Is trigonometry ever useful in real life?

Message about trigonometry in medicine. Trigonometry in the world around us and human life

Trigonometry is a section 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. Courses in school and self-employed work they will help you learn the basics of trigonometry and understand many periodic processes.

steps

Explore the basics of trigonometry

    Look at the concept of a triangle. Essentially, trigonometry is active in the study of various relationships in triangles. The triangle has three sides and three corners. The sum of the angles of each triangle is 180 degrees. When studying trigonometry, it is necessary to learn about the triangles and the concepts associated with them, e.g.

    • hypotenuse - the longest side of the rectangular triangle;
    • stupid angle - an angle greater than 90 degrees;
    • acute angle - an angle less than 90 degrees.
  1. Learn to build a single circle. A single circle allows any rectangular triangle to be built so that the hypotenuse is equal to one. It comes in handy when you are working with trigonometric functions such as sines and cosines. After mastering the unit circle, you can easily find values ​​of trigonometric functions for specific angles and solve problems where triangles appear with those angles.

    • Example 1. The sine angle of 30 degrees is 0.50. This means that the length of the opposite of this vertex of the category is half the length of the hypotenuse.
    • Example 2. Using this ratio, it is possible to calculate the length of the hypotenus of a triangle in which there is an angle of 30 degrees, and the length of the control of this corner of the category is 7 centimeters. In this case, the length of the hypotenuse is 14 centimeters.
  2. Familiarize yourself with trigonometric functions. There are six important trigonometric functions to know while learning trigonometry. These functions are the relationships between the different sides of the rectangular triangle and help understand the properties of each triangle. These are these six functions:

    • sinus (sin);
    • cosine (cos);
    • tangent (TG);
    • sec (sec);
    • cosnean (cosec);
    • cotangent (ctg).
  3. Remember the relationships between functions. When studying trigonometry, it is extremely important to understand that all of the trigonometric functions are related. Although sine, cosine, tangent, and other functions are used in different ways, they are widely used because they have certain relationships between them. These relationships are easy to understand individually. Learn to use a single circle, and with the help of the ratios it describes, you will be able to solve many problems.

    Application of trigonometry.

    1. Learn more about the main domain areas that use trigonometry. Trigonometry is useful in many sections of math and other accurate sciences. With the help of trigonometry you can find the values ​​of corners and straight segments. In addition, any cyclic process can be described by trigonometric functions.

      • For example, spring oscillations can be described with a sinusoidal function.
    2. Think about regular processes. Sometimes the abstract concepts of mathematics and other precise sciences are difficult to understand. However, they are present in the surrounding world and it can make them easier to understand. Close to periodic phenomena around you and try to tie them with trigonometry.

      • The moon has a predictable cycle that lasts about 29.5 days.
    3. Imagine learning natural cycles. When you understand that there are many regular processes in nature, think about how these processes can be studied. Imagine in your mind what the picture of such processes looks like in the graphic. The graph allows you to make an equation that describes the observed phenomenon. In this case, trigonometric functions are useful.

      • Imagine fields and foams on the coast. During the high tide, water rises to a certain level and then the high tide comes and the water level falls. After the tide is out, the tide should be back and the water level will rise. This cyclical process can go on indefinitely. It can be described by a trigonometric function like cosine.

    Learn the material in advance

    1. Read the appropriate section. Some people are difficult to learn the ideas of trigonometry from the first time. If you familiarize yourself with the relevant material before the class, it will be better to digest it. Try to repeat the topic more often - this way you will find more relationships between the different concepts and concepts of trigonometry.

      • In addition, you can reveal unclear moments in advance.
    2. Drive a summary. While the tagging textbook is better than nothing when studying trigonometry, a leisurely, thoughtful reading is required. When examining a partition, he deceived the detailed summary. Remember that knowledge of trigonometry gradually accumulates and new material draws on what has been previously studied, so the data sets that have already passed will help you move on.

      • Among other things, write down your questions to ask the teacher.
    3. Decide the tasks given in the textbook. Even if you give trigonometry easily, there are problems to be solved. To make sure that you really understand the material you are studying, try doing several tasks before class. If you have problems define what you need to find out in class.

      • Many textbooks at the end are answers to tasks. With their help, you can check whether you have solved the tasks correctly.
    4. Take everything you need for class. Don't forget your summary and solve problems. These infringement materials will help you replenish the storage that has already been put in and help you further investigate the material. Clarify any questions that arose from your preliminary reading of the textbook.

Community general education.

"High School №1"

"Trigonometry in Real Life"

information project.

Executed:

Krasnov Egor.

student 9A class.

Leader:

Borodkina Tatyana Ivanovna.

Zheleznogorsk.

      Introduction ................................................. ..................... 3

      Relevance ................................................. .................... ......... 3

      Goal ................................................. .................................... 4.

      Tasks ................................................. ................. .4.

1.4 Methods ................................................ ............... ... 4.

2. trigonometry and history of its development ... ................................ ..5

2.1.Rigonometry and stages of education .... ...................... 5

2.2.Rigonometry as a concept. Characteristic .................. 7.

2.3. The construction of Sinus ............................ ................ .. 7th

2.4. The use of cosines .......................................... 8

2.5. Tangent and Kotnenz ...... .................. .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 ... .............................................. ...................................

3.4.medicin, biology and biorhythms. ... ........................ ..... 15

3.5.music ............................................... ............... .... 19.

3.6. IT .. ............................................. ......... ... 21.

3.7 Shop construction and geodesy. .............................. ... 22

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

Conclusion. .................................................. .............. .. ..... 25

References. ................................................ .. ........... 27.

Annex 1. ... .... ........................................ ..................................... 29.

introduction

IN THE MODERN WORLD Significant attention is paid to the mathematics survey as one of the areas of scientific activity and study. As we know, one of the components of math is trigonometry. Trigonometry is a section of mathematics that studies trigonometric functions. I believe that this topic is relevant from a practical point of view first. We learn to learn at school and we understand that knowledge of trigonometry is necessary for many professions simply because it enables you to measure distances to nearby stars in astronomy, between benchmarks in geography, control satellite navigation systems. Principles of trigonometry, in areas such as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound testing (ultrasound) and computed tomography), pharmaceuticals, chemistry, number theory (and, as a consequence, cryptography), seismology , Meteorology, oceanology, cartography, many sections of physics, topography and geodesy, architecture, phonetics, economics, electronic equipment, mechanical engineering, computer graphics, crystallography.

Secondly, relevance Topics "Trigonometry in Real Life" It is that the knowledge of trigonometry opens new avenues to solve various tasks in many areas of science and to understand some aspects of various sciences.

This practice has long been established with schoolchildren facing three times using trigonometry. So we can say that trigonometry has three parts. These parts are interconnected and depend on the time. At the same time, they are absolutely different, do not have similar features as in the meaning that is put in the explanation of the basic concepts and functions.

The first acquaintance occurs in grade 8. This is the period when schoolchildren: "The relationship between the sides and angles of the rectangular triangle". In studying trigonometry, the concept of cosine, sine and tangent is given.

The next step is to continue acquaintance with trigonometry in Note 9. The level of complexity increases, methods and procedures for solving examples are changed. Now a circle and its abilities come to the place of cosines and tangents.

The last step is the 10th grade, when trigonometry becomes more complex, the ways of solving problems are changed. The concept of the radian measure of the angle is introduced. Diagrams of trigonometric functions are introduced. At this point, students begin to solve and study trigonometric equations. But no geometry. For a complete understanding of trigonometry, it will be necessary to get acquainted with the history of the event and its evolution. After acquaintance with the historical certificate and studying the activities of the works of great figures, mathematicians and scientists, we can understand how trigonometry affects our lives, as it helps to create new objects, create discoveries.

purposemy project is to study the influence of trigonometry on a person's life and the development of interest in it. After solving this goal, we can understand what place trigonometry occupies in our world, which decides practical tasks.

In order to solve the goal, we determined the following tasks:

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

2. Consider examples of the practical influence of trigonometry in different areas of activity;

3. Examples show the possibilities of trigonometry and its use in a person's life.

Methods:Find and collect information.

1. trigonometry and history of its development

What is trigonometry? This term implies a section in mathematics that deals with the study of the dependence between different angles, studies the length of the side of the triangle and the algebraic identities of trigonometric functions. It is difficult to imagine that this area of ​​mathematics is found in us everyday life.

1.1.Rigonometry and stages of its formation

Let's turn to the history of its development, levels of education. Since ancient times, trigonometry gained its primaries, developed and showed the first results. The first information about the origin and development of this field we can see in the manuscripts located in ancient Egypt, Babylon, ancient China. After studying the 56th task of Rinda Papyrus (II Millennium BC. ER), you can see that it offers to find the slope of the pyramid, the height of which is a 250-elbow height. The length of the base of the pyramid is 360 elbows (Fig. 1). It is curious that in solving this problem, the Egyptians used two measuring systems at the same time - "Elbows" and "Palm". Today, to solve this problem, today we would find a tangent angle: know half of the base and the apothem (Fig. 1).

The next step was the stage of the development of science, which began with the astronomer Aristarkh samocheogram in the III century BC. Is associated. e. The treatise, in view of the size and distance of the sun and moon, sets you in a definite task. It was expressed in the need to determine the distance from each heavenly body. To make such calculations it was necessary to calculate the relationships of the parties of the rectangular triangle with the known value of one of the corners. Aristarh looked at a rectangular triangle formed by the sun, moon, and earth during the quadrature. To calculate the size of the hypotenuse, which as the basis of the distance from the ground to the sun with the cathetus that dipped the distance from the ground to the moon, with the known value of the neighboring angle (87 °), which is equivalent to the value to calculate sin angle 3.. According to Aristarchus, this value is between 1/20 and 1/18. This suggests that the distance from the sun to the earth is twenty times longer than from the moon to the earth. However, we know that the sun is 400 times the location of the moon. The incorrect assessment was due to inaccuracies in the measurement of the angle.

A few decades later, Clawedy-Ptoleum offers a detailed account of trigonometric supplements in cartography, astronomy and mechanics in their own workmen "Ethnogeography", "Analema" and "Plasfeey". Among other things, a stereographic projection is shown, a number of actual problems have been investigated, for example: To adjust the height and angle of the heavenly lights according to their decline and the hourly corner. From a trigonometry point of view, this means that it is necessary to find the side of the spherical triangle according to other 2 faces and an opposite corner (Fig 2)

In the aggregate it can be indicated that trigonometry has been applied to:

Clear time setting;

Calculations of the imminent location of celestial luminars, episodes of their sunrise and sunset, the insolation of the sun and moon;

Find geographic coordinates of the current location;

Calculation of the distance between Megapolis with known geographical coordinates.

Gnomon-ancient astronomical mechanism, vertical subject (stele, column, pole), which allows you to determine the angular height of the sun (Fig. 3) with the smallest length of the shadow.

Thus, cotangent seemed to us to be the length of the shadow of the vertical gnomon height of 12 (sometimes 7) units. It should be noted that these definitions were used in the first embodiment to calculate the solar cycle. A shadow seemed to fall tangentially from the horizontal gnomon. Understood by Cosneis and Sessions are understood as hypotenuses that correspond to rectangular triangles.

1.2.Rigonometry as a term. Characteristic

The concrete term "trigonometry" is found for the first time in 1505. It was published and used in the book of the German theologian and mathematics Bartolomeuse Pitiscus. While science has already been used to solve astronomical architectural problems.

The term trigonometry is characterized by Greek roots. And consists of two parts: "triangle" and "measure". We can say that we are saying that we have a science that studies change in triangles. The appearance of trigonometry is associated with land, astronomy, and the building process.Although the name did not prevail for a relatively long time, many currently selectable definition trigonometries and dates were known earlier than 2000.

1.3. The emergence of sinus

The long story has a sine presentation. Indeed, a variety of relationships of the segments of the triangle and circle (and essentially trigonometric functions) are found earlier in 3V. Bc. In the works of the famous mathematicians of Ancient Greece - Euclida, Archimedes, Apollonia Perga. In the Roman time interval, these relationships were already regularly studied (century. AD), although they did not receive a special name. The modern sine of the angle α is examined, for example, as a semiord, which is based on the central angle of α or in a doubtful arc.

In the subsequent gap, mathematics, formed the fastest by Indian and Arab scholars for a long time. In 4-5 centuries there was in particular a previously special period in the work of astronomy of the famous Indian scientist Ariarabhat (476-ok. 550), the first Hindu satellite of the earth is named after it. The cut called Ardhajdi (Ardha-half, Jiva-Alvelz of a pause that resembles the axis). Later a more abbreviated name was given by Jiva. Arabic mathematics in IX. The term Jiva (or Dzhiba) has been replaced by the Arabic word Jaib (concave). Moving through the Arabic mathematical texts in Xiiv. This word has been replaced by Latinsinus (sinus bend) (Fig. 4).

1.4. The origin of cosine

The definition and origin of the term "cosine" is more short-term and not used. The cosine is understood as an "additional sine" (or otherwise "sine of an additional arc"; remember cosα \ u003d sin (90 ° - a)). Interesting fact is that the first ways to solve triangles based on the dependency between the parties and the corners of the triangle were made by an astronomer from ancient Greece HIPPUCH in the second century BC. This study was also engaged in Claudius Ptolemy. Gradually appeared new facts about the relationship between the relations of the parties of the triangle of the triangle and its corners, began to apply a new definition - trigonometric function.

Arabs Al-Batti (850-929) and Abu-L-Wafa, Muhamed-Ben mohamed (940-998), who collected the sine and tangent tables with the accuracy of 10 'with the accuracy of 1/604 brought to the formation of trigonometry. The sine theorem previously chose the Indian professor Bhaskar (R. 1114, which is the year of death, if not) and Azerbaijani astrologer and scientist Nasi-studied Tusi Mohamed (1201-1274). In addition, Nasasedin told Tusi in his own work "Work on the full four-sided" a straight and spherical trigonometry as an independent discipline (Fig. 4).

1.5. The origin of tangent and kotnän

Tangents arose in relationships with the completion of the task of adjusting the length of the shadow. Tangent (and also cotangenes) is established in the 10th century Arabic Arabic arithmic Abu-L-Wafa, which also compiled and initial tables for finding tangents and cotangents. However, the opening dates remained unknown to European scientists for a long time, and the tangents were recaptured only in the XIV century by German arithmetic, Astronomenremontane (1467). He argued the theorem of tangents. Regomonanta also made detailed trigonometric tables; Thanks to his works, flat and spherical trigonometry has become an independent discipline and in Europe.

The term "tangent", which comes from the Latin Tangier (affected), dates back to 1583. Tangents is translated as "affectation" (the line of tangents is tangential to a single circle).
Another formation of trigonometry found in the works of outstanding astrologers Nikolai Copernicus (1473-1543), Ruhiger Brage (1546-1601) and Iogana Kepler (1571-1630) and also in the works of mathematics, Francois Vieta (1540-1603) who completely solved the problem of determining absolutely all components of a flat or spherical triangle in three dates (Fig. 4).

1.6 Further development of trigonometry

For a long time, trigonometry was purely a geometric appearance; H. The data we are currently formulating in the definitions of trigonometric functions has been formulated and argued with the support of geometric concepts and allegations. So it existed in the middle century, although analytical methods were sometimes used, especially after the appearance of logarithms. Perhaps the maximum incentives for the formation of trigonometry appeared in relationships with the solution of astronomical problems, which gave a huge positive interest (for example, to deal with questions of determining the location of the ship, displacement prediction, etc.). Astroks occupied the relationship between the sides and angles of spherical triangles. And ancient arithmets were successfully coped with with raised questions.

Starting from the XVII century, the trigonometric functions began to solve equations, questions of mechanics, optics, electricity, radio engineering, to show oscillatory measures, the propagation of waves, displacement of different elements, for the study of the alternating galvanic current, etc. Because of this, trigonometric functions have been studied extensively and deeply, and they have been essential to all mathematics.

The analytical theory of trigonometric functions was created mainly by an eminent mathematician of the 18th century Leonard Euler (1707-1783) from a member of the Petersburg Academy. Science Euler's vast scientific legacy includes brilliant results relating to mathematical analysis, geometry, numbers, mechanics, and other math applications. It was Euler who first introduced the well-known definitions of trigonometric functions, began to consider the functions of any angle that received the formula of Bringings. According to Euler, trigonometry acquired the calculus form: various facts began to prove themselves through the formal use of trigonometry formulas, the proofs became much more compact, simpler,

Thus, trigonometry, which emerged as the science of solution science, evolved over the course of the science of trigonometric functions.

Later, a part of trigonometry that studies the properties of trigonometric functions and dependencies between them was called goniometry (science translated on the measurement of angles, from the Greek gwnia - angle, METEHR). Goniometrial term B. Practically not used recently.

2. Trigonometry and real life

Modern society is characterized by constant change, discovery, creating high-tech inventions that improve our lives. Trigonometry is done and interacts with physics, biology, mathematics, medicine, geophysics, navigation, computer science.

We will get to know each other with interaction in every industry.

2.1. Navigation.

The first point that explains us the uses and benefits of trigonometry is to communicate with navigation. Through navigation, we understand the science, its purpose, which is to study and create the most convenient and useful navigation wines in it. Therefore, scientists are developing a simple navigation that involves building a route from one point to another, evaluating it and choosing a better option from all. These routes are required by navigators who face many difficulties, barriers, questions about the rate of movement during their travels. Navigation is also necessary: ​​pilots who control complex, high-tech aircraft are sometimes aimed in very extreme situations; Cosmonauts whose work involves complex route construction and its development with risk. We learn in more detail in the following concepts and tasks. You can submit the following condition as a task: We know geographic coordinates: latitude and longitude between points A and in the earth's surface. It is necessary to find the shortest route between points A and in the surface of the earth (the radius of the earth is known to be: R \ u003d 6371 km).

We can also submit a solution to this problem, namely: First, we refine that the size of the angle formed by the radius om of the latter, where the mean of the earth is, with the plane of the equator: ≤ and the navra of the equator of latitude holds as positive and south negative. For the length of paragraph M we take the value of the Dugrani corner in the planes and sleep. By C we mean the north pole of the earth. As h we understand the point corresponding to the Greenwich Observatory: ≤ (east of the Greenwich meridian, the longitude is considered positive, to the west - negative). As we already know, the shortest distance between points A and on the surface of the earth is represented in the smallest arc of a large circle connecting A and V. This type of ARC can call orthodromesia. Translated from Greek, this term is understood by the direct angle. For this reason, our task is to define the length of the side of the AV spherical triangle in which the Northern Polis is understood.

An interesting example can describe the following. When creating a route with Naval, it is necessary to do accurate and tedious work. In order to place the ship's course on the map, which was carried out in the projection of Gerhard Mercator in 1569, there was an urgent need to determine the latitude. However, when entering the sea, in locations up to the XVII century, the latitude was not indicated by navigators. For the first time trigonometric calculations were used in navigation Edmond Gunter (1623).

With its help trigonometry, pilots could count wind errors for the most accurate and safest aircraft. In order to carry out these calculations, we appeal to the speed triangle. This triangle is expressed by the formed air speed (V), the wind vector (W), vector running speed (Vp). PU - travel angle, WC - wind angle, Bub - wind flow angle (Fig. 5).

To get acquainted with the nature of the dependency between the elements of the navigation triangle of speeds, it is necessary to take a look below:

Vp \ u003d v cos hs + w cos hc; Sin usd \ u003d * sin hc, tg wc

To solve the navigation triangle of speeds, countable devices are used that use the navigation line and counting in the mind.

2.2. Algebra.

The next area of ​​trigonometry interaction is algebra. It is due to trigonometric functions that are solved very complexly, which requires large calculations of the equation and tasks.

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

2.3 Fizika.

In addition to navigation and algebra, trigonometry has a direct impact and impact on physics. When objects are immersed in water, they do not change shape or volume. A complete secret is a visual effect that forces our vision to take the subject differently. Simple trigonometric formulas and sine values ​​of the angle of incidence and the refraction to the half-bypass offer the possibility to calculate the constant refractive index when moving the light beam from the sphere. For example, the rainbow appears on the fact that sunlight It is broken into water droplets, which, according to the refractive law, are suspended in the air:

sin α / sin β \ u003d n1 / n2

where: n1 is the index of refraction of the first medium; N2 is the index of refraction of the second neighborhood; α-angle of the due, β-angle angle of the light.

In a situation in the upper layers of the atmosphere of planets the charged elements of the solar wind is determined by the interaction of the earth's magnetic field with sunshine.

The force on the magnetic area that moves in the charged particle is called the Lorentz power. It is due to the charge of the particle and the vector product of the field and the speed of moving the particle.

We show the practical aspects of using trigonometry in physics an example. This task should be solved with trigonometric formulas and solutions. Conditions of the problem: in an inclined plane, the angle, the angle of which is 24.5o, is a body weighing 90 kg. It is necessary to find the force in which there is a body that is on the inclined plane (i.e. what pressure does the body have on this plane) (Fig. 6).

Remembering the x and y axes, let's start building the projection of the forces on the axis to start with this formula:

ma \ u003d n + mg, then look at the drawing,

X: ma \ u003d 0 + mg sin24.50

Y: 0 \ u003d n - mg cos24.50

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

Answer: 819 a

2.4. Medicine, Biology and Biorhythms

The fourth region where trigonometry has serious implications and support are two areas at the same time: medicine and biology.

One of the fundamental characteristics of wildlife is the cycle of most of the processes that occur in it. Between the movement of heavenly tel 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 precisely define the position of the sun, react 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 to change such changes is inherited and found almost all living organisms. They can be observed in individual cells, tissues and organs, whole organisms and populations. Biorhythms are divided by physiological, Periods from a fraction of a second to a few minutes and environmentaccording to the duration that coincides with a rhythm environment. These include daily, seasonal, annual, tidal, and lunar rhythms. The main rhythm of the terrestrial rhythm is due due to the rotation of the earth around its axis, therefore almost all processes in the living organism have daily periodicity.

Many environmental factors on our planet, first of all, the lighting mode, temperature, pressure and humidity, atmospheric and electromagnetic fields, sea areas and subsidence, are of course changed under the influence of this rotation.

We are seventy-five percent of the water, and when at the time of the full moon the waters of world mine rise 19 meters above sea level and the tide begins, then the water in our body rushes into the upper compartments of our body. And people with increased pressure in these times, often no worsening of the disease is observed, and naturalists collecting medicinal herbs know exactly in what phase of the moon "tops - (fruits)" and in what - "roots" collect.

Have you noticed that certain periods of time makes your life inexplicable leaps and bounds? Suddenly, from where you do not take - hit by the edge of the emotions. Increased sensitivity, which can suddenly change full apathy. Creative and barren days, happy and unhappy moments, sharp jumps in mood. It is noted that the possibilities of the human body change regularly. This knowledge is based on the "theory of three biorhythms".

Physical biorhythm - regulates physical activity. In the first half of the physical cycle, the person is energetic and gets the best results in his activities (the second half - the energy is inferior to the bond).

Emotional rhythm - sensitivity increases during activity, mood improves. A person becomes susceptible to various external disasters. When he's in a good mood, he builds castles in the air, dreams, dreams of falling in love and falling in love. With a decrease in emotional biorhythm, there is a decrease in soul forces, a desire disappears, joyful mood.

Intellectual biorhythm. - it has memory, the ability to learn, logical thinking. In the activity phase, the increase is observed, and in the second phase, the decrease in creative activity is not happiness and success.

Theory of three rhythms:

· Physical cycle -23 day. Determines energy, strength, endurance, coordination of movements

· Emotional cycle - 28 days. Condition nervous system and mood

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

Trigonometry is found in nature. Movement of fish in water occurs according to the law of sine or cosine, if you fix the point on the tail and then consider the trajectory of the movement. When swimming, the body of the fish takes the shape of the curve, which is similar to the graph of the function y \ u003d TGX.

When the bird flies, the flight path of the wings of the wings forms a sine wave.

Trigonometry in Medicine.As a result of a study conducted by the University of Iranian University, Shiraz Vakhid-Cut Abbashi, doctors first have the option of rationalizing information about the electrical activity of the heart, or in other words, electrocardiography.

The formula, called Tehran, was represented by the broad scientific community at the 14th Conference on Geographic Medicine and then at the 28th Conference on the Use of Computing in Cardiology in the Netherlands.

This formula is a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 of the main parameters, including several additional calculations in cases of arrhythmias. According to the doctors, this formula facilitates the process of describing the basic parameters of the heart, thereby accelerating the formulation of the diagnosis and the start of treatment itself.

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

Trigonometry helps our brain determine distances to objects. American scientists claim that the brain estimates the distance to objects, measures the angle between the plane of the earth and the plane of view. Such a conclusion came after a series of experiments whose participants were asked to look at the environment through the prisms that raise this angle.

Such a bias resulted in experimental carriers being recommended as closer than closer than closer and unable to cope with the simplest of tests. Some of the participants in the experiments even lean forward, trying to orient their bodies perpendicular to the false surface of the earth. However, after 20 minutes after the incident, they are used to distorted perception and all problems disappeared. This 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, for some time the opposite effect that was observed - reassessment of the distance.

The results of the new research, as it can be assumed, is not essential for engineers who create navigation systems for robots, as well as specialists who work to create the most realistic virtual models. Applications in the medical field are possible with the rehabilitation of patients with damage to certain areas of the brain.

2.5. Music.

The musical activity also interacts with trigonometry.

I present some interesting information to your attention about a particular method that precisely links trigonometry to music.

This method of analyzing musical works has been known as "geometric music theory". With its help, the most important musical structures and transformations are transferred into the language of modern geometry.

Each note within new theory It is represented as the logarithm of the frequency of the corresponding sound (note "to" of the first octave, for example corresponds to the number 60, octave number 12). The chord is thus presented as a point with specified 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 well-known types of musical transformations that were previously not considered in music theory in the classification of sound sequences - an octave permutation (O), permutation (P), transposition (T), inversion ( I) and modification of the calculation (c). All these transformations form, as the authors write, the so-called optic symmetry in n-dimensional space and store musical information about the chord - in which octave your notes are, in which order they are reproduced, how often they are repeated and so on. With the help of optical symmetries, similar but not identical chords and their sequences are classified.

The authors of the article show that different combinations of these 5 symmetries form many different musical structures, some of which are already known in music theory (for example, the order of the chords, are expressed in new terms as OPC) and others are fundamental new concepts, that can be attached to interchangeable components of the future.

As an example, the authors offer a geometric representation of different types of chords of four sounds - tetrahedra. The balls of the diagram represent the types of chords, the colors of the balls correspond to the size of the intervals between the sounds of the chord: blue - small spaces, warmer tones - more "economical" chord sounds. Red ball is the most harmonious chord with equal intervals between notes, which was popular in composers of the XIX century.

The "geometric" method of analyzing music, according to the study's authors, can lead to the creation of fundamentally new musical instruments and new ways to visualize music, as well as changes in modern methods of teaching music and ways to different musical styles (classics , Pop music, rock music, etc.). The new terminology will also help compare the musical works of composers from different eras and present the results of the studies in a more convenient mathematical form. In other words, it is suggested to assign her mathematical essence from the musical works.

Frequencies corresponding to the same note in the first second, etc. Octaves, belong as 1: 2: 4: 8 ... According to the legends that came from ancient times, the first to try to do it was Pythagoras and his students.

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

2.6. IT.

Tried trigonometry with its influence and computer science. Its functions are therefore applicable to precise calculations. Thank you very much. We can bring any (in some sense "good") function that put it out in a Fourier series:

a0 + A1 cos x + b1 sin x + a2 cos 2x + b2 sin 2x + a3 cos 3x + b3 sin 3x + ...

The number selection process is the most suitable number A0, A1, B1, A2, B2, ..., can be in the form of such a (infinite) set to represent almost all functions of the computer with the required accuracy.

Trigonometry has a huge role and development aid in working with graphical information. If you need to simulate the process, with a description in electronic form, with the rotation of a particular object around an axis. It's an angle on. To find the coordinates of the points, the sinuses and cosines need to multiply.

So you can bring the example of Justin Wyndel, a programmer and designer with the Google Grafika Laboratory. He posted a demo showing an example of using trigonometric functions to create dynamic animation.

2.7. Construction and geodesy

An interesting industry that interacts with trigonometry is the field of construction and geodasy. The lengths of the sides and the values ​​of the angles of any triangle on the plane are related by certain relationships, the main of which are called cosine theorems and sines. The formulas containing A, B, C that the letters are represented by the sides of the triangle that lie according to the angles A, B, B, C. These formulas allow three elements of the triangle - the lengths of the parties and the corners - to recreate the other three elements. They are used to solve practical tasks such as geodesy.

All "classical" geodesy is based on trigonometry. Since in ancient times there are "solid" triangles ".

The process of building structures, paths, bridges and other buildings comes from the survey and design work. Without exception, the construction was carried out with the support of geodesic devices such as a tachometer and a trigonometric level. In trigonometric leveling, the height difference between several points on the earth's surface is set.

2.8 Trigonometry in art and architecture

As the person became on earth, the basis for improving life and other areas of life has become science. The fundamentals of anything created by a person are different directions in natural and mathematical sciences. One of them is geometry. Architecture isn't the only area of ​​science that uses trigonometric formulas. Most of the composite decisions and samples of the drawings passed precisely with geometry. But little is known about theoretical data. Consider an example of building a sculpture by the French master of the Golden Century of Art.

The proportional relationship in building the statue was perfect. However, when she picked up a statue on a tall pedestal, it looked ugly. The sculptor did not take into account that in the future many details will decrease on the horizon, and on the bottom of the floor the impressions of his ideality. Lots of calculations were done so that the figure looked proportional with high height. They were mainly based on the vision method, that is, an approximate measurement, eye. However, the difference in the difference in certain proportions made it possible to create a figure close to the ideal. So knowing the approximate distance from the statue to the point of view, namely from the top of the statue to the human eye and the height of the statue, you can calculate the sine angle of the look on the table with the help of a table, where we find a position ( Fig. 9).

In Figure 10, the situation changes, as the statue is raised to the level of the AC and NS, calculate the cosine values ​​of the angle C, we find an angle of the falling angle on the table. In doing so, you can calculate a trigonometric identity as well as the sine corner C, which you can use to check the results using the main pagecos.2 A + SIN.2 a \ u003d 1.

By comparing the measurements of the A in the first and second cases, you can find the coefficient of proportionality. Then we get a drawing, and then a sculpture, if you have recorded what a visual figure is close to the ideal

Cult buildings around the world were designed based on mathematics, which can be considered a genius of architecture. Some famous examples of such buildings: Gaudi children's school in Barcelona, ​​Skyscraul Mary-Ex in London, Bodegas Isos winery in Spain, restaurant in Los Manantiaz in Argentina. When designing these buildings it doesn't cost without trigonometry.

Conclusion

After studying the theoretical and applied aspects of trigonometry, I realized that this industry is closely related to many sciences. In the beginning, trigonometry was required to make and take measurements between the angles. Later, however, the simple measurement of angles turned into full-fledged, fun, educational trigonometric functions. We can identify the following areas where there is a close relationship between trigonometry and the physics of architecture, nature, medicine, biology.

Due to trigonometric functions in medicine, the heart formula was thus opened, which is a complex algebraic-trigonometric equality consisting of 8 expressions, 32 coefficients and 33 of the main parameters, including the possibility of additional mismatches in the occurrence of arrhythmias. This discovery is helping physicians deliver more qualified, quality medical care.

Note too. That all classical geodesy is based on trigonometry. Indeed, from ancient times, GODESSISTS have been active in the fact that "solid" 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 with geodetic tools such as a theodolite and trigonometric level. In trigonometric leveling, the difference in height between several points on the earth's surface is determined.

We can introduce the influence into other areas, we can conclude that trigonometry actively influences human vital activity. Connecting math with the surrounding world enables you to "materialize" the knowledge of school children. Thanks to this, we can adequately perceive and assimilate the knowledge and information that we are taught in school.

The purpose of my project was successful. I studied the effect of trigonometry in life and the development of interest in it.

In order to achieve the goal, we performed the following tasks:

1. I became familiar with the history of formation and the development of trigonometry;

2. Consider examples of the practical implications of trigonometry in different areas of activity;

3. showed the examples, the possibilities of trigonometry and its use in a person's life.

Studying the history of the occurrence of this industry will help interest interest among school children to form a true worldview and increase the overall culture of the high school student.

This work will be useful for high school students who have not yet seen all of the beauty of trigonometry and are unfamiliar with the areas of use in the area.

Reference list

    Glaser g.i.

    Glaser g.i.

    Rybnikov Ka.

Reference list

    A. Kolmogorov, A.m. Abramov, Yu.p. Duditin and others. "Algebra and Beginning of Analysis" tutorial for 10-11 classes of general education, M., Enlightenment, 2013.

    Glaser g.i.History of Mathematics in School: vii-viii cl.- M .: Enlightenment, 2012.

    Glaser g.i.History of Mathematics in School: IX-X CL. - M .: Enlightenment, 2013.

    Rybnikov Ka.History of Mathematics: Tutorial. - M .: Publishing house of the Moscow State University, 1994. Ololand exercises for algebra, trigonometry and elementary functions / Åland, S.N. and. - M .: secondary school, 2016. - 134 C.

    Ololand, s.n. Exercises on algebra, trigonometry and elementary functions / S.N. Olochnik. - M .: High school, 2013. - 645 c.

    Potapov, M.K. Algebra, Trigonometry and Elementary Functions / M.K. Potapov. - M .: High school, 2014. - 586 c.

    Potapov, M.K. Algebra. Trigonometry and Elementary Functions / M.K. Potapov, v.v. Alexandrov, P.i. Pasichenko. - M .: [Not specified], 2015. - 762 c.

Annex 1

FigImage of the pyramid. Calculation of the slope.b./ H.

Corent Semic.

In general, the Egyptian formula for calculating the seed pyramid looks like

so:.

Ancient Egyptian Term " hald"Mix down the tilt angle. It was by the height divided by half the base.

"The length of the pyramid on the east side is 360 (elbows), the height is 250 (elbows). It is necessary to calculate the slope of the east side. To do this, take half of 360, that is, 180 parts You 180 on. To 250. You get:1 / 2 , 1 / 5 , 1 / 50 Elbow. Note that one elbow is 7 palm widths. Now multiply the numbers obtained on 7 as follows: "

Fig 2. Gnomon

Fig. 3 Determination of the angular height of the sun

Fig.4 The main formulas of trigonometry

Fig. 5 Navigation trigonometry.

Fig.6 Physics in trigonometry

Fig. 7 Theory of three rhythms

(Physical cycle -23 day. Determines energy, strength, endurance, coordination of movements; Emotional cycle - 28 days. State of the nervous system and mood; Intellectual cycle - 33 days. Determines the creative ability of the personality)

Cowardly. 8 Trigonometry in Music

Fig. 9, 10.Trigonometry in Architecture.

align \ u003d Center \ u003e

trigonometry - Michematician Micro-Site, in which the dependencies between the values ​​of the angles and the lengths of the sides of the triangles are examined, as well as algebraic identities of trigonometric functions.
There are many areas in which trigonometric and trigonometric functions are used. Trigonometry or trigonometric functions are used in astronomy, sea and air navigation, acoustics, optics, electronics, architecture and other areas.

The history of the creation of trigonometry

The history of trigonometry, as the science of the relationships between the angles and the sides of the triangle and other geometrical figures, covers more than two thousand years. Most of these relationships cannot be expressed using conventional algebraic operations, and so it was necessary to introduce special trigonometric functions that were originally created in the form of numeric tables.
Historians believe that trigonometry created ancient astronomers, it began to be used in architecture a little later. Over time, the scope of trigonometry has steadily expanded, today it includes almost everything science, equipment, and a number of other areas of activity.

Early century

From Babylonian mathematics the customary measurement of the angles of degrees, minutes and seconds (the introduction of these units into ancient Greek mathematics usually usually on, II century BC) becomes.

The main achievement of this period was the ratio of cathets and hypotenuses in a rectangular triangle, later received the name of the Pytagora theorem.

Ancient Greece

The general and logically related explanation of trigonometric ratios appeared in ancient Greek geometry. The Greek mathematicians did not isolate trigonometry as a separate science, for them it was part of astronomy.
The main achievement of an ancient trigonometric theory was the solution in the overall form of the problem of "solutions of triangles", that is, that is, unknown elements of the triangle based on the three elements that its elements contain is the party).
Applied trigonometric tasks are distinguished by a great variety, for example, the results of actions above the listed values ​​(for example, the sum of the angles or the ratio of the lengths of the sides) can be set in practice.
In parallel with the development of the trigonometry of the Greek airplane under the influence of astronomy, spherical trigonometry was well advanced. In the "beginning of" Euclida on this subject there is only the sentence in the treatment of spheres of different diameters, but the needs of astronomy and cartography caused rapid development of spherical trigonometry and related regions - systems of celestial coordinates, theory of cartographic projections, technology astronomical instruments.

middle Ages

In the 4th century, after the death of ancient science, the Mathematics Development Center moved to India. They changed concept separator a bit, brought it up to modernity: for example, they were first introduced to the use of cosines.

The first specialized treatise on trigonometry was an essay by the central Asian scientist (X-XI centuries) "Book of Keys to Astronomy" (995-996). The entire course of trigonometry contained the main work of Al-Biruni - "Canon Massday" (Book III). In addition to the sine tables (with 15 "increments), Al-Biruni produced tangent tables (with 1 ° fractions).

After Arabic treatises were translated into Latin in the XII-XIII centuries, many ideas of Indian and Persian mathematicians were in the presence of European science. Apparently, the first acquaintance of Europeans with trigonometry took place due to Zija, of which the two translations were fulfilled in the XII century.

The first European compositional composition devoted entirely to trigonometry is often referred to as the "four treatises on direct and addressed chords" by the English astronomer Richard Wallingford (c. 1320). Trigonometric tables, more often translated from Arabic but sometimes original, are included in the writings of a number of other authors of the XIV-XV centuries. Then trigonometry took place among the university courses.

New time

The development of trigonometry in the new age is extremely important not only for astronomy and astrology, but also for other applications, primarily artillery, optics and navigation with distant seabes. Therefore, after the sixteenth century, many outstanding scientists were engaged in this topic, including Nikolai Copernicus, Johann Kepler, Francois Viet. Copernicus devoted two chapters to trigonometry in his treatise "on the rotation of the celestial spheres" (1543). Soon (1551) there were 15-digit trigonometric tables from Retik, the pupil of Copernicus. Kepler published the work "Optical part of astronomy" (1604).

Wits in the first part of his "mathematical canon" (1579) placed a multitude of tables, including trigonometric, and in the second part gave a detailed and systematic, if without evidence, the presentation of flat and spherical trigonometry. In 1593 Viet prepared an extended publication of this capital work.
Thanks to the works of Albrecht Dürer, a sinusoid appeared in the world.

XVIII century

The modern type of trigonometry existed. In the treatise "Introduction to the Analysis of the Infinite" (1748) Euler made the definition of trigonometric functions that correspond to the modern one, and accordingly identified the inverse functions.

Euler considered allowable negative angles and angles, large 360 ​​°, which made it possible to identify trigonometric functions on an entire real numerical line, and then continue them into a complex plane. When the question of the distribution of trigonometric functions on stupid angles arose, the signs of these functions were often chosen in error up to Euler; For example, many mathematicians thought cosine and a dumb angle with tangent to be positive. Euler defined these signs for angles in different coordinate quadrants, based on the equation formulas.
The general theory of trigonometric series Euler was not indented, and the convergence of the received series did not investigate, but received several important results. In particular, he brought about the decomposition of the total degrees of sine and cosine.

Application of trigonometry.

Those who say trigonometry is not needed in real life. Now what are their usual application tasks? Measure the distance between inaccessible objects.
The technique of triangulation is of great importance, whereby distances to the nearby stars in astronomy between benchmarks in geography, satellite navigation systems can control. There should also be the use of trigonometry in areas such as navigational engineering, music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound research (ultrasound) and computed tomography), pharmaceuticals, chemistry, the theory of numbers (and as Result, cryptography), seismology, meteorology, oceanology, cartography, many sections of physics, topography, architecture, phonetics, economics, electronic devices, mechanical engineering, computer graphics, crystallography, etc.
Output: Trigonometry is a great assistant in our daily life.

    Trigonometry in Astronomy:

    The need to solve triangles was previously discovered in astronomy; Therefore, trigonometry developed for a long time and therefore studied as one of the sections of astronomy.

    Compiled from the hypoche table of the positions of the sun and moon, it allowed the moments of the eclipse to occur (with an error of 1-2 hours). The Hypoche first began to use spherical trigonometry methods in astronomy. It increased the accuracy of the observations, as a result of which the cross of the threads on the luster appeared on the lights - sestructural efforts and quadrants. At that time, the scientist took a large catalog of the determinations of 850 stars and divided them into 6 degrees (star values) in gloss. Hipparchus introduced geographic coordinates - latitude and longitude, and it can be considered the founder of mathematical geography. (Ok 190 BC E. - OK. 120 BC He)


    Complete solution of the problem of determining all elements of flat or spherical triangles in three dates, important decomposition of Sin BC and COS PSI into COS X and SINX degrees. Knowledge of the formula of sines and cosines of several arcs made it possible to solve the 45th degree equation proposed by mathematics A. Rosome; Vieta showed that solving this equation is reduced to separating one angle in 45 equal parts and that there are 23 positive roots of this equation. Viet decided Apollonia's problem with a ruler and circulation.
    Solving spherical triangles is one of the tasks of astronomy to compute the parties and angles of a spherical triangle in three appropriately determined parties or corners that allow theorems: (the sine theorem) (cosine theorem for angles) (cosine theorem for the parties)).

    Trigonometry in Physics:

    types of vibration phenomena.

    Harmonic oscillation is a phenomenon of periodic changes in any value in which the dependence on the argument has the character of the sine or cosine function. For example, the value varies over time, harmonized as follows:

    Where X is the value of a variable value, T time, A - the amplitude of vibrations, ω is the cyclic frequency of vibrations, the total phase of vibrations, R is the initial phase of vibrations.

    Mechanical vibrations . Mechanical vibrations

    Trigonometry in nature.

    We often ask a question

  • One of basic properties
  • - These are more or less regular changes in the nature and intensity of biological processes.
  • Basic Earth Rhythm. - Every day.

Trigonometry in Biology.

  • Trigonometry plays an important role in medicine. With his help, Iranian scientists discovered the formula of the heart - a complex algebraic-trigonometric equation, consisting of 8 expressions, 32 coefficients and 33 of the main parameters, including several additional calculations in cases of arrhythmias.
  • diatonic gamma 2: 3: 5

Trigonometry in Architecture.

  • Swiss Re Insurance Corporation in London
  1. interpretation

We have only brought a small part of it, on which you can perform trigonometric functions. We found out

We have proven that trigonometry is closely related to physics, nature, medicine. You can continuously bring many examples of periodic processes of living and unnecessary nature. All periodic processes can be described with trigonometric functions and displayed for time schedules.

We believe that trigonometry was reflected in our life and the sphere,

in which it plays an important role, it will expand.

  • Found outThat trigonometry was brought to life with the need to measure corners, but evolved over time into the science of trigonometric functions.
  • Proven
  • Thought

Look at the contents of the document
"Danilova t.v. scenario"

MKOU "NENETS Community Secondary School - Boarding School. A.p. pipherki"

Training project

" "

Danilova Tatyana Wladimirovna.

Math teacher

    Justification of the relevance of the project.

Trigonometry is a section of mathematics that studies trigonometric functions. It's hard to imagine, but with this science we stand not only in the teachings of mathematics but also in our daily lives. You could not have guessed it, but the trigonometry is in the sciences such as physics, biology, it also plays in medicine, and that the most interesting, without it, does not cost in music and architecture.
The word trigonometry appears for the first time in 1505 in the title book of German mathematics by Pitiscus.
Trigonometry - Greek word and literally means to measure triangles (Trigonan - triangle, Metreo - measure).
The appearance of trigonometry was closely related to land, astronomy, and the construction business. ...

The student in 14-15 years doesn't always know where to go next? Learn and where will work.
For some professions, his knowledge is necessary because Allows you to measure distances to nearby stars in astronomy, between benchmarks in geography, control satellite navigation systems. Principles of trigonometry, in areas such as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound testing (ultrasound) and computed tomography), pharmaceuticals, chemistry, number theory (and, as a consequence, cryptography), seismology , Meteorology, oceanology, cartography, many sections of physics, topography and geodesy, architecture, phonetics, economics, electronic equipment, mechanical engineering, computer graphics, crystallography.

    Determination of the research object

3. Project goals.

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

    hypothesis

    Check hypothesis.

trigonometry (from Greek.trigonon. - triangle,metro. - Metry) -

History of trigonometry:

Ancient people calculated the height of the tree by comparing the length of the shadow to the shadow length of the pole whose height was known. The stars calculated the position of the ship to the sea.

The next step in the development of trigonometry was taken by the Indians in the period from V to XII. Century made.

The concept of the cosine itself appeared for the first time at the end of the XVI V. In the so-called "Supplement Sinus", plus Sinus angle that supplements this angle to 90 °. Sinus food supplement or (Latin) Sinus-Sinus-PleaseVenti is reduced to absorb as Sinus CO or CO -Sinus.

In the XVII-XIX centuries. Trigonometry becomes one of the brains of mathematical analysis.

It is widely used in mechanics, physics and technology, especially in the study of vibrational movements and other periodic processes.

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

to the system of mathematical analysis.

Where trigonometry is used

Trigonometric calculations are used in almost all spheres of human vital activity. Applications in such areas should be noted: astronomy, physics, nature, biology, music, medicine and many others.

Trigonometry in Astronomy:

The need to solve triangles was previously discovered in astronomy; Therefore, trigonometry developed for a long time and therefore studied as one of the sections of astronomy.

The need to solve triangles was previously discovered in astronomy; Therefore, trigonometry developed for a long time and therefore studied as one of the sections of astronomy.

Achievements of Vieta in trigonometry
Complete solution of the problem of determining all elements of flat or spherical triangles in three dates, important decomposition of Sin BC and COS PSI into COS X and SINX degrees. Knowledge of the formula of sines and cosines of several arcs made it possible to solve the 45th degree equation proposed by mathematics A. Rosome; Vieta showed that solving this equation is reduced to separating one angle in 45 equal parts and that there are 23 positive roots of this equation. Viet decided Apollonia's problem with a ruler and circulation.
Solving spherical triangles is one of the tasks of astronomy to calculate the parties and angles of a spherical triangle in three correspondingly determined parties or corners).

Trigonometry in Physics:

In the world around us it is necessary to deal with periodic processes that are repeated at the same intervals. These processes are called oscillatory. The oscillatory phenomena of various physical nature obey general patterns and are described by the same equations. There are different types of vibration phenomena.

Harmonic oscillation. - the phenomenon of periodic changes in any value in which the dependence on the argument has the character of a sine or cosine function. For example, the value varies over time, harmonized as follows:

Where X is the value of a variable value, T time, A - the amplitude of vibrations, ω is the cyclic frequency of vibrations, the total phase of vibrations, R is the initial phase of vibrations.

Generalized harmonic oscillation in the differential form x '' + ω²x \ u003d 0.

Mechanical vibrations . Mechanical vibrationscalled the movements of bodies that are repeated at exactly the same time intervals. Graphic image This function gives a visual idea of ​​the flow of the oscillatory process in time. Examples of simple mechanical oscillatory systems can serve as a spring or mathematical pendulum.

Trigonometry in nature.

We often ask a question "Why do we sometimes see what is really not?". The following questions are suggested for research: "How do rainbows form? Northern lights?", "What is optical illusion?" "How can trigonometry help to find answers to these questions?"

For the first time, the theory of the rainbow was given to René Descartes in 1637. He explained the rainbow as a phenomenon associated with the reflection and refraction of light in the rainpalls.

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

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

    American scientists claim that the brain estimates the distance to objects, measures the angle between the plane of the earth and the plane of view.

    In addition, biology uses such a concept as Sinus Sleepy Sinus, Carotid Sinus, and Venous or Cavernous Sinus.

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

    One of basic properties Wildlife is the cycle of most of the processes that occur in it.

    Biological rhythms, biorhythms

    Basic Earth Rhythm. - Every day.

    The model of biorhythms can be created with trigonometric functions.

Trigonometry in Biology.

What biological processes are associated with trigonometry?

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

    Biological rhythms, biorhythms are associated with trigonometry

    The model of biorhythms can be created with trigonometric functions. To do this, you need to enter a person's date of birth (day, month, year) and the duration of the forecast

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

The emergence of musical harmony

    According to the legends that came from ancient times, the first to try to do this was Pythagoras and his disciples.

    Frequencies corresponding to the same note in the first second, etc. Octaves, belong as 1: 2: 4: 8 ...

    diatonic gamma 2: 3: 5

Trigonometry in Architecture.

    Gaudi children's school in Barcelona

    Swiss Re Insurance Corporation in London

    Felix Candela Restaurant in Los Manantiales

    interpretation

We have only brought a small part of it, on which you can perform trigonometric functions. We found that trigonometry was brought to life with the need to measure the corners, but evolved over time into the science of trigonometric functions.

We have proven that trigonometry is closely related to physics, nature, medicine. You can continuously bring many examples of periodic processes of living and unnecessary nature. All periodic processes can be described with trigonometric functions and displayed for time schedules.

We believe that trigonometry was reflected in our life and the sphere,

in which it plays an important role, it will expand.

    Found outThat trigonometry was brought to life with the need to measure corners, but evolved over time into the science of trigonometric functions.

    ProvenThis trigonometry is closely related to physics, occurs in nature, music, astronomy and medicine.

    ThoughtThis trigonometry was reflected in our lives, and the spheres in which it plays an important role will expand.

7. Literature.

    MAPLE6 program that implements graphs

    Wikipedia.

    Study.ru.

    Math.ru "library"

Look at the content of the presentation
"Danilova t.v."

" Trigonometry in the world around us and human life "



Research goals:

Communication trigonometry with real life.


Problematic question. 1. What are the concepts of trigonometry most common 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 physical phenomena of nature, physiological processes, patterns in music and art can be described with trigonometry and trigonometric functions.


What is trigonometry ???

trigonometry(from Greek. Trigonon - triangle, Metro - Metron) - Mathematics microgenic, in which the dependencies between the values ​​of the angles and the lengths of the parties of the triangles are examined, as well as algebraic identities of trigonometric functions.



History of trigonometry.

The origins of trigonometry date back to ancient Egypt, Babylonia and the Indus Valley more than 3000 years ago.

The word separator is first found in 1505 in the title of the book of German mathematics by Pitiscus.

For the first time the solutions of triangles, based on dependencies between the parties and the triangle angles, were found by the ancient Greek astronomers of Hippuch and Ptolemy.

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

The stars calculated the position of the ship to the sea.


The next step in the development of trigonometry was taken by the Indians in the period from V to XII. Century made.

IN THEdiffered from the Greeks by Indnytsy.began to consider and use in calculations, no longer chord mmof the corresponding central angle, but only half of the MR, i.e. H. Sinus- half central angle.

The term cosine itself first appeared at the end of the XVI much later in the works of European scientists. « sinus supplements» . Sine angle that complements this angle to 90. « Sinus supplements» or (in Latin) Sinus-Sinus-Nunpradeventi began to record as Sinus CO or Co-Sinus.

Along with Sinus, the Indians were introduced to trigonometrycosineMore precisely, they used the cosine line in their calculations. They also known COS ratios\ u003d sin (90)- ) and sin.2 + Cos.2 \ u003d R.2 as well as the formula for sine quantities and differences between two angles.


In the XVII-XIX centuries. Trigonometry will.

one of the heads of mathematical analysis.

It finds a good use in the mechanic,

physics and technology, especially when studying

oscillatory movements and others

regular processes.

The properties of the frequency of trigonometric functions knew more, with the earliest mathematical studies addressed in trigonometry.

Proved that every periodic

movement can be

represented (with any degree

accuracy) in the form of the ordinary amount

harmonic vibrations.


founderanalytically

theory

trigonometric.functions.

Leonard Euler.

In the "Introduction to the Analysis of Infinity" (1748 g)

truck sine, cosine, etc. not like

trigonometric lines necessarily

connected with a circle and how

trigonometric functions he

regarded as the relationship of the parties

rectangular triangle like numeric

values.

Excluded from his formulas

R is a whole sine, takes

R \ u003d 1 and simplifies such

recording and calculations.

Development of a teaching

on trigonometric functions

any argument.


In the XIX century it was continued

development of theory

trigonometric.

functions.

N.i. Lobachevsky.

"Geometric considerations", writes Lobachevsky, are necessary up to the beginning of trigonometry, as long as they do not serve as distinguishing properties of trigonometric functions ... From here on, trigonometry is made completely independent of geometry and has all the analytical advantages. "


Trigonometry development stages:

  • Trigonometry came to life with the need to measure angles.
  • The first steps in trigonometry was to establish connections between the angle value and the posture of the specially constructed segments of the straight lines. The result is the ability to solve flat triangles.
  • The need to publish the values ​​of the managed trigonometric functions.
  • Trigonometric functions have been converted into independent research objects.
  • In the XVIII century trigonometric functions were added

to the system of mathematical analysis.


Where trigonometry is used

Trigonometric calculations are used in almost all spheres of human vital activity. Applications in such areas should be noted: astronomy, physics, nature, biology, music, medicine and many others.


Trigonometry in Astronomy.

The need to solve triangles was previously discovered in astronomy; Therefore, trigonometry developed for a long time and therefore studied as one of the sections of astronomy.

Significant heights reached trigonometry and Indian medieval astronomers.

The main goal of the Indian astronomers was the replacement of the chord

sinus, which made it possible to connect various functions

with the sides and corners of the rectangular triangle.

This is how trigonometry began in India

as lessons on trigonometric values.


Compiled from the hypoche table of the positions of the sun and moon, it allowed the moments of the eclipse to occur (with an error of 1-2 hours). The Hypoche first began to use spherical trigonometry methods in astronomy. It increased the accuracy of the observations, as a result of which the cross of the threads on the luster appeared on the lights - sestructural efforts and quadrants. At that time, the scientist took a large catalog of the determinations of 850 stars and divided them into 6 degrees (star values) in gloss. Hipparchus introduced geographic coordinates - latitude and longitude, and it can be considered the founder of mathematical geography. (Ok 190 BC E. - OK. 120 BC He)

Hipparchus



Trigonometry in Physics.

In the world around us it is necessary to deal with periodic processes that are repeated at the same intervals. These processes are called oscillatory. The oscillatory phenomena of various physical nature obey general patterns and are described by the same equations. There are different types of oscillatory phenomena, for example:

Mechanical vibrations

Harmonic oscillations.


Harmonic oscillations.

Harmonic oscillation. - the phenomenon of periodic changes in any value in which the dependence on the argument has the character of a sine or cosine function. For example, the value varies over time, harmonized as follows:

or

Where X is the value of a variable value, T time, A - the amplitude of vibrations, ω is the cyclic frequency of vibrations, the total phase of vibrations, R is the initial phase of vibrations.

Generalized harmonic oscillation in the differential form x '' + ω²x \ u003d 0.


Mechanical vibrations

Mechanical vibrations Called the movements of bodies that are repeated at exactly the same time intervals. The graphic image of this function gives a visual idea of ​​the flow of the oscillatory process in time.

Examples of simple mechanical oscillatory systems can serve as a spring or mathematical pendulum.


Mathematical pendulum.

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