A message on the topic of trigonometric functions in life. Trigonometry in the world around us and human life. Trigonometry and stages of its formation

The term itself, which gave its name to this branch of mathematics, was first discovered in the title of a book authored by the German mathematician Pitiscus in 1505. Word " trigonometry" is of Greek origin and means " measuring a triangle».

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 were used to calculate the location of a ship at sea.

2. 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 natures 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 patterns allows us to consider oscillatory processes of various natures from a single point of view. Along with the translational and rotational motions of bodies in mechanics, oscillatory motions are also of significant interest.

Mechanical vibrations are movements of bodies that repeat exactly (or approximately) at equal intervals of time. The law of motion of a body oscillating is specified using a certain periodic function of time x = f(t). A graphical representation of this function gives a visual representation of the course of the oscillatory process over time. An example of a wave of this kind is 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 vibrations, like oscillatory processes of any other physical nature, can be free and forced. Free vibrations occur under the influence of the internal forces of the system, after the system has been brought out of equilibrium. Oscillations of a weight on a spring or oscillations of a pendulum are free oscillations. Oscillations that occur under the influence of external periodically changing forces are called forced.

3. Trigonometry in astronomy

4. Trigonometry in medicine

One of the fundamental properties of living nature is the cyclical nature of most 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.

Biorhythms are divided into physiological, having periods from fractions of a second to several minutes and environmental, duration coinciding with any rhythm of the environment. These include daily, seasonal, annual, tidal and lunar rhythms. The main earthly rhythm is daily, determined by 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 light conditions, 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 moment 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 to 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 what phase of the moon to collect “ tops – (fruits)", and which one - " roots».

Have you noticed that at certain periods your life takes inexplicable leaps? Suddenly, out of nowhere, emotions overflow. Sensitivity increases, which can suddenly give way to complete apathy. Creative and fruitless days, happy and unhappy moments, sudden mood swings. It has been noted that the capabilities of the human body change periodically. This knowledge underlies " 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 activities (the second half - energy gives way to laziness).

Emotional rhythm– during periods of its activity, sensitivity increases and mood improves. A person becomes excitable to various external disasters. If he is in a good mood, he builds castles in the air, dreams of falling in love and falls in love. When the emotional biorhythm decreases, mental strength declines, desire and joyful mood disappear.

Intellectual biorhythm - it controls memory, the ability to learn, and logical thinking. In the activity phase there is a rise, and in the second phase there is a decline in creative activity, there is no luck and success.

Three Rhythms Theory

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

The emotional cycle is 28 days. State of the nervous system and mood

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

Trigonometry also occurs in nature. 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 fish's body takes the shape of a curve that resembles the graph of the function y=tgx.

When a bird flies, the trajectory of the flapping wings forms a sinusoid.

American scientists claim that the brain estimates the distance to objects by measuring the angle between the plane of the earth and the plane of vision. As a result of a study conducted by Iranian Shiraz University student Vahid-Reza Abbasi, doctors for the first time were able to organize information related to the electrical activity of the heart, or in other words, electrocardiography.

The 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 heart activity, thereby speeding up the diagnosis and the start of treatment itself.

MUNICIPAL EDUCATIONAL INSTITUTION

"GYMNASIUM No. 1"

"TRIGONOMETRY IN REAL LIFE"

information project

Completed:

Krasnov Egor

student of class 9A

Supervisor:

Borodkina Tatyana Ivanovna

Zheleznogorsk

Introduction……………………………………………………..……3

Relevance…………………………………………………………….3

Goal………………………………………………………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. Characteristics……………….7

2.3. Occurrence of sine……………………….……………….7

2.4. The appearance 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

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

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

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

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

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

3.7. Construction sector and geodesy.…………………………...22

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

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

References.………………………….…………….……………27

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

Introduction

In the modern world, considerable 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 the branch of mathematics that studies trigonometric functions. I believe that this topic, firstly, is relevant from a practical point of view. We are finishing our studies at 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, and 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 consequence, 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 theme of "Trigonometry in Real Life" is that knowledge of trigonometry will open up new ways of solving various problems in many fields of science and simplify the understanding of certain aspects of various sciences.

It has long been an established practice that schoolchildren encounter 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, 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 learn: “Relationships between the sides and angles of a right triangle.” In the process of studying trigonometry, the concepts of cosine, sine and tangent are given.

The next step is to continue learning about trigonometry in 9th grade. 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 capabilities.

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

Purpose 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 influence 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 refers to a branch of mathematics that studies the relationship between different angle sizes, studies the lengths of the sides of a triangle and 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 stages of its formation

Let's turn to the history of its development, the stages of formation. Since ancient times, trigonometry has gained its rudiments, developed and shown its first results. We can see the very first information about the emergence and development of this area in manuscripts that are located in ancient Egypt, Babylon, and Ancient China. Having studied the 56th problem from the Rhindus papyrus (2nd millennium BC), one can see that it proposes to find the inclination of a 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 in solving this problem the Egyptians 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 the 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 3rd century BC. e. The treatise, considering the magnitude and distance of the Sun and 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 necessary to calculate the ratio of the sides of a right triangle with a known value of one of the angles. Aristarchus considered the right triangle formed by the Sun, Moon and Earth during a quadrature. To calculate the value of the hypotenuse, which serves as the basis for the distance from the Earth to the Sun, using the leg, which serves as the basis for the distance from the Earth to the Moon, with a known value of the adjacent angle (87°), which is equivalent to calculating the value sin of 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 Moon's location. The error of judgment arose due to inaccuracy in the measurement of the angle.

Several decades later, Claudius Ptolemy, in his own works Ethnogeography, Analemma and Planispherium, provides a detailed account of trigonometric additions to cartography, astronomy and mechanics. Among other things, a stereographic projection is depicted, a number of factual issues are studied, for example: to establish the height and angle of the 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)

Taken together, it can be noted that trigonometry was used for the purpose of:

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;

Calculating the distance between megacities with known geographic coordinates.

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

Thus, the cotangent was represented to us as the length of the shadow from a vertical gnomon with a height of 12 (sometimes 7) units. Note that in the original version, these definitions were used to calculate sundials. 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” appears in 1505. It was published and used in a book by the German theologian and mathematician Bartholomeus Pitiscus. At that time, science was already used to solve astronomical and 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 of triangles. The appearance of trigonometry is associated with land surveying, astronomy and the construction process. Although the name appeared relatively recently, many definitions and data currently classified as trigonometry were known before 2000.

1.3. Occurrence of sinus

The sine representation has a long history. In fact, various relationships between segments of a triangle and a circle (and, in essence, trigonometric functions) were found earlier in the 3rd century. BC. in the works of famous mathematicians of Ancient Greece - Euclid, Archimedes, Apollonius of Perga. During 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 double arc.

In the subsequent period, mathematics for a long time was most rapidly formed by Indian and Arab scientists. In the 4th-5th centuries, in particular, a previously special term arose in the works on astronomy of the famous Indian scientist Aryabhata (476-c. 550), after whom the first Hindu satellite of the Earth was named. He called the segment ardhajiva (ardha-half, jiva-string, a break that resembles an axis). Later, the more abbreviated name jiva was adopted. Arab mathematicians in the 9th century. the term jiva (or jiba) was replaced by the Arabic word jaib (concavity). During the transition of Arabic mathematical texts to the 12th century. this word was replaced by the Latin sinus (sinus-bend) (Fig. 4).

1.4. The appearance of cosine

The definition and origin of the term “cosine” is more short-term and short-term in nature. By cosine we mean “additional sine” (or otherwise “sine of the additional arc”; remember cosα= sin(90° - a)). An interesting fact is that the first methods for solving triangles, which are based on the relationship between the sides and angles of a triangle, were found by the Ancient Greek astronomer Hipparchus 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, and 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, Muhamed bin Muhamed (940-998), who collected 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, year of death unknown) and the Azerbaijani astrologer and scientist Nasireddin Tusi Muhamed (1201-1274). In addition, Nasireddin Tusi, in his own 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. Tangent (and also cotangent) was established in the 10th century by the Arabian arithmetician Abu-l-Wafa, who compiled the initial tables for finding tangents and cotangents. But these discoveries remained unknown to European scientists for a long time, and tangents were rediscovered only in the 14th century by the German arithmetician and astronomer Regimontanus (1467). He argued the tangent theorem. Regiomontanus also compiled detailed trigonometric tables; Thanks to his works, plane and spherical trigonometry became an independent discipline in Europe.

The designation “tangent”, which comes from the Latin tanger (to touch), arose in 1583. Tangens is translated as “touching” (the line of tangents is a tangent to the unit circle).
Trigonometry was further developed in the works of 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 using 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. In this way, it existed back in the Middle Ages, although sometimes analytical methods were also used in it, especially after the emergence of logarithms. Perhaps, the maximum incentives for the formation of trigonometry appeared in connection with the solution of astronomy problems, which gave enormous positive interest (for example, in order to solve problems of determining the location of a ship, forecasting blackout, etc.). Astrologers were interested in the relationships between the sides and angles of spherical triangles. And the arithmeticians of antiquity successfully coped with the questions posed.

Starting from the 17th century, trigonometric functions began to be used to solve 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 received significant significance for the whole of mathematics.

The analytical theory of trigonometric functions was mainly created by the outstanding 18th century mathematician Leonhard Euler (1707-1783), a member of the St. Petersburg Academy of Sciences. Euler's enormous scientific legacy includes brilliant results related to mathematical analysis, 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 proven through the formal application of trigonometry formulas, proofs became much more compact and simpler,

Thus, trigonometry, which originated 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 dependencies between them, began to be called goniometry (translated as the science of measuring angles, from the Greek gwnia - angle, metrew - I measure). The term goniometry has hardly been used recently.

2. Trigonometry and real life

Modern society is characterized by constant changes, discoveries, and 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 take a look at the interactions in each industry in order.

2.1.Navigation

The first point that explains to us the use and benefits of trigonometry is its connection with navigation. By navigation we mean a science whose goal is to study and create the most convenient and useful ways of navigation. Thus, scientists are developing simple navigation, which involves constructing a route from one point to another, evaluating it and choosing the best option from all those proposed. These routes are necessary for seafarers who, during their journey, face many difficulties, obstacles, and questions regarding the course of travel. Navigation is also necessary: pilots who fly complex, high-tech aircraft navigate, sometimes in very extreme situations; cosmonauts whose work involves risk to life, complex route construction and its development. Let's study the following concepts and tasks in more detail. As a problem, 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 imagine a solution to this problem, namely: first we clarify that the latitude of a point M on 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 equatorial plane: ≤ , and to the north of the equator the latitude is considered positive, and to the south – negative. For the longitude of point M we will take the value of the dihedral angle passing in the COM and SON planes. 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 arc of the great circle that connects A and B. We can call this type 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 side AB of the spherical triangle ABC, where C refers to the northern polis.

An interesting example is the following. When creating a route by sailors, precise and painstaking work is necessary. So, to plot 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. Edmond Gunther (1623) was the first to use trigonometric calculations in navigation.

With its help, trigonometry, pilots could calculate wind errors for the most accurate and safe control of the aircraft. In order to carry out these calculations, we refer to the velocity triangle. This triangle expresses the resulting air speed (V), wind vector (W), and ground speed vector (Vp). PU is the heading angle, UV is the wind angle, KUV is the wind heading angle (Fig. 5).

To familiarize yourself with the type of relationship between the elements of the navigation triangle of speeds, you need to look below:

Vp =V cos US + W cos UV; sin CV = * sin CV, tg CV

To solve the navigation triangle of speeds, calculating devices are used using a navigation ruler and mental calculations.

2.2.Algebra

The next area of interaction between trigonometry is algebra. It is thanks to trigonometric functions that very complex equations and problems 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 doesn’t matter what it is: acoustics, optics or the swing of a pendulum.

2.3.Physics

Besides 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 complete secret is a visual effect that forces our vision to perceive an object differently. Simple trigonometric formulas and the values of the sine of the angle of incidence and refraction of a half-line make it possible to calculate the constant refractive index when a light ray 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:

sin α / sin β = 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.

The entry of charged solar wind elements into the upper layers of the atmosphere of planets is caused by 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 proportional to the charge of the particle and the vector product of the field and the speed of the particle.

Revealing the practical aspects of the use of trigonometry in physics, we will give an example. This problem must be solved using trigonometric formulas and solution methods. Problem conditions: a body weighing 90 kg is located on an inclined plane with an angle of 24.5°. It is necessary to find what force the body has exerting 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 begin to build projections of forces on the axis, first using this formula:

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

X: ma = 0 + mg sin24.50

Y: 0 = N – mg cos24.50

We substitute the mass and find that the force is 819 N.

Answer: 819 N

2.4.Medicine, biology and biorhythms

The fourth area where trigonometry has a major impact and assistance is in two areas: medicine and biology.

Biological rhythms, biorhythms, are more or less regular changes in the nature and intensity of biological processes. The ability to make such changes in life activity is inherited and is 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, duration coinciding with any rhythm of the environment. These include daily, seasonal, annual, tidal and lunar rhythms. The main earthly rhythm is daily, determined by the rotation of the Earth around its axis, therefore almost all processes in a living organism have a daily periodicity.

We are seventy-five percent water, and if at the moment 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 to 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 to collect “roots”.

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

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

Intellectual biorhythm - it controls memory, the ability to learn, and logical thinking. In the activity phase there is a rise, and in the second phase there is a decline in creative activity, there is no luck and success.

Three Rhythms Theory:

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

· Emotional cycle - 28 days. State of the nervous system and mood

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

Trigonometry also occurs 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 fish's body takes the shape of a curve that resembles the graph of the function y=tgx.

When a bird flies, the trajectory of the flapping wings forms a sinusoid.

Trigonometry in medicine. As a result of a study conducted by Iranian Shiraz University student Vahid-Reza Abbasi, doctors for the first time were able to organize 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 use 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 heart activity, thereby speeding up the diagnosis and the start of treatment itself.

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

Trigonometry helps our brain determine distances to objects. American scientists claim that the brain estimates the distance to objects by measuring the angle between the plane of the earth 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 increased this angle.

This distortion led to the fact that experimental prism carriers perceived 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 incorrectly imagined surface of the earth. However, after 20 minutes they got used to the distorted perception, and all the 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 one might assume, will be of interest to engineers who design navigation systems for robots, as well as specialists who work on creating the most realistic virtual models. Applications in the field of medicine are also possible, 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 music theory.” With its help, basic musical structures and transformations are translated into the language of modern geometry.

Each note within the framework of the new theory is represented as a logarithm of the frequency of the corresponding sound (the note “C” of the first octave, for example, corresponds to the number 60, the octave to the number 12). The chord is thus represented as a point with given coordinates in geometric space. The 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 change in cardinality (C) . All these transformations, as the authors write, form so-called OPTIC symmetries in n-dimensional space and store musical information about the chord - in which octave its notes are located, in what sequence they are played, how many times they are repeated, etc. 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 chord sequence, for example, will be expressed in new terms as OPC), while others are fundamentally new concepts that , perhaps, will be adopted by 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 the sounds of the chord: blue - small intervals, warmer tones - more “sparse” sounds of the chord. 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 make changes to modern methods of teaching music and ways of studying various musical styles (classical, pop, rock). music, etc.). The new terminology will also help to more deeply compare musical works of composers from different eras and present research results in a more convenient mathematical form. In other words, it is proposed to isolate 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 legends that have come down from ancient times, the first who tried to do this were Pythagoras and his disciples.

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

2.6.Informatics

Trigonometry, with its influence, did not bypass computer science. Thus, its functions are applicable for accurate calculations. Thanks to this point, we can approximate any (in a 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, the 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 plays a serious role and assistance in the development and 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. A rotation occurs at a certain angle. To determine the coordinates of points, you will have to multiply by sines and cosines.

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

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 values of the angles of an arbitrary triangle on the plane are related to each other by certain relationships, the most important of which are called the theorems of cosines and sines. 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 the three elements of the triangle - the lengths of the sides and the angles - to restore the remaining three elements. 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 interested in “solving” triangles.

The process of erecting buildings, tracks, 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 trigonometric level. When trigonometric leveling, the height difference between several points on the earth's surface is established.

2.8 Trigonometry in art and architecture

Since the time man began to exist on earth, science has become the basis for improving everyday life and other areas of life. The foundations of everything created by man are various areas 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 means little. Let's consider an example of the construction of one sculpture by a French master of the Golden Age of art.

The proportional relationship in the construction of the statue was ideal. However, when the statue was raised on a high pedestal, it looked ugly. The sculptor did not take into account that in perspective, towards the horizon, many details are reduced and when looking from the bottom up, the impression of its ideality is no longer created. Many calculations were made to ensure that the figure from a great height looked proportional. They were mainly based on the method of sighting, that is, approximate measurement by eye. However, the difference coefficient 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 person’s eyes and the height of the statue, we can calculate the sine of the angle of incidence of the view using a table, thereby finding the point of view (Fig. 9).

In Figure 10, the situation changes, since the statue is raised to a height AC and NS increases, we can calculate the values of the cosine of angle C, and from the table we will find the angle of incidence of the gaze. In the process, you can calculate AN, as well as the sine of the 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 AN measurements in the first and second cases, one can find the proportionality coefficient. Subsequently, we will receive a drawing, and then a sculpture, when lifted, the figure will be visually closer to the ideal

Iconic buildings all over the world were designed thanks to mathematics, which can be considered the genius of architecture. Some famous examples of such buildings: Gaudi Children's School in Barcelona, Mary Ax Skyscraper in London, Bodegas Isios Winery in Spain, Restaurant in Los Manantiales in Argentina. When designing these buildings, trigonometry was involved.

Conclusion

Having studied the theoretical and applied aspects of trigonometry, I realized that this branch is closely related to many sciences. In the very beginning, trigonometry was needed to create and take measurements between angles. However, subsequently the simple measurement of angles grew into a full-fledged science studying 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.

Thus, thanks to trigonometric functions in medicine, the heart formula was discovered, which is a complex algebraic-trigonometric equality, which consists of 8 expressions, 32 coefficients and 33 basic parameters, including the possibility of additional calculations when arrhythmia occurs. This discovery helps doctors 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 constructing buildings, roads, bridges and other structures begins with survey and design work. All measurements at a construction site are carried out using surveying instruments such as theodolite and trigonometric level. With trigonometric leveling, the difference in height between several points on the earth's surface is determined.

Getting acquainted with its influence in other areas, we can conclude that trigonometry actively influences human life. The connection between mathematics and the outside world allows us 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 was completed successfully. 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 influence 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 this industry will help arouse interest among schoolchildren, form the right worldview and improve the general culture of high school students.

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 life around them.

Bibliography

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 general education institutions, M., Prosveshchenie, 2013.

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

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

Rybnikov K.A. History of mathematics: Textbook. - M.: Moscow State University Publishing House, 1994. Olehnik Problems in algebra, trigonometry and elementary functions / Olehnik, S.N. And. - M.: Higher School, 2016. - 134 p.

Olehnik, S.N. Problems in algebra, trigonometry and elementary functions / S.N. Olehnik. - 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. Alexandrov, P.I. Pasichenko. - M.: [not specified], 2015. - 762 p.

Annex 1

Fig.1Pyramid image. Slope calculation b / h.

Goniometer Seked

In general, the Egyptian formula for calculating the sekeda of the pyramid looks like

So:.

Ancient Egyptian term " second" indicated the angle of inclination. It was located across the height, divided by half the base.

"The length of the pyramid on the eastern 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 will get: 1 / 2 , 1 / 5 , 1 / 50 elbow. Keep in mind that one cubit is equal to 7 palm widths. Now multiply the resulting numbers by 7 as follows: "

Fig.2Gnomon

Fig.3 Determining 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

( Physical cycle - 23 days. Determines energy, strength, endurance, coordination of movement; The emotional cycle is 28 days. 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

Rodikova Valeria, Tipsin Eldar

The first mathematical knowledge appears in ancient times (IV-III centuries BC) in Ancient Greece. In the 17th-18th centuries, the fundamental content of science took place. Scientists from different countries at different periods of the development of civilization contributed to the development of modern mathematics. The branch of mathematics that studies trigonometric functions is called trigonometry. People from all walks of life use elements of trigonometry in their work. These are researchers in various scientific and applied fields, physicists, designers, computer technology specialists, designers, authors of multimedia presentations, doctors, and specialists in various fields. This project explored the application of trigonometry in architecture.

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The work was carried out by: Rodikova Valeria, Tipsin Eldar, students of class 10 “A” of MBOU “Beloyarsk Secondary School No. 1” Supervisor: Zhelnirovich N.V., mathematics teacher Trigonometry in architecture 2013 Regional research conference of students “Future elite of Verkhneketye”

TRIGONOMETRY - (from the Greek trigwnon - triangle and metrew - measure) - a science that studies the relationships between the angles and sides of triangles and trigonometric functions.

We assumed that trigonometry is used not only in the principles of analysis and algebra, but also in many other sciences, for example in architecture. Hypothesis

Introduction to the areas of application of trigonometry in architecture. Goals of work

Learn how trigonometry is used in architecture Explore the application of trigonometry in this problem area

Zaha Hadid Zaha Hadid (31 October 1950, Baghdad, Iraq) is a British architect of Arab origin. Representative of deconstructivism. In 2004, she became the first female architect in history to be awarded the Pritzker Prize. Deconstructivism is a trend in modern architecture. Deconstructivist projects are characterized by visual complexity, unexpected broken and deliberately destructive forms, as well as a pointedly aggressive invasion of the urban environment.

Sheikh Zayed Bridge in Abu Dhabi, UAE

Antoni Placid Guillem Gaudí i Curnet is a Spanish architect, most of whose whimsical and fantastic works were erected in Barcelona. The style in which Gaudi worked is classified as Art Nouveau. However, in his work he used elements of a wide variety of styles, subjecting them to processing. Modern is an artistic movement in art, its distinctive features are the rejection of straight lines and angles in favor of more natural, “natural” lines.

Gaudi Children's School in Barcelona, Spain

Gaudí surfaces k =1, a =1

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Santiago Calatrava Valls is a Spanish architect and sculptor, the author of many futuristic buildings in different countries of the world.

Bodegas Isios Winery Spain

CANDELA Felix (1910-1997), Mexican architect and engineer. Creator of various reinforced concrete shell vaults; developed thin-walled coatings in the form of hyperbolic paraboloids.

Restaurant in Los Manantiales, Argentina [ a d cos (t) + d d t , b d sin (t), c d t + e d t 2 ]

Swiss Re Insurance Corporation in London, UK x = λ y = f (λ) cos θ z = f (λ) sin θ

Gothic architecture Notre Dame Cathedral 1163 – mid-14th century.

Berlin sine waves, Germany

RESULTS Project “Schools of the Future”

: We found out that trigonometry is used not only in algebra and the principles of analysis, but also in many other sciences. Trigonometry is the basis for the creation of many masterpieces of art and architecture. We learned to see trigonometry in the construction of building models. Conclusion

Thank you for your attention!

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 the positions of the Sun and Moon compiled by Hipparchus made it possible to pre-calculate the moments of the onset of eclipses (with an error of 1-2 hours). Hipparchus was the first to use spherical trigonometry methods in astronomy. He increased the accuracy of his observations by using a cross of threads in goniometric instruments—sextants and quadrants—to point at the luminary. The scientist compiled a huge catalog of the positions of 850 stars for those times, dividing them by brightness into 6 degrees (stellar 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 the elements of a plane or spherical triangle from three given elements, important expansions of sinпх and cosпх in powers of cos x and sinx. Knowledge of the formula for sines and cosines of multiple arcs enabled Viet to solve the 45th degree equation proposed by the mathematician A. Roomen; Viète showed that the solution to this equation is reduced to dividing the angle into 45 equal parts and that there are 23 positive roots of this equation. Vieth solved Apollonius' problem using a ruler and compass.
Solving spherical triangles is one of the problems of astronomy. The following theorems allow us to calculate the sides and angles of any spherical triangle from three appropriately specified sides or angles: (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 any quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity oscillates harmoniously and changes over time as follows:

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 the question

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

Trigonometry in biology

Trigonometry plays an important role in medicine. With its help, Iranian scientists discovered the heart formula - a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 basic 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

Interpretation

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

We have proven that trigonometry is closely related to physics and is found in nature and medicine. One can give endlessly many examples of periodic processes of living 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

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"Danilova T.V.-script"

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

Educational project

" "

Danilova Tatyana Vladimirovna

Mathematic teacher

Justification of the relevance of the project.

Trigonometry is the branch of mathematics that studies trigonometric functions. It’s hard to imagine, but we encounter this science not only in mathematics lessons, but also in our everyday life. You might not have suspected it, but trigonometry is found in such sciences as physics, biology, it plays an important role in medicine, and, most interestingly, even music and architecture cannot 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 translated means the measurement of triangles (trigonan - triangle, metreo - I measure).
The emergence of trigonometry was closely related to land surveying, astronomy and construction.…

A schoolchild 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, and 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 consequence, 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.

Problematic question
1. Which trigonometry concepts 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 related?

Hypothesis

Hypothesis testing

Trigonometry (from Greektrigonon - triangle,metro – metric) –

History of trigonometry:

The next step in the development of trigonometry was made 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 “sine of the complement”, i.e. sine of the angle that complements the given angle to 90°. “Sine of the 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 wide application in mechanics, physics and technology, especially in the study of oscillatory movements 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 that it is used 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.

Vieta's achievements in trigonometry
A complete solution to the problem of determining all the elements of a plane or spherical triangle from three given elements, important expansions of sinпх and cosпх in powers of cos x and sinx. Knowledge of the formula for sines and cosines of multiple arcs enabled Viet to solve the 45th degree equation proposed by the mathematician A. Roomen; Viète showed that the solution to this equation is reduced to dividing the angle into 45 equal parts and that there are 23 positive roots of this equation. Vieth solved Apollonius' problem using a ruler and compass.
Solving spherical triangles is one of the problems of astronomy. The following theorems allow us to calculate the sides and angles of any spherical triangle from three appropriately specified sides or angles: (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 various physical natures obey general laws and are described by the same equations. There are different types of oscillatory phenomena.

Harmonic oscillation- the phenomenon of periodic change of any quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity oscillates harmoniously and changes over time as follows:

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

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

Trigonometry in nature.

We often ask the 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 proposed in 1637 by Rene Descartes. He explained rainbows as a phenomenon related to the reflection and refraction of light in raindrops.

Northern Lights The penetration of charged solar wind particles into the upper layers of the 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 speed of the particle.

American scientists claim that the brain estimates the distance to objects by measuring the angle between the plane of the earth and the plane of vision.

In addition, in biology such concepts as carotid sinus, carotid sinus and venous or cavernous sinus are used.

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

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

Biological rhythms, biorhythms

Basic earth rhythm– daily allowance.

A model of biorhythms can be built using trigonometric functions.

Trigonometry in biology

What biological processes are associated with trigonometry?

Biological rhythms, biorhythms are associated with trigonometry

A model of biorhythms can be built using graphs of trigonometric functions. To do this, you need to enter the person’s date of birth (day, month, year) and forecast duration

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 legends that have come down from ancient times, the first to try 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

Gaudi 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 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"

Ucheba.ru

Math.ru "library"

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"Danilova T.V."

" Trigonometry in the world around us and human life "

Research objectives:

The connection between trigonometry and real life.

Problematic question 1. Which trigonometry concepts 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 related?

Hypothesis

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

What is trigonometry???

Trigonometry (from the Greek trigonon - triangle, metro - metric) - microsection of mathematics, which studies the relationships between the values of angles and the lengths of the sides of triangles, as well as algebraic identities of trigonometric functions.

History of trigonometry

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

The word trigonometry first appears 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 were used to calculate the location of a ship at sea.

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

IN difference from the Greeks yians began to consider and use in calculations no longer the whole chord of MM  the corresponding central angle, but only its half MR, i.e. sine  - half of the central angle.

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's complement » , i.e. sine of the angle that complements the given angle to 90  . « Sine complement » or (in Latin) sinus complementi began to be 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 relations 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 wide application in mechanics,

physics and technology, especially when studying

oscillatory movements and others

periodic processes.

Viète, whose first mathematical studies related to trigonometry, knew about the properties of periodicity of trigonometric functions.

Proved that every periodic

movement may be

presented (with any degree

accuracy) in the form of a sum of primes

harmonic vibrations.

Founder analytical

theories

trigonometric functions .

Leonard Euler

In "Introduction to the Analysis of Infinites" (1748)

interprets sine, cosine, etc. not like

trigonometric lines, required

related to the circle, and how

trigonometric functions that he

viewed as a relationship between the parties

right triangle like numbers

quantities.

Excluded from my formulas

R – whole sine, taking

R = 1, and simplified it like this

way of recording and calculation.

Develops doctrine

about trigonometric functions

any argument.

Continued in the 19th century

theory development

trigonometric

functions.

N.I.Lobachevsky

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

Stages of development of trigonometry:

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

The first steps of trigonometry were to establish connections between the magnitude of the angle and the ratio of specially constructed straight line segments. The result is the ability to solve planar triangles.

The need to tabulate the values of entered trigonometric functions.

Trigonometric functions turned into independent objects of research.

In the 18th century trigonometric functions were included

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 that it is used 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 significant 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

with the sides and angles of a right triangle.

Thus, the beginning of trigonometry was laid in India

as the study of trigonometric quantities.

The tables of the positions of the Sun and Moon compiled by Hipparchus made it possible to pre-calculate the moments of the onset of eclipses (with an error of 1-2 hours). Hipparchus was the first to use spherical trigonometry methods in astronomy. He increased the accuracy of observations by using a cross of threads in goniometric instruments - sextants and quadrants - to point at the luminary. The scientist compiled a huge catalog of the positions of 850 stars for those times, dividing them by brightness into 6 degrees (stellar 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

Mechanical vibrations

Harmonic vibrations

Harmonic vibrations

Harmonic oscillation - the phenomenon of periodic change of any quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity oscillates harmoniously and changes over time as follows:

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

Mechanical vibrations

Mechanical vibrations are movements of bodies that repeat at exactly equal intervals of time. A graphical representation of this function gives a visual representation of the course of the oscillatory process over time.

Examples of simple mechanical oscillatory systems are a weight on a spring or a mathematical pendulum.

Math 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

The figure shows that the projections of the vectors on the X and Y axes are respectively equal

υ x = υ o cos α

υ y = υ o sin α

Trigonometry in nature

Optical illusions

natural

artificial

mixed

Rainbow theory

Rainbows occur when sunlight is refracted by water droplets suspended in the air. law of refraction:

The rainbow theory was first proposed in 1637 by Rene Descartes. He explained rainbows as a phenomenon related to the reflection and refraction of light in raindrops.

sin α /sin β = n 1 /n 2

where n 1 =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 refraction of light.

Northern lights

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

American scientists claim that the brain estimates the distance to objects by measuring the angle between the plane of the earth and the plane of vision.

In addition, in biology such concepts as carotid sinus, carotid sinus and venous or cavernous sinus are used.

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

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

Biological rhythms, biorhythms– these are more or less regular changes in the nature and intensity of biological processes.

Basic earth rhythm– daily allowance.

A 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 heart formula - a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 basic 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 need to 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 fish's body takes the shape of a curve that resembles the graph of the function y=tgx.

The emergence of musical harmony

According to legends that have come down from ancient times, the first to try 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

AC– the distance from the top of the statue to the person’s eyes,

AN– height of the statue,

sin C- sine of the angle of incidence of gaze.

Trigonometry in architecture

Gaudi 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 Guide to Mathematics"
Maple6 program that implements the image of graphs
"Wikipedia"
Ucheba.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 September 1, 1998.

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

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

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

Story

It is unknown at what point in time humanity began to create the future trigonometry from scratch. However, it is documented that already in the second millennium BC, the Egyptians were familiar with the basics of this science: archaeologists found a papyrus with a task in which it was required to find the angle of inclination of the pyramid on two known sides.

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

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

Name

Literally, the term “trigonometry” can be translated as “measurement of triangles.” The main object of study within this section of science for many centuries was the right triangle, or more precisely, the relationship between the magnitudes of the angles and the lengths of its sides (today, the study of trigonometry from scratch begins with this section). There are often situations in life when it is practically impossible to measure all the required parameters of an object (or the distance to the object), and then it becomes necessary to obtain the missing data through calculations.

For example, in the past, people could not measure the distance to space objects, but attempts to calculate these distances occurred long before the advent of our era. Trigonometry also played a crucial role in navigation: with some knowledge, the captain could always navigate by the stars at night and adjust the course.

Basic Concepts

Mastering trigonometry from scratch requires understanding and remembering several basic terms.

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

Tangent is the ratio of the opposite side to the adjacent side (or, which is the same, the ratio of sine to cosine). Cotangent is the unit divided by the tangent.

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

Popular mistakes

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

First, when solving geometry problems, you must remember that the use of sines and cosines is only possible in a right triangle. It happens that a student “automatically” takes the longest side of a triangle as the hypotenuse and gets incorrect calculation results.

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

Thirdly, until the problem is completely solved, you should not round any values, extract roots, or write a common fraction as a decimal. Often students strive to get a “beautiful” number in a trigonometry problem and immediately extract the root of three, although after exactly one action this root can be reduced.

Etymology of the word "sine"

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

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

Tables of values

There are tables that contain numerical values for sines, cosines and tangents of all possible angles. Below we present data for angles of 0, 30, 45, 60 and 90 degrees, which must be learned as a mandatory section of trigonometry for “dummies”; fortunately, they are quite easy to remember.

If it happens that the numerical value of the sine or cosine of an angle “got out of your head,” there is a way to derive it yourself.

Geometric representation

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

This method is very relevant if you forgot the required value, for example, during an exam, and you don’t have a trigonometry textbook at hand. You won’t get an exact number this way, but you will definitely see the difference between ½ and 1.73/2 (sine and cosine of an angle of 30 degrees).

Application

Some of the first experts to use trigonometry were sailors who had no other reference point on the open sea except the sky above their heads. Today, captains of ships (airplanes and other modes of transport) do not look for the shortest path using the stars, but actively resort to GPS navigation, which would be impossible without the use of trigonometry.

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

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

Repeatability

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

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

Finally

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

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

2. Trigonometry in physics

3. Trigonometry in astronomy

4. Trigonometry in medicine

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Story

Name

Basic Concepts

Popular mistakes

Etymology of the word "sine"

Tables of values

Geometric representation

Application

Repeatability

Finally

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