A message on the topic of trigonometry in medicine. Trigonometry in the world around us and human life

Trigonometry is a branch of mathematics that studies trigonometric functions and their use in geometry. Trigonometric functions are used to describe the properties of various angles, triangles, and periodic functions. Studying trigonometry will help you understand these properties. School activities and independent work will help you learn the basics of trigonometry and understand many periodic processes.

Steps

Learn the basics of trigonometry

Familiarize yourself with the concept of a triangle. In essence, trigonometry deals with the study of various relationships in triangles. A triangle has three sides and three angles. The sum of the angles of any triangle is 180 degrees. When learning trigonometry, it is necessary to become familiar with triangles and related concepts, such as:

hypotenuse is the longest side right triangle;
obtuse angle - an angle of more than 90 degrees;
an acute angle is an angle less than 90 degrees.

Learn to draw a unit circle. The unit circle makes it possible to construct any right triangle so that the hypotenuse is equal to one. This is useful when working with trigonometric functions such as sine and cosine. Having mastered the unit circle, you can easily find the values of trigonometric functions for certain angles and solve problems in which triangles with these angles appear.
- Example 1. The sine of an angle of 30 degrees is 0.50. This means that the length of the leg opposite the given angle is equal to half the length of the hypotenuse.
- Example 2. Using this ratio, you can calculate the length of the hypotenuse of a triangle in which there is an angle of 30 degrees, and the length of the leg opposite this angle is 7 centimeters. In this case, the length of the hypotenuse will be 14 centimeters.
Familiarize yourself with trigonometric functions. There are six basic trigonometric functions that you need to know when learning trigonometry. These functions represent the relationships between different sides of a right triangle and help you understand the properties of any triangle. These six functions are:
- sine (sin);
- cosine (cos);
- tangent (tg);
- secant (sec);
- cosecant (cosec);
- cotangent (ctg).
Remember the relationships between functions. When studying trigonometry, it is extremely important to understand that all trigonometric functions are interconnected. Although sine, cosine, tangent and other functions are used in different ways, they are widely used due to the fact that there are certain relationships between them. These relationships are easy to understand using unit circle. Learn to use the unit circle, and with the help of the relationships it describes, you will be able to solve many problems.

Application of trigonometry
1. Learn about the main areas of science that use trigonometry. Trigonometry is useful in many branches of mathematics and other exact sciences. Trigonometry can be used to find angles and line segments. In addition, trigonometric functions can describe any cyclic process.
  - For example, the vibrations of a spring can be described by a sinusoidal function.
2. Think about batch processes. Sometimes the abstract concepts of mathematics and other exact sciences are difficult to understand. However, they are present in the outside world, and this can make them easier to understand. Take a closer look at the periodic phenomena around you and try to connect them with trigonometry.
  - The moon has a predictable cycle of about 29.5 days.
3. Imagine how natural cycles can be studied. When you understand that there are many periodic processes in nature, think about how you can study these processes. Mentally imagine what the image of such processes looks like on a graph. Using a graph, you can write an equation that describes the observed phenomenon. This is where trigonometric functions come in handy.
  - Imagine the ebb and flow of the sea. At high tide, the water rises to a certain level, and then the tide is low, and the water level drops. After low tide, high tide follows again, and the water level rises. This cyclical process can continue indefinitely. It can be described by a trigonometric function, such as cosine.
Study the material ahead of time
1. Read the relevant section. Some people find it difficult to grasp the ideas of trigonometry the first time. If you familiarize yourself with the relevant material before class, you will better absorb it. Try to repeat the subject being studied more often - this way you will find more relationships between different concepts and concepts of trigonometry.
  - In addition, it will allow you to identify unclear points in advance.
2. Keep an outline. While skimming through a textbook is better than nothing, learning trigonometry requires slow, thoughtful reading. When studying any section, keep a detailed note. Remember that knowledge of trigonometry is accumulated gradually, and new material builds on what you've learned so far, so writing down what you've learned will help you move forward.
  - Among other things, write down questions you have to ask your teacher later.
3. Solve the problems given in the textbook. Even if trigonometry is easy for you, you need to solve problems. To make sure you really understand what you've learned, try solving a few problems before class. If you have problems doing this, you will determine what exactly you need to find out during the classes.
  - In many textbooks, answers to problems are given at the end. With their help, you can check whether you have solved the problems correctly.
4. Take everything you need to class. Don't forget your notes and problem solving. These handy materials will help you brush up on what you've already learned and move forward with your study of the material. Also clarify any questions that you have while reading the textbook.

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 modern world significant attention is paid to mathematics, as one of the areas 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 areas such 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 topics "Trigonometry in real life"is that the knowledge of trigonometry will open up new ways of solving various problems in many fields of science and will 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 to know historical reference 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, 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, found by an astronomer from Ancient Greece 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, 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, i.e., 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.

Since the 17th century, trigonometric functions have been used to solve equations, questions of mechanics, optics, electricity, radio engineering, in order to display oscillatory actions, wave propagation, displacement different elements, for the study of 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 analytic theory of trigonometric functions was mainly created by the outstanding 18th century mathematician Leonhard Euler (1707-1783) member Petersburg Academy 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 in Lately practically not used.

2. Trigonometry and real life

Modern society 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 geographical coordinates: latitude and longitude between points A and B 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), vector ground speed(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 because sunlight experiences refraction in water droplets suspended in 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.

The entry of charged elements of the solar wind into the upper layers of the atmosphere of the planets is determined by the interaction magnetic field land with 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 task must be solved using trigonometric formulas and methods of solution. Task conditions: inclined plane, the angle of which is 24.5o, 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. Between movement celestial bodies and living organisms on Earth there is a connection. 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 some kind of rhythm 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.

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

We are seventy-five percent water, and if at the time of the full moon the waters of the 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 exacerbations of the disease are often observed 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 in certain periods Is your life making 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 the best 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 has 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 manages 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. State nervous system and mood

· Intellectual cycle - 33 days. Defines creativity personalities

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 reached after a series of experiments in which participants were asked to look at the world through prisms that increase this angle.

Such a distortion led to the fact that the experimental carriers of prisms 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 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 some method that accurately provides a link 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 new theory is represented as the logarithm of the frequency of the corresponding sound (the note “do” of 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), while 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 "sparse" 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 to visualize 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, 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 present moment, 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 relationships, the most important of which are called the cosine and sine theorems. The formulas containing a, b, c imply that the letters are represented by the sides of the triangle, which lie respectively against 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 fascinated by the fact that they "solve" 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 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 human eyes 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 main 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.

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.: graduate 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. Alexandrov, P.I. Pasichenko. - M.: [not specified], 2015. - 762 p.

Annex 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 east 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

align=center>

Trigonometry- 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.
There are many areas where trigonometry and trigonometric functions are applied. Trigonometry or trigonometric functions are used in astronomy, marine and air navigation, acoustics, optics, electronics, architecture and other fields.

The history of the creation of trigonometry

The history of trigonometry, as a science of the relationship between the angles and sides of a triangle and others geometric shapes spans over 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.

Early centuries

From Babylonian mathematics, we are accustomed to measuring angles in degrees, minutes and seconds (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, later called the Pythagorean theorem.

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 solution in a general form of the problem of "solving triangles", that is, finding unknown elements of a triangle, based on three given elements (of which at least one is a side).
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 volumes of balls of different diameters, but the needs of astronomy and cartography caused fast development spherical trigonometry and related areas - systems celestial coordinates, the theory of cartographic projections, technologies of astronomical instruments.

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 the English astronomer Richard of Wallingford (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 development of trigonometry in modern times has become extremely important not only for astronomy and astrology, but also for other applications, primarily artillery, optics and navigation during long-distance sea voyages. Therefore, after the 16th century, many prominent scientists dealt with this topic, including Nicolaus Copernicus, Johannes Kepler, Francois Viet. Copernicus devoted two chapters to trigonometry in his treatise On the Revolutions of the Celestial Spheres (1543). Soon (1551) 15-digit trigonometric tables of Rheticus, a student of Copernicus, appeared. Kepler published Optical Astronomy (1604).

Vieta in the first part of his "Mathematical Canon" (1579) placed various tables, including trigonometric ones, and in the second part he gave a detailed and systematic, although without proof, presentation of plane and spherical trigonometry. In 1593 Vieta prepared an expanded edition of this capital work.
Thanks to the work of Albrecht Dürer, a sinusoid was born.

18th century

He gave a modern look to trigonometry. In the treatise Introduction to the Analysis of Infinites (1748), Euler gave a definition of trigonometric functions equivalent to the modern one and defined inverse functions accordingly.

Euler considered negative angles and angles greater than 360° as admissible, which made it possible to determine trigonometric functions on the entire real number line, and then extend them to the complex plane. When the question arose of extending trigonometric functions to obtuse angles, the signs of these functions before Euler were often chosen erroneously; many mathematicians considered, for example, the cosine and tangent of an obtuse angle to be positive. Euler determined these signs for angles in different coordinate quadrants based on reduction formulas.
Euler did not study the general theory of trigonometric series and did not investigate the convergence of the obtained series, but he obtained several important results. In particular, he derived the expansions of integer powers of sine and cosine.

Application of trigonometry

Those who say that trigonometry is not needed in real life are right in their own way. Well, what are its usual applied tasks? Measure the distance between inaccessible objects.
Of great importance is the triangulation technique, which makes it possible to measure the distances to nearby stars in astronomy, between landmarks in geography, and to control satellite navigation systems. Also of note is the application of trigonometry in such areas as navigation technology, 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, etc.
Conclusion: trigonometry is a huge helper in our daily life.

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 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.

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

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

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"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 schoolboy at the age of 14-15 does not always know where will he go to study and where to 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 areas such 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.

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:

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. Graphic image 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 is 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?

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 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.

IN difference from the Greeks eytsy began to consider and use in calculations not the whole chord MM  the corresponding central angle, but only its half MP, i.e. the 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,” writes Lobachevsky, “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

sinuses, which made it possible to enter various functions related

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

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:

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.

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

AC- 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 early XIX 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.

Municipal budgetary educational institution

average comprehensive school №10

with in-depth study of individual subjects

The project was completed by:

Pavlov Roman

10b grade student

Supervisor:

mathematic teacher

Boldyreva N. A

Yelets, 2012

1. Introduction.

3. The world of trigonometry.

· Trigonometry in physics.

· Trigonometry in planimetry.

· Trigonometry in art and architecture.

· Trigonometry in medicine and biology.

3.2 Graphical representations of the transformation of "little interesting" trigonometric functions into original curves (using computer program"Functions and Graphs").

· Curves in polar coordinates (rosettes).

· Curves in Cartesian coordinates (Lissajous curves).

· Mathematical ornaments.

4. Conclusion.

5. List of references.

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. It plays an important role in medicine, and, most interestingly, even music and architecture could not do without it.

Object of study - trigonometry

Subject of study - applied orientation of trigonometry; graphs of some functions, using trigonometric formulas.

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.

Hypothesis - assumptions: 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 methods - analysis of mathematical literature on the topic; selection of specific tasks of an applied nature on this topic; computer simulation based on a computer program. open mathematics"Functions and Graphs" (Physicon).

1. Introduction

“One thing remains clear that the world is arranged

terrible and wonderful."

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. 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. Problems with practical content play a significant role in developing the skills to apply in practice the theoretical knowledge gained in the study of mathematics. Every student of mathematics is interested in how and where the acquired knowledge is applied. This work provides an answer to this question.

2.History of the development of trigonometry.

Word trigonometry was composed of two Greek words: τρίγονον (trigonon-triangle) and and μετρειν (meter - to measure) in literal translation means triangle measurement.

It is this task - the measurement of triangles or, as they say now, the solution of triangles, i.e., the determination of all sides and angles of a triangle according to its three known elements (a side and two angles, two sides and an angle or three sides) - from ancient times was basis of practical applications of trigonometry.

Like any other science, trigonometry has grown out of human practice, in the process of solving specific practical problems. The first stages in the development of trigonometry are closely related to the development of astronomy. A great influence on the development of astronomy and trigonometry, closely related to it, was exerted by the needs of developing navigation, which required the ability to correctly determine the course of a ship on the high seas by the position of heavenly bodies. A significant role in the development of trigonometry was played by the need to compile geographical maps and the closely related need to correctly determine large distances on the earth's surface.

The works of the ancient Greek astronomer were of fundamental importance for the development of trigonometry in the era of its inception. Hipparchus(mid-2nd century BC). Trigonometry as a science, in the modern sense of the word, was absent not only from Hipparchus, but also from other scientists of antiquity, since they still had no idea about the functions of angles and did not even raise the question of the relationship between the angles and sides of a triangle in a general form. But in essence, using the means of elementary geometry known to them, they solved the problems that trigonometry deals with. At the same time, the main means of obtaining desired results there was an ability to calculate the lengths of circular chords based on the known relationships between the sides of a regular three-, four-, five- and decagon and the radius of the circumscribed circle.

Hipparchus compiled the first tables of chords, that is, tables expressing the length of the chord for various central angles in a circle of constant radius. These were, in essence, tables of double sines of half a central angle. However, the original tables of Hipparchus (like almost everything written by him) have not come down to us, and we can form an idea of them mainly from the composition “Great Construction” or (in Arabic translation) “Almagest” by the famous astronomer Claudius Ptolemy who lived in the middle of the 2nd century AD. e.

Ptolemy divided the circumference into 360 degrees and the diameter into 120 parts. He considered the radius to be 60 parts (60¢¢). He divided each of the parts into 60¢, every minute into 60¢¢, every second into 60 thirds (60¢¢¢), etc., using the indicated division, Ptolemy expressed the side of a regular inscribed hexagon or a chord subtracting an arc of 60 ° in the form of 60 parts of a radius (60h), and he equated the side of an inscribed square or a chord of 90 ° to the number 84h51¢10². , equal to the diameter of a circle, he wrote on the basis of the Pythagorean theorem: (chord a) 2 + (chord | 180-a |) 2 \u003d (diameter) 2, which corresponds to the modern formula sin2a + cos2a \u003d 1.

The Almagest contains a table of chords at half a degree from 0° to 180°, which from our modern point of view represents a table of sines for angles from 0° to 90° every quarter of a degree.

The basis of all trigonometric calculations among the Greeks was Ptolemy's theorem known to Hipparchus: "a rectangle built on the diagonals of a quadrilateral inscribed in a circle is equal to the sum of the rectangles built on opposite sides" (i.e., the product of the diagonals is equal to the sum of the products of the opposite sides). Using this theorem, the Greeks were able (with the help of the Pythagorean theorem) to calculate the chord of the sum (or the chord of the difference) of these angles or the chord of half of a given angle from the chords of two angles, i.e. they were able to obtain the results that we now obtain using the formulas for the sine of the sum (or difference) of two angles or half an angle.

New steps in the development of trigonometry are associated with the development of the mathematical culture of peoples India, Central Asia and Europe (V-XII).

An important step forward in the period from the 5th to the 12th century was made by the Hindus, who, unlike the Greeks, began to consider and use in calculations not the whole chord MM¢ (see drawing) of the corresponding central angle, but only its half MP, i.e. what we now call the line of the sine of the a-half of the central angle.

Along with the sine, the Indians introduced the cosine into trigonometry, more precisely, they began to use the cosine line in their calculations. (The term cosine itself appeared much later in the works of European scientists for the first time at the end of the 16th century from the so-called “complement sine”, that is, the sine of the angle that complements a given angle up to 90 °. “Sine of the complement” or (in Latin) sinus complementi began to be abbreviated as sinus co or co-sinus).

They also knew the ratios cosa=sin(90°-a) and sin2a+cos2a=r2 , as well as formulas for the sine of the sum and difference of two angles.

The next stage in the development of trigonometry is associated with countries

Central Asia, Middle East, Transcaucasia(VII-15th century)

Developing in close connection with astronomy and geography, Central Asian mathematics had a pronounced "computational character" and was aimed at solving applied problems of measuring geometry and trigonometry, and trigonometry was formed into a special mathematical discipline to a large extent precisely in the works of Central Asian scientists. Among the most important successes they made, first of all, we should note the introduction of all six trigonometric lines: sine, cosine, tangent, cotangent, secant and cosecant, of which only the first two were known to the Greeks and Hindus.

https://pandia.ru/text/78/114/images/image004_97.gif" width="41" height="44"> =a×ctgj of a pole of a certain length (a=12) for j=1°,2 °,3°……

Abu-l-Wafa from Khorasan, who lived in the 10th century (940-998), compiled a similar "table of tangents", i.e. calculated the length of the shadow b=a×=a×tgj cast by a horizontal pole of a certain length (a=60) on a vertical wall ( see drawing).

It should be noted that the terms "tangent" (in literal translation - "touching") and "cotangent" themselves originated from Latin and appeared in Europe much later (XVI-XVII centuries). The Central Asian scientists called the corresponding lines "shadows": cotangent - "first shadow", tangent - "second shadow".

Abu-l-Wafa gave an absolutely precise geometric definition of the tangent line in a trigonometric circle and added the lines of secant and cosecant to the lines of tangent and cotangent. He also expressed (verbally) algebraic relationships between all trigonometric functions and, in particular, for the case when the radius of a circle is equal to one. This extremely important case was considered by European scientists 300 years later. Finally, Abu-l-Wafa compiled a table of sines every 10¢.

In the works of Central Asian scientists, trigonometry turned from a science serving astronomy into a special mathematical discipline of independent interest.

Trigonometry separates from astronomy and becomes independent science. This branch is usually associated with the name of the Azerbaijani mathematician Nasiraddin Tusi ().

For the first time in European science, a harmonious presentation of trigonometry is given in the book "On Triangles of Different Kinds", written by Johann Müller, better known in mathematics as Regiomontana(). It generalizes in it methods for solving right triangles and gives tables of sines with an accuracy of 0.0000001. At the same time, it is remarkable that he assumed the radius of the circle to be equal, i.e., he expressed the values of trigonometric functions in decimal fractions, actually moving from the sexagesimal number system to decimal.

English scholar of the 14th century Bradwardine() he was the first in Europe to introduce a cotangent called the "direct shadow" and a tangent called the "reverse shadow" into trigonometric calculations.

On the threshold of the XVII century. In the development of trigonometry, a new direction is outlined - analytical. If before that the main goal of trigonometry was considered to be the solution of triangles, the calculation of the elements of geometric shapes and the doctrine of trigonometric functions was based on geometric basis, then in the XVII-XIX centuries. trigonometry gradually becomes one of the chapters of mathematical analysis. I also knew about the properties of the periodicity of trigonometric functions viet, the first mathematical studies of which were related to trigonometry.

Swiss mathematician Johann Bernoulli () already used the symbols of trigonometric functions.

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

Of great importance in the history of trigonometry was the work of the famous Petersburg academician Leonhard Euler(), he gave all trigonometry a modern look.

In his work "Introduction to Analysis" (1748), Euler developed trigonometry as a science of trigonometric functions, gave it an analytical presentation, deriving the entire set of trigonometric formulas from a few basic formulas.

Euler owns the final solution of the question of the signs of trigonometric functions in all quarters of the circle, the derivation of reduction formulas for general cases.

Having introduced new functions into mathematics - trigonometric ones, it became expedient to raise the question of expanding these functions into an infinite series. It turns out that such expansions are possible:

Sinx=x-https://pandia.ru/text/78/114/images/image008_62.gif" width="224" height="47">

These series make it much easier to compile tables of trigonometric quantities and to find them with any degree of accuracy.

The analytical construction of the theory of trigonometric functions, begun by Euler, was completed in the works , Gauss, Cauchy, Fourier and others.

Nowadays, trigonometry is no longer considered as an independent branch of mathematics. Its most important part, the doctrine of trigonometric functions, is part of a more general doctrine of functions studied in mathematical analysis, built from a unified point of view; the other part - the solution of triangles - is considered as the head of geometry.

3. The world of trigonometry.

3.1 Application of trigonometry in various sciences.

Trigonometric calculations are used in almost all areas of geometry, physics and engineering.

Of great importance is the technique of triangulation, which makes it possible to measure the distances to nearby stars in astronomy, between landmarks in geography, and to control satellite navigation systems. It should be noted the use of trigonometry in the following areas: navigation technology, music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound), computed tomography, pharmaceuticals, chemistry, number theory, seismology, meteorology, oceanology, cartography, many branches of physics, topography, geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography.

Trigonometry in physics.

Harmonic vibrations.

When a point moves in a straight line alternately in one direction or the other, then they say that the point makes fluctuations.

One of the simplest types of oscillations is the movement along the projection axis of the point M, which rotates uniformly around the circumference. The law of these oscillations has the form x=Rcos(https://pandia.ru/text/78/114/images/image010_59.gif" width="19" height="41 src="> .

Usually, instead of this frequency, one considers cyclic frequencyw=, indicating the angular velocity of rotation, expressed in radians per second. In these notations we have: x=Rcos(wt+a). (2)

Number a called the initial phase of the oscillation.

The study of oscillations of any kind is important already for the mere fact that we encounter oscillatory movements or waves very often in the world around us and use them with great success (sound waves, electromagnetic waves).

Mechanical vibrations.

Mechanical oscillations are the movements of bodies that repeat exactly (or approximately) at regular intervals. Examples of simple oscillatory systems are a weight on a spring or a pendulum. Take, for example, a weight suspended on a spring (see Fig.) and push it down. The kettlebell will begin to oscillate up and down..gif" align="left" width="132 height=155" height="155">.gif" width="72" height="59 src=">.jpg" align= "left" width="202 height=146" height="146"> The swing graph (2) is obtained from the swing graph (1) by shifting to the left

on . The number a is called the initial phase.

https://pandia.ru/text/78/114/images/image020_33.gif" width="29" height="45 src=">), where l is the length of the pendulum, and j0 is the initial deflection angle. The longer the pendulum, the slower it swings. (This is clearly seen in Fig. 1-7 appendix VIII). Figure 8-16, Appendix VIII clearly shows how a change in the initial deviation affects the amplitude of the pendulum oscillations, while the period does not change. By measuring the period of oscillation of a pendulum of known length, one can calculate the acceleration of the earth's gravity g at various points on the earth's surface.

Capacitor discharge.

Not only many mechanical vibrations occur according to a sinusoidal law. And sinusoidal oscillations occur in electrical circuits. So in the circuit depicted on the right upper corner models, the charge on the capacitor plates varies according to the law q \u003d CU + (q0 - CU) cos ωt, where C is the capacitance of the capacitor, U is the voltage at the current source, L is the inductance of the coil, https://pandia.ru/text/78 /114/images/image022_30.jpg" align="left" width="348" height="253 src=">Thanks to the capacitor model available in the Functions and Graphs program, you can set the parameters of the oscillatory circuit and build corresponding graphs g (t) and I(t). Graphs 1-4 clearly show how the voltage affects the change in the current strength and charge of the capacitor, while it is clear that with a positive voltage, the charge also takes on positive values. Figure 5-8 of Appendix IX shows that when the capacitance of the capacitor changes (when the inductance of the coil changes in Fig. 9-14 of Appendix IX) and the remaining parameters remain unchanged, the period of oscillation changes, i.e., the frequency of current oscillations in the circuit changes and the capacitor charge frequency changes .. (see. Annex IX).

How to connect two pipes.

The examples given may give the impression that sinusoids occur only in connection with oscillations. However, it is not. For example, sinusoids are used when connecting two cylindrical pipes at an angle to each other. To connect two pipes in this way, you need to cut them obliquely.

If you unfold a pipe cut obliquely, then it will be bounded from above by a sinusoid. This can be verified by wrapping the candle with paper, cutting it obliquely and unfolding the paper. Therefore, in order to obtain an even cut of the pipe, you can first cut the metal sheet from above along the sinusoid and roll it into a pipe.

rainbow theory.

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 n1=1, n2≈1.33 are respectively the refractive indices of air and water, 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 force Lorenz. It is proportional to the charge of the particle and the vector product of the field and the speed of the particle

Problems in trigonometry with practical content.

https://pandia.ru/text/78/114/images/image026_24.gif" width="25" height="41">.

Determination of the coefficient of friction.

A body of weight P is placed on an inclined plane with an inclination angle a. The body under the influence of its own weight has accelerated the path S in t seconds. Determine the coefficient of friction k.

Body pressure force on an inclined plane =kPcosa.

The force that pulls the body down is F=Psina-kPcosa=P(sina-kcosa).(1)

If the body moves along an inclined plane, then the acceleration is a=https://pandia.ru/text/78/114/images/image029_22.gif" width="20" height="41">==gF ;hence 2)

From equalities (1) and (2) it follows that g(sina-kcosa)=https://pandia.ru/text/78/114/images/image032_21.gif" width="129" height="48"> =gtga-.

Trigonometry in planimetry.

Basic formulas for solving problems in geometry using trigonometry:

sin²α=1/(1+ctg²α)=tg²α/(1+tg²α); cos²α=1/(1+tg²α)=ctg²α/(1+ctg²α);

sin(α±β)=sinα*cosβ±cosα*sinβ; cos(α±β)=cosα*cos+sinα*sinβ.

The ratio of sides and angles in a right triangle:

1) The leg of a right triangle is equal to the product of the other leg and the tangent of the opposite angle.

2) The leg of a right triangle is equal to the product of the hypotenuse and the sine of the included angle.

3) The leg of a right triangle is equal to the product of the hypotenuse and the cosine of the included angle.

4) The leg of a right triangle is equal to the product of the other leg and the cotangent of the included angle.

Task 1:On the sides AB and CD isosceles trapezoidABCD points M andN in such a way that the lineMN is parallel to the bases of the trapezium. It is known that in each of the formed small trapeziumsMBCN andAMND it is possible to inscribe a circle, and the radii of these circles are equalr andR respectively. Find groundsAD andBC.

Given: ABCD-trapezoid, AB=CD, MєAB, NєCD, MN||AD, in the trapezoids MBCN and AMND one can inscribe a circle with radius r and R respectively.

Find: AD and BC.

Solution:

Let O1 and O2 be the centers of circles inscribed in small trapezoids. Direct O1K||CD.

In ∆O1O2K cosα =O2K/O1O2 = (R-r)/(R+r).

Since ∆O2FD is rectangular, then O2DF = α/2 => FD=R*ctg(α/2). Because AD=2DF=2R*ctg(α/2),

similarly BC = 2r*tan(α/2).

cos α = (1-tg²α/2)/(1+tg²(α/2)) => (R-r)/(R+r)= (1-tg²(α/2))/(1+tg²(α /2)) => (1-r/R)/(1+r/R)= (1-tg²α/2)/(1+tg²(α/2)) => tg (α/2)=√ (r/R) => ctg(α/2)= √(R/r), then AD=2R*ctg(α/2), BC=2r*tg(α/2), we find the answer.

Answer : AD=2R√(R/r), BC=2r√(r/R).

Task2:In a triangle ABC known parties b, c and the angle between the median and height emanating from the vertex A. Calculate the area of a triangle ABC.

Given: ∆ ABC, AD-height, AE-median, DAE=α, AB=c, AC=b.

Find: S∆ABC.

Solution:

Let CE=EB=x, AE=y, AED=γ. By the law of cosines in ∆AEC b²=x²+y²-2xy*cosγ(1); and in ∆ACE, by the cosine theorem c²=x²+y²+2xy*cosγ(2). Subtracting equalities 2 from 1 we get c²-b²=4xy*cosγ(3).

Since S∆ABC=2S∆ACE=xy*sinγ(4), then dividing 3 equality by 4 we get: (c²-b²)/S=4*ctgγ, but ctgγ=tgαb, therefore S∆ABC= ( c²-b²)/4*tgα.

Answer: (s²- b² )/4*tg α .

Trigonometry in art and architecture.

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. I want to give 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 (we can do the same with the lower point of view), thereby finding the point vision (Fig. 1)

The situation is changing (Fig. 2), 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 2a+sin 2a = 1.

https://pandia.ru/text/78/114/images/image037_18.gif" width="162" height="101">.gif" width="108 height=132" height="132">

Trigonometry in medicine and biology.

Biorhythm model

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).

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.

Heart formula

As a result of a study conducted by an Iranian university student Shiraz Wahid-Reza Abbasi, for the first time, physicians 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.

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. Strictly speaking, the idea of "measuring angles" is not new. More artists Ancient China drew distant objects higher in the field of view, somewhat neglecting the laws of perspective. Alhazen, an Arab scientist of the 11th century, formulated the theory of determining distance by estimating angles. After a long oblivion in the middle of the last century, the idea was revived by psychologist James Gibson, who based his conclusions on the basis of experience with pilots. military aviation. However, after talking about the theory

forgotten again.

3.2 Graphical representations of the transformation of "little interesting" trigonometric functions into original curves.

Curves in polar coordinates.

With. 16is. 19 Sockets.

In polar coordinates, a single segment is selected e, pole O and polar axis Ox. The position of any point M is determined by the polar radius OM and the polar angle j formed by the beam OM and the beam Ox. The number r expressing the length of the OM in terms of e(OM=re) and the numerical value of the angle j, expressed in degrees or in radians, are called the polar coordinates of the point M.

For any point other than O, we can assume 0≤j<2p и r>0. However, when constructing curves corresponding to equations of the form r=f(j), it is natural to assign any values to the variable j (including negative ones and those exceeding 2p), and r can turn out to be both positive and negative.

In order to find the point (j, r), we draw a ray from the point O, forming an angle j with the Ox axis, and plot on it (for r>0) or on its continuation in the opposite direction (for r>0) the segment ½ r ½e.

Everything will be greatly simplified if you first construct a coordinate grid consisting of concentric circles with radii e, 2e, 3e, etc. (centered at the pole O) and rays for which j = 0 °, 10 °, 20 °, ... ,340°,350°; these rays will also be suitable for j<0°, и при j>360°; for example, at j=740° and at j=-340° we will hit a beam for which j=20°.

The study of these graphs helps computer program Functions and Graphs. Using the capabilities of this program, we explore some interesting graphs of trigonometric functions.

1 .Consider the curves given by the equations:r=a+sin3j

I. r=sin3j (shamrock ) (fig.1)

II. r=1/2+sin3j (Fig. 2), III. r=1+ sin3j (fig.3), r=3/2+ sin3j (fig.4) .

Curve IV has the smallest value r=0.5 and the petals have an unfinished appearance. Thus, when a > 1, the shamrock petals have an unfinished look.

2. Consider the curveswhen a=0; 1/2; 1;3/2

At a=0 (Fig. 1), at a=1/2 (Fig. 2), at a=1 (Fig. 3) the petals are finished, at a=3/2 there will be five unfinished petals., (Fig. .4).

3. In general, the curver=https://pandia.ru/text/78/114/images/image042_15.gif" width="45 height=41" height="41">), because in this sector 0°≤≤180 °..gif" width="20" height="41">.gif" width="16" height="41"> a single petal would require a "sector" greater than 360°.

Figure 1-4 shows the appearance of the petals with =https://pandia.ru/text/78/114/images/image044_13.gif" width="16" height="41 src=">.gif" width="16" height="41 src=">.

4. Equations found by a German naturalist mathematician Habenicht For geometric shapes found in the plant world. For example, the equations r=4(1+cos3j) and r=4(1+cos3j)+4sin23j correspond to the curves shown in Figure 1.2.

Curves in Cartesian coordinates.

Lissajous curves.

Many interesting curves can also be constructed in Cartesian coordinates. Particularly interesting are the curves whose equations are given in parametric form:

Where t is an auxiliary variable (parameter). For example, consider Lissajous curves, characterized in the general case by the equations:

If we take time as the parameter t, then the Lissajous figures will be the result of the addition of two harmonic oscillatory movements performed in mutually perpendicular directions. In the general case, the curve is located inside a rectangle with sides 2a and 2c.

Let's take a look at the following examples

I.x=sin3t; y=sin 5t (fig.1)

II. x=sin3t; y=cos 5t (fig.2)

III. x=sin3t; y=sin 4t. (Fig. 3)

Curves can be closed or open.

For example, replacement of equations I by equations: x=sin 3t; y=sin5(t+3) turns an open curve into a closed curve. (Fig. 4)

Interesting and peculiar are the lines corresponding to equations of the form

at=arcsin(sin k(x-a)).

From the equation y=arcsin(sinx) follows:

1) and 2) siny=sinx.

Under these two conditions, the function y=x satisfies. Graph it in the interval (-;https://pandia.ru/text/78/114/images/image053_13.gif" width="77" height="41"> we will have y=p-x, since sin( p-x)=sinx and in this interval

. Here the graph will be represented by the segment BC.

Since sinx is a periodic function with a period of 2p, the broken line ABC constructed in the interval (,) will be repeated in other sections.

The equation y=arcsin(sinkx) will correspond to a broken line with a period https://pandia.ru/text/78/114/images/image058_13.gif" width="79 height=48" height="48">

satisfies the coordinates of points that lie simultaneously above the sinusoid (for them y>sinx) and below the curve y=-sinx, i.e., the “solution area” of the system will consist of areas shaded in Fig. 1.

2. Consider the inequalities

1) (y-sinx)(y+sinx)<0.

To solve this inequality, we first build graphs of functions: y=sinx; y=-sinx.

Then we paint over the areas where y>sinx and at the same time y<-sinx; затем закрашиваем области, где y< sinx и одновременно y>-sinx.

This inequality will satisfy the areas shaded in Fig. 2

2)(y2-arcsin2(sinx))(y2-arcsin2(sin(x+)))<0

Let's move on to the next inequality:

(y-arcsin(sinx))(y+arcsin(sinx))( y-arcsin(sin(x+)))(y+arcsin(sin(x+))}<0

To solve this inequality, we first build function graphs: y=±arcsin(sinx); y=±arcsin(sin(x+ )) .

Let's make a table of possible solutions.

1 multiplier has a sign	2 multiplier has a sign	3 multiplier has a sign	4 multiplier has a sign

Then we consider and paint over the solutions of the following systems.

)| and |y|>|sin(x-)|.

2) The second multiplier is less than zero, i.e.gif" width="17" height="41">)|.

3) The third factor is less than zero, i.e. |y|<|sin(x-)|, другие множители положительны, т. е. |y|>|sinx| and |y|>|sin(x+Academic disciplines" href="/text/category/uchebnie_distciplini/" rel="bookmark">academic disciplines, technology, everyday life.

The use of the modeling program "Functions and Graphs" significantly expanded the possibilities of conducting research, made it possible to materialize knowledge when considering the applications of trigonometry in physics. Thanks to this program, laboratory computer studies of mechanical oscillations were carried out using the example of pendulum oscillations, and oscillations in an electric circuit were considered. The use of a computer program made it possible to investigate interesting mathematical curves defined using trigonometric equations and plotting in polar and Cartesian coordinates. The graphical solution of trigonometric inequalities led to the consideration of interesting mathematical ornaments.

5. List of used literature.

1. ., Atanasov of mathematical problems with practical content: Book. for the teacher.-M.: Education, p.

2. .Vilenkin in nature and technology: Book. for extracurricular reading IX-X cells - M .: Education, 5s (World of Knowledge).

3. Household games and entertainment. State. ed. physics and mathematics lit. M, 9str.

4. .Kozhurov trigonometry for technical schools. State. ed. technical-theoretical lit. M., 1956

5. Book. For extracurricular reading mathematics in high school. State. educational-ped. ed. Min. Prosv. RF, M., p.

6. Tarakanov trigonometry. 10 cells ..-M .: Bustard, p.

7. About trigonometry and not only about it: a guide for students in grades 9-11. -M .: Education, 1996-80s.

8. Shapiro problems with practical content in teaching mathematics. Book. for the teacher.-M.: Education, 1990-96s.

A message on the topic of trigonometry in medicine. Trigonometry in the world around us and human life

Steps

Learn the basics of trigonometry

Application of trigonometry

Study the material ahead of time

The history of the creation of trigonometry

Early centuries

Ancient Greece

Middle Ages

new time

18th century

Application of trigonometry

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