, a rainbow occurs due to the fact that sunlight is refracted in water droplets suspended in air according to the law of refraction:

sin α / sin β =n 1 /n 2

where:

n 1 - refractive index of the first medium
n 2 - refractive index of the second medium

α -angle of incidence, β is the angle of refraction of light.

Penetration of charged particles of the solar wind into the upper atmosphere of planets is determined by the interaction of the planet's magnetic field with the 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.

As a practical example, consider a physical problem that is solved using trigonometry.

Task. On an inclined plane making an angle of 24.5 with the horizon about , there is a body of mass 90 kg. Find the force with which this body presses on the inclined plane (i.e., what pressure does the body exert on this plane).

Decision:

Having designated the X and Y axes, we will begin to build projections of forces on the axes, first using this formula:

ma = N + mg , then look at the picture,

X : ma = 0 + mg sin24.5 0

Y: 0 = N - mg cos24.5 0

N = mg cos 24,5 0

we substitute the mass, we find that the force is 819 N.

Answer: 819 N

Trigonometry in medicine and biology

One of fundamental propertiesliving nature is the cyclicity of most of the processes occurring in it.

Biological rhythms, biorhythmsare more or less regular changes in the nature and intensity of biological processes.

Basic earth rhythm- daily.

The model of biorhythms can be built using trigonometric functions.

To build a model of biorhythms, you must enter the date of birth of a person, the date of reference (day, month, year) and the duration of the forecast (number of days).

Even some parts of the brain are called sinuses.

The walls of the sinuses are formed by a dura mater lined with endothelium. The lumen of the sinuses gapes, the valves and the muscular membrane, unlike other veins, are absent. In the cavity of the sinuses there are fibrous septa covered with endothelium. From the sinuses, blood enters the internal jugular veins; in addition, there is a connection between the sinuses and the veins of the outer surface of the skull through reserve venous graduates.

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.

When swimming, the body of the fish takes the form of a curve that resembles a graph.

functions y= tgx.

Trigonometry in music

We listen to musicmp3.

An audio signal is a wave, here is its “graph”.

As you can see, although it is very complex, it is a sinusoid that obeys the laws of trigonometry.

In the Moscow Art Theater in the spring of 2003, the presentation of the album "Trigonometry" by the group "Night Snipers", soloist Diana Arbenina took place. The content of the album reveals the original meaning of the word "trigonometry" - the measurement of the Earth.

Trigonometry in computer science

Trigonometric functions can be used for precise calculations.

Using trigonometric functions, you can approximate any

(in a sense, "good") function by expanding it into a Fourier series:

a 0 + a 1 cos x + b 1 sin x + a 2 cos 2x + b 2 sin 2x + a 3 cos 3x + b 3 sin 3x + ...

Picking the right numbers a 0 , a 1 , b 1 , a 2 , b 2 , ..., it is possible to represent almost any functions in a computer with the required accuracy in the form of such an (infinite) sum.

Trigonometric functions are useful when working with graphical information. It is necessary to simulate (describe in a computer) the rotation of some object around some axis. There is a rotation through a certain angle. To determine the coordinates of the points, you will have to multiply by the sines and cosines.

Justin Windell, programmer and designer fromGoogle graphics Lab , published a demo showing examples of using trigonometric functions to create dynamic animations.

Trigonometry in construction and geodesy

The lengths of the sides and the angles of an arbitrary triangle on the plane are interconnected by certain relations, the most important of which are called the cosine and sine theorems.

2ab

= =

In these formulas,b, c- the lengths of the sides of the triangle ABC, lying respectively opposite the angles A, B, C. These formulas allow us to restore the remaining three elements from the three elements of the triangle - the lengths of the sides and the angles. They are used in solving practical problems, for example, in geodesy.

All "classical" geodesy is based on trigonometry. Since, in fact, since ancient times, surveyors have been engaged in "solving" triangles.

The process of building buildings, roads, bridges and other structures begins with survey and design work. All measurements at the construction site are carried out using surveying instruments such as theodolite and trigonometric level. With trigonometric leveling, the height difference between several points on the earth's surface is determined.

Conclusion

Trigonometry was brought to life by the need to measure angles, but eventually developed into the science of trigonometric functions.

Trigonometry is closely related to physics, found in nature, music, architecture, medicine and technology.

Trigonometry is reflected in our lives, and the areas in which it plays an important role will expand, so everyone needs to know its laws.

The connection of mathematics with the outside world allows you to "materialize" the knowledge of schoolchildren. This helps us to better understand the vital need for knowledge acquired in school.

By a mathematical problem with practical content (task of an applied nature), we mean a problem whose plot reveals the applications of mathematics in related academic disciplines, technology, and everyday life.

The story about the historical reasons for the emergence of trigonometry, its development and practical application encourages our schoolchildren to be interested in the subject being studied, forms our worldview and improves our general culture.

This work will be useful for high school students who have not yet seen the beauty of trigonometry and are not familiar with the areas of its application in the surrounding life.

Bibliography:

Introduction

The real processes of the surrounding world are usually associated with a large number of variables and dependencies between them. These dependencies can be described using functions. The concept of "function" has played and still plays a big role in the knowledge of the real world. Knowledge of the properties of functions allows us to understand the essence of ongoing processes, predict the course of their development, and manage them. Function learning is relevant always.

Target: reveal the connection of trigonometric functions with the phenomena of the surrounding world and show that these functions are widely used in life.

tasks:

1. Study the literature and remote access resources on the topic of the project.

2. Find out what laws of nature are expressed by trigonometric functions.

3. Find examples of the use of trigonometric functions in the world around.

4. Analyze and systematize the available material.

5. Prepare the designed material in accordance with the requirements of the information project.

6. Develop an electronic presentation in accordance with the content of the project.

7. Speak at the conference with the results of the work done.

At the preparatory stage I found material on this topic and got acquainted with it put forward hypotheses formulated the purpose of my project. I started searching for the necessary information, studied the literature on my topic and materials of remote access resources.

At the main stage, information on the topic was selected and accumulated, the materials found were analyzed. I figured out the main areas of application of trigonometric functions. All data were summarized and systematized. Then a holistic final version of the information project was developed, a presentation was made on the topic of the study.

At the final stage the presentation of the work for the competition was analyzed. At this stage, activities were also assumed to implement all the tasks set, summing up, that is, assessing one's activities.

Sunrise and sunset, the change in the phases of the moon, the alternation of the seasons, the beating of the heart, the cycles in the life of the body, the rotation of the wheel, the tides of the sea - the models of these diverse processes are described by trigonometric functions.

Trigonometry in physics.

In technology and the world around us, we often have to deal with periodic (or almost periodic) processes that repeat at regular intervals. Such processes are called oscillatory. Oscillatory phenomena of various physical nature are subject to general laws. For example, current oscillations in an electrical circuit and oscillations of a mathematical pendulum can be described by the same equations. The commonality of oscillatory regularities makes it possible to consider oscillatory processes of various nature from a single point of view. Along with the translational and rotational motions of bodies in mechanics, oscillatory motions are also of considerable interest.

Mechanical vibrations called the movements of bodies that repeat exactly (or approximately) at regular intervals. The law of motion of an oscillating body is given by some periodic function of time x = f(t). The graphic representation of this function gives a visual representation of the course of the oscillatory process in time. An example of a wave of this kind can be waves traveling along a stretched rubber band or along a string.

Examples of simple oscillatory systems are a load on a spring or a mathematical pendulum (Fig. 1).

Fig.1. Mechanical oscillatory systems.

Mechanical oscillations, like oscillatory processes of any other physical nature, can be free and forced. Free vibrations are made under the action of the internal forces of the system, after the system has been brought out of equilibrium. The oscillations of a weight on a spring or the oscillations of a pendulum are free oscillations. Oscillations occurring under the action of external periodically changing forces are called forced.

Figure 2 shows the graphs of the coordinates, velocity and acceleration of a body that performs harmonic oscillations.

The simplest type of oscillatory process is simple harmonic oscillations, which are described by the equation:

x = m cos (ωt + f 0).

Figure 2- Graphs of coordinate x(t), speed υ(t)

and acceleration a(t) of a body performing harmonic oscillations.

sound waves or simply sound, it is customary to call the waves perceived by the human ear.

If oscillations of particles are excited in some place of a solid, liquid or gaseous medium, then due to the interaction of atoms and molecules of the medium, oscillations begin to be transmitted from one point to another with a finite speed. The process of propagation of oscillations in a medium is called a wave.

Of considerable interest for practice are simple harmonic or sinusoidal waves. They are characterized by particle vibration amplitude A, frequency f, and wavelength λ. Sinusoidal waves propagate in homogeneous media at a certain constant speed υ.

If people's vision had the ability to see sound, electromagnetic and radio waves, then we would see numerous sinusoids of all kinds around.

Surely, everyone has observed the phenomenon more than once, when objects lowered into the water immediately change their sizes and proportions. An interesting phenomenon, you immerse your hand in water, and it immediately turns into the hand of some other person. Why is this happening? The answer to this question and a detailed explanation of this phenomenon, as always, is given by physics - a science that can explain almost everything that surrounds us in this world.

So, in fact, when immersed in water, objects, of course, do not change either their size or their outlines. This is just an optical effect, that is, we visually perceive this object in a different way. This happens because of the properties of the light beam. It turns out that the speed of light propagation is greatly affected by the so-called optical density of the medium. The denser this optical medium, the slower the light beam propagates.

But the change in the speed of a beam of light still does not fully explain the phenomenon we are considering. There is another factor as well. So, when a light beam passes the boundary between a less dense optical medium, such as air, and a denser optical medium, such as water, part of the light beam does not penetrate into the new medium, but is reflected from its surface. The other part of the light beam penetrates inside, but already changing direction.

This phenomenon is called the refraction of light, and scientists have long been able to not only observe, but also accurately calculate the angle of this refraction. It turned out that the simplest trigonometric formulas and knowledge of the sine of the angle of incidence and the angle of refraction make it possible to find out the constant refractive index for the transition of a light beam from one particular medium to another. For example, the refractive index of air is extremely small and is 1.0002926, the refractive index of water is slightly higher - 1.332986, diamond refracts light with a coefficient of 2.419, and silicon - 4.010.

This phenomenon underlies the so-called rainbow theories. 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.

A rainbow occurs due to the fact that sunlight is refracted in water droplets suspended in the air according to the law of refraction:

where n 1 \u003d 1, n 2 ≈1.33 are the refractive indices of air and water, respectively, α is the angle of incidence, and β is the angle of light refraction.

Application of trigonometry in art and architecture.

From the time that man began to exist on earth, science has become the basis for improving everyday life and other areas of life. The foundations of everything that is created by man are various directions in the natural and mathematical sciences. One of them is geometry. Architecture is not the only field of science in which trigonometric formulas are used. Most of the compositional decisions and construction of drawings took place precisely with the help of geometry. But theoretical data mean little. Consider an example of the construction of one sculpture by the French master of the Golden Age of Art.

The proportional relationship in the construction of the statue was perfect. However, when the statue was raised to a high pedestal, it looked ugly. The sculptor did not take into account that many details are reduced in perspective towards the horizon, and when viewed from the bottom up, the impression of its ideality is no longer created. A lot of calculations were carried out so that the figure from a great height looked proportional. Basically, they were based on the method of sighting, that is, an approximate measurement, by eye. However, the coefficient of difference of certain proportions made it possible to make the figure closer to the ideal. Thus, knowing the approximate distance from the statue to the point of view, namely from the top of the statue to the eyes of a person and the height of the statue, we can calculate the sine of the angle of incidence of the gaze using the table, thereby finding the point of view (Fig. 4).

In Figure 5, the situation changes, since the statue is raised to the height of AC and HC increase, you can calculate the cosine of angle C, using the table we find the angle of incidence of the gaze. In the process, you can calculate AH, as well as the sine of angle C, which will allow you to check the results using the basic trigonometric identity cos 2 a + sin 2 a = 1.

By comparing the measurements of AH in the first and second cases, one can find the proportionality coefficient. Subsequently, we will receive a drawing, and then a sculpture, when raised, the figure will be visually close to the ideal.

Iconic buildings around the world have been designed with mathematics that can be considered the genius of architecture. Some famous examples of such buildings are the Gaudí Children's School in Barcelona, ​​the Mary Ax in London, the Bodegas Isios Winery in Spain, and the Los Manantiales Restaurant in Argentina. The design of these buildings was not without trigonometry.

Trigonometry in biology.

One of the fundamental properties of living nature is the cyclicity of most of the processes occurring in it. There is a connection between the movement of celestial bodies and living organisms on Earth. Living organisms not only capture the light and heat of the Sun and Moon, but also have various mechanisms that accurately determine the position of the Sun, respond to the rhythm of the tides, the phases of the Moon and the movement of our planet.

Biological rhythms, biorhythms, are more or less regular changes in the nature and intensity of biological processes. The ability for such changes in vital activity is inherited and found in almost all living organisms. They can be observed in individual cells, tissues and organs, whole organisms and populations. Biorhythms are divided into physiological, having periods from fractions of a second to several minutes and environmental, in duration coinciding with any rhythm of the environment. These include daily, seasonal, annual, tidal and lunar rhythms. The main earthly rhythm is daily, due to the rotation of the Earth around its axis, therefore, almost all processes in a living organism have a daily periodicity.

Many environmental factors on our planet, primarily the light regime, temperature, air pressure and humidity, atmospheric and electromagnetic fields, sea tides, naturally change under the influence of this rotation.

We are seventy-five percent water, and if at the time of the full moon the waters of the world's oceans rise 19 meters above sea level and the tide begins, then the water in our body also rushes into the upper parts of our body. And people with high blood pressure often experience exacerbations of the disease during these periods, and naturalists who collect medicinal herbs know exactly in which phase of the moon to collect "tops - (fruits)", and in which - "roots".

Have you noticed that at certain periods your life makes inexplicable jumps? Suddenly, out of nowhere - emotions overflow. Sensitivity increases, which can suddenly be replaced by complete apathy. Creative and barren days, happy and unhappy moments, mood swings. It is noted that the capabilities of the human body change periodically. This knowledge underlies the "theory of three biorhythms".

Physical biorhythm- regulates physical activity. During the first half of the physical cycle, a person is energetic, and achieves better results in his activity (the second half - energy is inferior to laziness).

Emotional Rhythm- during periods of its activity, sensitivity increases, mood improves. A person becomes excitable to various external cataclysms. If he is in a good mood, he builds castles in the air, dreams of falling in love and falls in love. With a decrease in the emotional biorhythm, a decline in mental strength occurs, desire and joyful mood disappear.

Intelligent biorhythm - he controls memory, the ability to learn, logical thinking. In the activity phase, there is an increase, and in the second phase, a decline in creative activity, there is no luck and success.

Theory of three rhythms.

· The physical cycle is 23 days. Determines energy, strength, endurance, coordination of movement

Emotional cycle - 28 days. The state of the nervous system and mood

· Intellectual cycle - 33 days. Determines the creative ability of the individual

Trigonometry is also found in nature. The movement of fish in the 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. When swimming, the body of the fish takes the form of a curve that resembles the graph of the function y=tgx.

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

Trigonometry in medicine.

As a result of a study conducted by a student at the Iranian University of Shiraz, Wahid-Reza Abbasi, physicians for the first time were able to streamline information related to the electrical activity of the heart, or, in other words, electrocardiography.

The formula, called Tehran, was presented to the general scientific community at the 14th Conference of Geographical Medicine and then at the 28th Conference on the Application of Computer Technology in Cardiology, held in the Netherlands.

This formula is a complex algebraic-trigonometric equation, consisting of 8 expressions, 32 coefficients and 33 main parameters, including several additional ones for calculations in cases of arrhythmia. According to doctors, this formula greatly facilitates the process of describing the main parameters of the activity of the heart, thereby speeding up the diagnosis and the start of the actual treatment.

Many people have to do an ECG of the heart, but few know that the ECG of the human heart is a sine or cosine plot.

Trigonometry helps our brain determine the distances to objects. American scientists claim that the brain estimates the distance to objects by measuring the angle between the ground plane and the plane of vision. This conclusion was made after a series of experiments in which participants were asked to look at the world around them through prisms that increase this angle.

This distortion caused experimental prism wearers to perceive distant objects as closer and could not cope with the simplest tests. Some of the participants in the experiments even leaned forward, trying to align their bodies perpendicular to the misrepresented ground surface. However, after 20 minutes, they got used to the distorted perception, and all problems disappeared. This circumstance indicates the flexibility of the mechanism by which the brain adapts the visual system to changing external conditions. It is interesting to note that after the prisms were removed, the opposite effect was observed for some time - an overestimation of the distance.

The results of the new study, as you might expect, will be of interest to engineers designing navigation systems for robots, as well as specialists who are working on creating the most realistic virtual models. Applications are also possible in the field of medicine, in the rehabilitation of patients with damage to certain areas of the brain.

Conclusion

At present, trigonometric calculations are used in almost all areas of geometry, physics and engineering. Of great importance is the triangulation technique, which makes it possible to measure distances to nearby stars in astronomy, between landmarks in geography, and to control satellite navigation systems. Also of note are the applications of trigonometry in areas such as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, medicine (including ultrasound and computed tomography), pharmaceuticals, chemistry, number theory, seismology, meteorology, oceanology, cartography, many branches of physics, topography and geodesy, architecture, economics, electronic engineering, mechanical engineering, computer graphics, crystallography.

Findings:

· We found that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.

· We have proved that trigonometry is closely related to physics, biology, occurs in nature, architecture and medicine.

· We think that trigonometry is reflected in our lives, and the areas in which it plays an important role will expand.

Literature

1. Alimov Sh.A. et al. "Algebra and the beginning of analysis" Textbook for grades 10-11 of educational institutions, M., Education, 2010.

2. Vilenkin N. Ya. Functions in nature and technology: Book. for out-of-class reading IX-XX cl. - 2nd ed., Rev.-M: Enlightenment, 1985.

3. Glazer G.I. History of mathematics at school: IX-X cells. - M.: Enlightenment, 1983.

4. Maslova T.N. "Student's Handbook of Mathematics"

5. Rybnikov K.A. History of Mathematics: Textbook. - M.: Publishing House of Moscow State University, 1994.

6. Study.ru

7. Math.ru "library"

MKOU "Nenets secondary school - boarding school named after. A.P. Pyrerki"

Educational project

" "

Danilova Tatyana Vladimirovna

Mathematic teacher

2013

Rationale for the relevance of the project.

Trigonometry is a branch of mathematics that studies trigonometric functions. It is hard to imagine, but we encounter this science not only in mathematics lessons, but also in our daily life. You might not be aware of this, but trigonometry is found in such sciences as physics, biology, it plays an important role in medicine, and, most interestingly, even music and architecture could not do without it.
The word trigonometry first appears in 1505 in the title of a book by the German mathematician Pitiscus.
Trigonometry is a Greek word, and literally means the measurement of triangles (trigonan - triangle, metreo - I measure).
The emergence of trigonometry was closely connected with land surveying, astronomy and construction.

A student at the age of 14-15 does not always know where he will go to study and where he will work.
For some professions, its knowledge is necessary, because. allows you to measure distances to nearby stars in astronomy, between landmarks in geography, control satellite navigation systems. The principles of trigonometry are also used in such areas as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound and computed tomography), pharmaceuticals, chemistry, number theory ( and, as a result, cryptography), seismology, meteorology, oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography.

Definition of the subject of research

Why knowledge of trigonometry is necessary for a modern person?

3.Project goals.

The connection of trigonometry with real life.

problem question
1. What concepts of trigonometry are most often used in real life?
2. What role does trigonometry play in astronomy, physics, biology and medicine?
3. How are architecture, music and trigonometry connected?

Hypothesis

Most of the physical phenomena of nature, physiological processes, patterns in music and art can be described using trigonometry and trigonometric functions.

Hypothesis testing

Trigonometry (from Greek. trigonon - triangle, metro - meter) - a microsection of mathematics that studies the relationship between the angles and the lengths of the sides of triangles, as well as the algebraic identities of trigonometric functions.

The beginnings of trigonometric knowledge originated in antiquity. At an early stage, trigonometry developed in close connection with astronomy and was its auxiliary section.

History of trigonometry:

The origins of trigonometry go back to ancient Egypt, Babylonia and the Indus Valley over 3,000 years ago.

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

For the first time, methods for solving triangles based on the dependencies between the sides and angles of a triangle were found by the ancient Greek astronomers Hipparchus and Ptolemy.

Ancient people calculated the height of a tree by comparing the length of its shadow with the length of the shadow of a pole whose height was known. The stars calculated the location of the ship at sea.

The next step in the development of trigonometry was taken by the Indians in the period from the 5th to the 12th centuries.

The term cosine itself appeared much later in the works of European scientists for the first time at the end of the 16th century from the so-called “complement sine”, i.e. the sine of the angle that complements the given angle up to 90°. "Sine complement" or (in Latin) sinus complementi began to be abbreviated as sinus co or co-sinus.

AT 17th - 19th centuries trigonometry becomes one of the chapters of mathematical analysis.

It finds great application in mechanics, physics and technology, especially in the study of oscillatory motions and other periodic processes.

Jean Fourier proved that any periodic motion can be represented (with any degree of accuracy) as a sum of simple harmonic oscillations.

Trigonometry development stages:

Trigonometry was brought to life by the need to measure angles.

The first steps in trigonometry were establishing relationships between the magnitude of the angle and the ratio of specially constructed line segments. The result is the ability to solve flat triangles.

The need to tabulate the values ​​of the introduced trigonometric functions.

Trigonometric functions turned into independent objects of research.

In the XVIII century. trigonometric functions have been enabled

into the system of mathematical analysis.

Where is trigonometry used?

Trigonometric calculations are used in almost all areas of human life. It should be noted the application in such areas as: astronomy, physics, nature, biology, music, medicine and many others.

Trigonometry in astronomy:

The need for solving triangles was first discovered in astronomy; therefore, for a long time trigonometry was developed and studied as one of the branches of astronomy.

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. He improved the accuracy of observations by using threads in goniometric instruments - sextants and quadrants - to point the star at the star. The scientist compiled a catalog of the positions of 850 stars, huge at that time, dividing them by brightness into 6 degrees (magnitudes). Hipparchus introduced geographical coordinates - latitude and longitude, and he can be considered the founder of mathematical geography. (c. 190 BC - c. 120 BC)

Vieta's achievements in trigonometry
A complete solution to the problem of determining all elements of a flat or spherical triangle from three given elements, important expansions of sin nx and cos nx in powers of cos x and sinx. Knowing the formula for the sines and cosines of multiple arcs enabled Viet to solve the 45th degree equation proposed by the mathematician A. Roomen; Viet showed that the solution to this equation comes down to dividing the angle into 45 equal parts and that there are 23 positive roots of this equation. Viet solved Apollonius' problem with a ruler and a compass.
Solving spherical triangles is one of the tasks of astronomy. Calculate the sides and angles of any spherical triangle from three suitably given sides or angles using the following theorems: (sine theorem) (cosine theorem for angles) (cosine theorem for sides).

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. There are different types of oscillatory phenomena.

harmonic oscillation- the phenomenon of a periodic change in a quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

Where x is the value of the changing quantity, t is the time, A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, r is the initial phase of the oscillations.

Generalized harmonic oscillation in differential form x'' + ω²x = 0.

Mechanical vibrations . Mechanical vibrations called movements of bodies that repeat exactly at the same intervals of time. 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.

Trigonometry in nature.

We often ask a question Why do we sometimes see things that aren't really there?. The following questions are proposed for research: “How does a rainbow appear? Northern Lights?”, “What are optical illusions?” ,"How can trigonometry help answer these questions?".

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 of charged particles of the solar wind into the upper atmosphere of planets is determined by the interaction of the planet's magnetic field with the 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.

Multifunctional trigonometry

American scientists claim 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 carotid sinus, carotid sinus and venous or cavernous sinus.

Trigonometry and trigonometric functions in medicine and biology.

One of fundamental properties 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.

Basic earth rhythm- daily.

The model of biorhythms can be built using trigonometric functions.

Trigonometry in biology

What biological processes are associated with trigonometry?

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 associated with trigonometry

Connection of biorhythms with trigonometry

A model of biorhythms can be built using graphs of trigonometric functions. To do this, you must enter the date of birth of the person (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.

The emergence of musical harmony

According to the legends that have come down from antiquity, the first who tried to do this were Pythagoras and his students.

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

diatonic scale 2:3:5

Trigonometry in architecture

Gaudí Children's School in Barcelona

Swiss Re Insurance Corporation in London

Felix Candela Restaurant in Los Manantiales

Interpretation

We have given only a small part of where trigonometric functions can be found .. We found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.

We proved that trigonometry is closely related to physics, occurs in nature, medicine. It is possible to give infinitely many examples of periodic processes of animate and inanimate nature. All periodic processes can be described using trigonometric functions and depicted on graphs

We think that trigonometry is reflected in our lives, and the spheres

in which it plays an important role will expand.

Conclusion

Found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.

Proved that trigonometry is closely related to physics, found in nature, music, astronomy and medicine.

We think that trigonometry is reflected in our lives, and the areas in which it plays an important role will expand.

7. Literature.

Maslova T.N. "Student's Handbook of Mathematics"

Maple6 program that implements the image of graphs

"Wikipedia"

Studies. en

Math.ru "library"

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.

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

Stories about applied mathematics//Moscow, 1979. A. V. Voloshinov. Mathematics and Art // Moscow, 1992. Newspaper Mathematics. Supplement to the newspaper dated 1.09.98.

Trigonometry in astronomy:

The need for solving triangles was first discovered in astronomy; therefore, for a long time trigonometry was developed and studied as one of the branches of astronomy.

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. He improved the accuracy of observations by using threads in goniometric instruments—sextants and quadrants—to point the star at the star. The scientist compiled a catalog of the positions of 850 stars, huge at that time, dividing them by brightness into 6 degrees (magnitudes). Hipparchus introduced geographical coordinates - latitude and longitude, and he can be considered the founder of mathematical geography. (c. 190 BC - c. 120 BC)


A complete solution to the problem of determining all elements of a flat or spherical triangle from three given elements, important expansions of sin nx and cos nx in powers of cos x and sinx. Knowing the formula for the sines and cosines of multiple arcs enabled Viet to solve the 45th degree equation proposed by the mathematician A. Roomen; Viet showed that the solution to this equation comes down to dividing the angle into 45 equal parts and that there are 23 positive roots of this equation. Viet solved Apollonius' problem with a ruler and a compass.
Solving spherical triangles is one of the tasks of astronomy. Calculate the sides and angles of any spherical triangle from three suitably given sides or angles using the following theorems: (sine theorem) (cosine theorem for angles) (cosine theorem for sides).

Trigonometry in physics:

types of oscillatory phenomena.

Harmonic oscillation is a phenomenon of periodic change of some quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

Where x is the value of the changing quantity, t is time, A is the amplitude of oscillations, ω is the cyclic frequency of oscillations, is the full phase of oscillations, r is the initial phase of oscillations.

Mechanical vibrations . Mechanical vibrations

Trigonometry in nature.

We often ask a question

• One of fundamental properties
• are more or less regular changes in the nature and intensity of biological processes.
• Basic earth rhythm- daily.

Trigonometry in biology

• 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.

• diatonic scale 2:3:5

Trigonometry in architecture

• Swiss Re Insurance Corporation in London

1. Interpretation

We have given only a small part of where you can find trigonometric functions .. We found out

We proved that trigonometry is closely related to physics, occurs in nature, medicine. It is possible to give infinitely many examples of periodic processes of animate and inanimate nature. All periodic processes can be described using trigonometric functions and depicted on graphs

We think that trigonometry is reflected in our lives, and the spheres

in which it plays an important role will expand.

• Found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.
• Proved
• We think

View document content
"Danilova T.V.-scenario"

MKOU "Nenets secondary school - boarding school named after. A.P. Pyrerki"

Educational project

" "

Danilova Tatyana Vladimirovna

Mathematic teacher

Rationale for the relevance of the project.

Trigonometry is a branch of mathematics that studies trigonometric functions. It is hard to imagine, but we encounter this science not only in mathematics lessons, but also in our daily life. You might not be aware of this, but trigonometry is found in such sciences as physics, biology, it plays an important role in medicine, and, most interestingly, even music and architecture could not do without it.
The word trigonometry first appears in 1505 in the title of a book by the German mathematician Pitiscus.
Trigonometry is a Greek word, and literally means the measurement of triangles (trigonan - triangle, metreo - I measure).
The emergence of trigonometry was closely connected with land surveying, astronomy and construction.

A student at the age of 14-15 does not always know where he will go to study and where he will work.
For some professions, its knowledge is necessary, because. allows you to measure distances to nearby stars in astronomy, between landmarks in geography, control satellite navigation systems. The principles of trigonometry are also used in such areas as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound and computed tomography), pharmaceuticals, chemistry, number theory ( and, as a result, cryptography), seismology, meteorology, oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography.

Definition of the subject of research

3. Project goals.

problem question
1. What concepts of trigonometry are most often used in real life?
2. What role does trigonometry play in astronomy, physics, biology and medicine?
3. How are architecture, music and trigonometry connected?

Hypothesis

Hypothesis testing

Trigonometry (from Greek.trigonon - triangle,metro - meter) -

History of trigonometry:

Ancient people calculated the height of a tree by comparing the length of its shadow with the length of the shadow of a pole whose height was known. The stars calculated the location of the ship at sea.

The next step in the development of trigonometry was taken by the Indians in the period from the 5th to the 12th centuries.

The term cosine itself appeared much later in the works of European scientists for the first time at the end of the 16th century from the so-called “complement sine”, i.e. the sine of the angle that complements the given angle up to 90°. "Sine complement" or (in Latin) sinus complementi began to be abbreviated as sinus co or co-sinus.

In the XVII - XIX centuries. trigonometry becomes one of the chapters of mathematical analysis.

It finds great application in mechanics, physics and technology, especially in the study of oscillatory motions and other periodic processes.

Jean Fourier proved that any periodic motion can be represented (with any degree of accuracy) as a sum of simple harmonic oscillations.

into the system of mathematical analysis.

Where is trigonometry used?

Trigonometric calculations are used in almost all areas of human life. It should be noted the application in such areas as: astronomy, physics, nature, biology, music, medicine and many others.

Trigonometry in astronomy:

The need for solving triangles was first discovered in astronomy; therefore, for a long time trigonometry was developed and studied as one of the branches of astronomy.

The need for solving triangles was first discovered in astronomy; therefore, for a long time trigonometry was developed and studied as one of the branches of astronomy.

Vieta's achievements in trigonometry
A complete solution to the problem of determining all elements of a flat or spherical triangle from three given elements, important expansions of sin nx and cos nx in powers of cos x and sinx. Knowing the formula for the sines and cosines of multiple arcs enabled Viet to solve the 45th degree equation proposed by the mathematician A. Roomen; Viet showed that the solution to this equation comes down to dividing the angle into 45 equal parts and that there are 23 positive roots of this equation. Viet solved Apollonius' problem with a ruler and a compass.
Solving spherical triangles is one of the tasks of astronomy. Calculate the sides and angles of any spherical triangle from three suitably given sides or angles using the following theorems: (sine theorem) (cosine theorem for angles) (cosine theorem for sides).

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. There are different types of oscillatory phenomena.

harmonic oscillation- the phenomenon of a periodic change in a quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

Where x is the value of the changing quantity, t is the time, A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, r is the initial phase of the oscillations.

Generalized harmonic oscillation in differential form x'' + ω²x = 0.

Mechanical vibrations . Mechanical vibrations called movements of bodies that repeat exactly at the same intervals of time. 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.

Trigonometry in nature.

We often ask a question Why do we sometimes see things that aren't really there?. The following questions are proposed for research: “How does a rainbow appear? Northern Lights?”, “What are optical illusions?” ,"How can trigonometry help answer these questions?".

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 of charged particles of the solar wind into the upper atmosphere of planets is determined by the interaction of the planet's magnetic field with the 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.

American scientists claim 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 carotid sinus, carotid sinus and venous or cavernous sinus.

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.

One of fundamental properties living nature is the cyclicity of most of the processes occurring in it.

Biological rhythms, biorhythms

Basic earth rhythm- daily.

The model of biorhythms can be built using trigonometric functions.

Trigonometry in biology

What biological processes are associated with trigonometry?

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 associated with trigonometry

A model of biorhythms can be built using graphs of trigonometric functions. To do this, you must enter the date of birth of the person (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.

The emergence of musical harmony

According to the legends that have come down from antiquity, the first who tried to do this were Pythagoras and his students.

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

diatonic scale 2:3:5

Trigonometry in architecture

Gaudí Children's School in Barcelona

Swiss Re Insurance Corporation in London

Felix Candela Restaurant in Los Manantiales

Interpretation

We have given only a small part of where trigonometric functions can be found .. We found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.

We proved that trigonometry is closely related to physics, occurs in nature, medicine. It is possible to give infinitely many examples of periodic processes of animate and inanimate nature. All periodic processes can be described using trigonometric functions and depicted on graphs

We think that trigonometry is reflected in our lives, and the spheres

in which it plays an important role will expand.

Found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.

Proved that trigonometry is closely related to physics, found in nature, music, astronomy and medicine.

We think that trigonometry is reflected in our lives, and the areas in which it plays an important role will expand.

7. Literature.

Maple6 program that implements the image of graphs

"Wikipedia"

Study.ru

Math.ru "library"

View presentation content
"Danilova T.V."

" Trigonometry in the world around us and human life "

Research objectives:

The connection of trigonometry with real life.

problem question 1. What concepts of trigonometry are most often used in real life? 2. What role does trigonometry play in astronomy, physics, biology and medicine? 3. How are architecture, music and trigonometry connected?

Hypothesis

Most of the physical phenomena of nature, physiological processes, patterns in music and art can be described using trigonometry and trigonometric functions.

What is trigonometry???

Trigonometry (from Greek trigonon - triangle, metro - meter) - a microsection of mathematics that studies the relationship between the angles and the lengths of the sides of triangles, as well as the algebraic identities of trigonometric functions.

History of trigonometry

The origins of trigonometry go back to ancient Egypt, Babylonia and the Indus Valley over 3,000 years ago.

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

For the first time, methods for solving triangles based on the dependencies between the sides and angles of a triangle were found by the ancient Greek astronomers Hipparchus and Ptolemy.

Ancient people calculated the height of a tree by comparing the length of its shadow with the length of the shadow of a pole whose height was known.

The stars calculated the location of the ship at sea.

The next step in the development of trigonometry was taken by the Indians in the period from the 5th to the 12th centuries.

AT difference from the Greeks eytsy began to consider and use in calculations not the whole chord MM  corresponding central angle, but only its half MP, i.e. sine  half of the central corner.

The term cosine itself appeared much later in the works of European scientists for the first time at the end of the 16th century from the so-called « sine supplement » , i.e. sine of the angle complementing the given angle to 90  . « Sinus addition » or (in Latin) sinus complementi became abbreviated as sinus co or co-sinus.

Along with the sine, the Indians introduced into trigonometry cosine , more precisely, they began to use the cosine line in their calculations. They also knew the ratios cos  =sin(90  -  ) and sin 2  + cos 2  =r 2 , as well as formulas for the sine of the sum and difference of two angles.

In the XVII - XIX centuries. trigonometry becomes

one of the chapters of mathematical analysis.

It finds great application in mechanics,

physics and technology, especially when studying

oscillatory movements and other

periodic processes.

Viet knew about the properties of the periodicity of trigonometric functions, the first mathematical studies of which were related to trigonometry.

Proved that every periodic

movement can be

presented (with any degree

accuracy) as a sum of simple

harmonic vibrations.

Founder analytical

theories

trigonometric functions .

Leonhard Euler

In "Introduction to the analysis of the infinite" (1748)

treats sine, cosine, etc. not like

trigonometric lines, required

related to the circle, but how

trigonometric functions, which

viewed as a relationship

right triangle as numeric

quantities.

Excluded from my formulas

R is a whole sine, taking

R = 1, and simplified like this

way of writing and calculating.

Develops a doctrine

about trigonometric functions

any argument.

In the 19th century continued

theory development

trigonometric

functions.

N.I. Lobachevsky

“Geometric considerations,” Lobachevsky writes, “are necessary until at the beginning of trigonometry, until they serve to discover a distinctive property of trigonometric functions ... Hence, trigonometry becomes completely independent of geometry and has all the advantages of analysis.”

Trigonometry development stages:

• Trigonometry was brought to life by the need to measure angles.

• The first steps in trigonometry were establishing relationships between the magnitude of the angle and the ratio of specially constructed line segments. The result is the ability to solve flat triangles.

• The need to tabulate the values ​​of the introduced trigonometric functions.

• Trigonometric functions turned into independent objects of research.

• In the XVIII century. trigonometric functions have been enabled

into the system of mathematical analysis.

Where is trigonometry used?

Trigonometric calculations are used in almost all areas of human life. It should be noted the application in such areas as: astronomy, physics, nature, biology, music, medicine and many others.

Trigonometry in astronomy

The need for solving triangles was first discovered in astronomy; therefore, for a long time trigonometry was developed and studied as one of the branches of astronomy.

Trigonometry also reached considerable heights among Indian medieval astronomers.

The main achievement of Indian astronomers was the replacement of chords

sines, which made it possible to introduce various functions related to

with sides and angles of a right triangle.

Thus, in India, the beginning of trigonometry was laid.

as the doctrine of trigonometric quantities.

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. He improved the accuracy of observations by using threads in goniometric instruments - sextants and quadrants - to point the star at the star. The scientist compiled a catalog of the positions of 850 stars, huge at that time, dividing them by brightness into 6 degrees (magnitudes). Hipparchus introduced geographical coordinates - latitude and longitude, and he can be considered the founder of mathematical geography. (c. 190 BC - c. 120 BC)

Hipparchus

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. There are different types of oscillatory phenomena, for example:

Mechanical vibrations

Harmonic vibrations

Harmonic vibrations

harmonic oscillation - the phenomenon of a periodic change in a quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

or

Where x is the value of the changing quantity, t is the time, A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, r is the initial phase of the oscillations.

Generalized harmonic oscillation in differential form x'' + ω²x = 0.

Mechanical vibrations

Mechanical vibrations called movements of bodies that repeat exactly at the same intervals of time. 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.

Mathematical pendulum

The figure shows the oscillations of a pendulum, it moves along a curve called cosine.

Bullet trajectory and vector projections on the X and Y axes

It can be seen from the figure that the projections of the vectors on the X and Y axes, respectively, are equal to

υ x = υ o cos α

υ y = υ o sin α

Trigonometry in nature

We often ask a question Why do we sometimes see things that aren't really there?. The following questions are proposed for research: “How does a rainbow appear? Northern Lights?”, “What are optical illusions?” ,"How can trigonometry help answer these questions?".

optical illusions

natural

artificial

mixed

rainbow theory

A rainbow is formed due to the fact that sunlight is refracted by water droplets suspended in the air along refraction law:

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.

sin α / sin β =n 1 /n 2

where n 1 \u003d 1, n 2 ≈1.33 are the refractive indices of air and water, respectively, α is the angle of incidence, and β is the angle of light refraction.

Northern lights

Penetration of charged particles of the solar wind into the upper atmosphere of planets is determined by the interaction of the planet's magnetic field with the 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.

• American scientists claim 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 carotid sinus, carotid sinus and venous or cavernous sinus.

• 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.

• One of fundamental properties 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.

• Basic earth rhythm- daily.

• The model of biorhythms can be built using trigonometric functions.

Trigonometry in biology

What biological processes are associated with trigonometry?

• 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.

Trigonometry in biology

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.

When swimming, the body of the fish takes the form of a curve that resembles the graph of the function y=tgx.

The emergence of musical harmony

• According to the legends that have come down from antiquity, the first who tried to do this were Pythagoras and his students.

• Frequencies corresponding

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

• diatonic scale 2:3:5

Music has its own geometry

Tetrahedron of different types of chords of four sounds:

blue - small intervals;

warmer tones - more "discharged" chord sounds; the red sphere is the most harmonious chord with equal intervals between notes.

cos 2 C + sin 2 C = 1

AU- the distance from the top of the statue to the eyes of a person,

AN- the height of the statue,

sin C is the sine of the angle of incidence.

Trigonometry in architecture

Gaudí Children's School in Barcelona

Swiss Re Insurance Corporation in London

y = f (λ) cos θ

z = f(λ)sin θ

Felix Candela Restaurant in Los Manantiales

• Found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.

• Proved that trigonometry is closely related to physics, found in nature, music, astronomy and medicine.

• We think that trigonometry is reflected in our lives, and the areas in which it plays an important role will expand.

Trigonometry has come a long way in development. And now, we can say with confidence that trigonometry does not depend on other sciences, and other sciences depend on trigonometry.

• Maslova T.N. "Student's Handbook of Mathematics"
• Maple6 program that implements the image of graphs
• "Wikipedia"
• Study.ru
• Math.ru "library"
• 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.
• Predecessors of Modern Mathematics// ed. S. N. Niro. Moscow, 1983 A. N. Tikhonov, D. P. Kostomarov.
• Stories about applied mathematics//Moscow, 1979. A. V. Voloshinov. Mathematics and Art // Moscow, 1992. Newspaper Mathematics. Supplement to the newspaper dated 1.09.98.