Report trigonometry in real life. Additional applications of trigonometry in life

Trigonometry in medicine

Head: Kozlova Lyudmila Vasilievna

Purpose of work: To study the use of trigonometry in medicine. After the work done, I studied the use of trigonometry in medicine: compiling human biorhythms, cardiology. It provides the basis for compiling formulas of human organs, which will subsequently help to treat any diseases. This work tells in which areas of medicine the knowledge of trigonometry is applied. Thanks to this work, I found out the basic principles of reading an electrocardiogram and I can independently distinguish between a normal examination result and bright deviations.

INTRODUCTION

Relevance: For the first time I encountered trigonometry in the eighth grade, when we began to study the basics of this section of mathematics. The simplest rules for determining sine and cosine seemed to me very easy, so they did not arouse much interest. Later, when I started studying in the tenth grade, it was immediately clear that trigonometry is a huge branch of mathematics, combining a large amount of knowledge and theory. Later I found out that the knowledge of trigonometry is very universal for all fields of activity. They are widely used in astronomy, geography, music theory, financial market analysis, electronics, probability theory, statistics, biology, medicine, pharmaceuticals, chemistry, cryptography and many others.

Trigonometry (from the Greek τρίγωνον (triangle) and Greek μέτρεο (measure), that is, the measurement of triangles) is a branch of mathematics that studies trigonometric functions and their use in geometry.

The term "trigonometry" was introduced in 1595 by the German mathematician and theologian Bartholomew Pitisk, the author of a textbook on trigonometry and trigonometric tables. By the end of the 16th century most of the trigonometric functions were already known, although the concept itself did not yet exist.

Scientists processed measurement data in order to keep a calendar and correctly determine the start time of sowing and harvesting, the dates of religious holidays. The stars were used to calculate the location of a ship at sea or the direction of a caravan in the desert. As you know, trigonometry is used not only in mathematics, but also in other areas of science. This work tells in which areas of medicine knowledge of geometry is applied.

One of the main applications is cardiology. ECG devices take a cardiogram from people, fixing heart beats. After speaking with an electrocardiogram reader, I found thatthe graph is a modified sinusoid. And here every unevenness of the schedule is important. The number of intervals and teeth, the maximum and minimum jumps, the length of the periods: all this plays an important role in determining the diagnosis and the correctness of the treatment.

MAIN CONTENT

OBJECTIVE: To study the use of trigonometry in medicine.

TASKS:

Learn the history of trigonometry.

Find out in which areas of medicine trigonometry is used.

Perform the practical part of the work, find out the principle on which cardiologists rely when reading the electrocardiogram graph.

1.2.HISTORY

The first trigonometric tables were apparently compiled by Hipparchus, who is now known as the "father of trigonometry".

Ancient Greek mathematicians in their constructions related to the measurement of arcs of a circle used the technique of chords. The perpendicular to the chord, dropped from the center of the circle, bisects the arc and the chord resting on it. Half a bisected chord is the sine of half an angle, and so the sine function is also known as "half a chord". To compensate for the lack of a table of chords in mathematics, from the time of Aristarchus, a well-known theorem was sometimes used, in modern notation -

where 0°< β < α < 90°,

The first trigonometric tables were probably compiled by Hipparchus of Nicaea (180-125 BC). Hipparchus was the first to tabulate the corresponding magnitudes of arcs and chords for a series of angles. The systematic use of the full circle of 360° was established mainly by Hipparchus.

Later, Claudius Ptolemy (90 - 168 AD) expanded the Hipparchus' "Chords in a circle" in the Almagest. The thirteen books of the Almagest are the most significant trigonometric work of all antiquity. Later, Ptolemy derived the formula for the half angle. Ptolemy used these results to create his trigonometric tables, which have not survived to this day.

The replacement of chords by sines was the main achievement of medieval India. Since the 8th century, scientists from the countries of the Near and Middle East have developed trigonometry. After the treatises of Muslim scholars were translated into Latin, many ideas became the property of European and world science.

2. TRIGONOMETRY IN MEDICINE

2.1.BIORHYTHMS

Biorhythms are periodically repeating changes in the nature and intensity of biological processes and phenomena. They are characteristic of living matter at all levels of its organization, from molecular to the biosphere. Some biological rhythms are relatively independent (heart rate, respiration), others are associated with the adaptation of organisms to geophysical cycles - daily (fluctuations in the intensity of cell division, metabolism).

The man from his birthday is in three, biorhythms: physical, emotional and intellectual.

The physical cycle is 23 days. It determines the energy of a person, his strength, endurance, coordination of movement.

The emotional cycle (28 days) determines the state of the nervous system and mood.

The intellectual cycle (33 days) determines the creative ability of the individual.

Any of the cycles consists of two half-cycles, positive and negative.

During the first half of the physical cycle, a person is energetic and achieves better results in his activities; in the second half of the cycle, energy yields to laziness.

In the first half of the emotional cycle, a person is cheerful, aggressive, optimistic, overestimates his capabilities, in the second half he is irritable, easily excitable, underestimates his capabilities, pessimistic, and critically analyzes everything.

Fig.1. Biorhythms

The model of biorhythms is built using graphs of trigonometric functions. There are a huge number of sites on the Internet that are engaged in the calculation of biorhythms. To do this, you must enter the person's date of birth (day, month, year) and the duration of the forecast.

2.2. HEART FORMULA

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

The formula, called Tehran,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, speeding up the diagnosis and initiation of treatment..

At the moment, the exact information on the issue is not known, active work and research on this topic is underway.

Russian scientists have developed a mathematical formula for the heart. Thanks to these equations, any heart disease can be calculated, predicted and prevented. The only laboratory of mathematical physiology in Russia operates at the Yekaterinburg Institute of Immunology and Physiology.

The problem of mathematical descriptions of the physiological functions of the body is the second most important problem after the problem of human DNA. In the future, the formulas of other human organs will be calculated, and doctors using elementary equations will be able to predict and treat any disease.

Man is a complex mechanism in which physical and chemical processes are continuously taking place. If all processes are translated into the language of equations, then it will be possible to derive a single formula for a person.

Mathematicians have created a model of the heart muscle, which biologists have virtually connected with real living tissue. In a computer program, scientists put different loads on the heart and observe how it behaves. By studying all sorts of algorithms that mimic the activity of the heart, scientists will be able to make real predictions.

2. 3. ELECTROCARDIOGRAM

Applied for practical purposes in the 70s of the 19th century by the Englishman A. Waller, an apparatus that records the electrical activity of the heart continues to serve a person to this day. An electrocardiograph can detect obvious deviations from the normal heart rhythm, such as myocardial infarction, ischemic heart disease, sinus bradycardia, tachycardia, arrhythmia, sick sinus syndrome, etc. How to distinguish normal ECG images from pronounced diseases?

3. PRACTICAL PART OF THE WORK

After I was able to talk to the cardiogram interpretation specialist at our hospital, I learned a lot of useful information for my research work.

The electrocardiogram graph is a modified sinusoid. And here every unevenness of the schedule is important. The number of intervals and teeth, the maximum and minimum jumps, the length of the periods: all this plays an important role in determining the diagnosis and the correctness of the treatment. Therefore, the ECG graph is always printed on graph paper.

When deciphering the results of the ECG, the duration of the intervals between its components is measured. This calculation is necessary to assess the frequency of the rhythm, where the shape and size of the teeth in different leads will be an indicator of the nature of the rhythm, the electrical phenomena occurring in the heart and the electrical activity of individual sections of the myocardium, that is, the electrocardiogram shows how our heart works in a given period.

A more rigorous interpretation of the ECG is carried out by analyzing and calculating the area of \u200b\u200bthe teeth using special leads, however, in practice, they manage with an indicator of the direction of the electrical axis, which is a total vector.

There are different ways to decipher the ECG. Some specialists rely on formulas and calculate everything according to them; so the heart rate can be calculated by the formula: whereR- Rthe duration of the interval, and some use ready-made data, which also does not prohibit domestic medicine. Figure 2 shows the results of heart rate calculations depending on the interval.

Fig.2

Fig.2. Estimated NPV

Fig.3. Types of cardiograms

Figure 3 shows three types of cardiogram. The first cardiogram of a healthy person, the second, of the same person, only with sinus tachycardia, after exercise, and the third cardiogram of a sick person with sinus arrhythmia.

OUTPUT:

After the work done, I studied the use of trigonometry in medicine: compiling human biorhythms, cardiology. It provides the basis for compiling formulas of human organs, which will subsequently help to treat any diseases. Thanks to this work, I found out the basic principles of reading an electrocardiogram and I can independently distinguish between a normal examination result and bright deviations.

REFERENCES

Electrocardiography: Textbook. allowance. -5th edition. - M.: MEDpress-inform, 2001. - 312 p., ill.

Internet sources: Anatomy of the coronary valve / Prof. Dr. med. Sciences Yu.P. Ostrovsky

Repeat the basic formulas of trigonometry and consolidate their knowledge during the exercises;
Develop self-control skills, the ability to work with a computer presentation.
Education of a responsible attitude to educational work, will and perseverance to achieve the final results.

Equipment: Computers, computer presentation.

Expected Result:

Each student should know trigonometry formulas and be able to apply them to transform trigonometric expressions at the level of required results.
Know the derivation of these formulas and be able to apply them to convert trigonometric expressions.
Know the formulas of trigonometry, be able to derive these formulas and apply them to more complex trigonometric expressions.

The main stages of the lesson:

The message of the topic, purpose, objectives of the lesson and the motivation of educational activities.
Verbal counting
Message from the history of mathematics
Repetition (from grade 9) of trigonometry formulas using a computer presentation
Applying trigonometric formulas to converting expressions
Test execution
Summing up the lesson
Setting a task at home

During the classes

I. Organizing time.

Reporting the topic, goals, objectives of the lesson and motivation for learning activities

II. Oral work (tasks are pre-printed for each student):

The radian measure of two angles of a triangle is and . Find the measure of each angle of the triangle. Answer: 60, 30, 90

Find the radian measure of the angles of a triangle if their ratio is 2:3:4. Answer: , ,

Can the cosine be equal to: a), b), c), d), e) -2? Answer: a) yes; b) no; c) no; d) yes; e) yes.

Can the sine be equal to: a) -3, 7 b), c)? Answer: a) no; b) yes; c) no.

For what values of a and b are the following equalities true: a) cos x = ; b) sin x=; c) cosx= ; d) tg x= ; e) sin x = a? Answer: a) /a/ 7; b) /a/ ; c) 0 d) b – any number; e) -

III. Message from the history of trigonometry (brief historical background):

Trigonometry arose and developed in antiquity as one of the branches of astronomy, as its computing apparatus that meets the practical needs of man.

Some trigonometric information was known to the ancient Babylonians and Egyptians, but the foundations of this science were laid in ancient Greece.

Greek astronomer Hipparchus in the 2nd century. BC e. compiled a table of numerical values of chords, depending on the magnitude of the arcs contracted by them. More complete information from trigonometry is contained in the famous "Almagest" of Ptolemy. The calculations made allowed Ptolemy to compile a table that contained chords from 0 to 180.

The names of the sine and cosine lines were first introduced by Indian scientists. They also compiled the first tables of sines, although less accurate than the Ptolemaic ones.

In India, in essence, the doctrine of trigonometric quantities begins, later called goniometry (from “gonia” - angle and “metrio” - I measure).

On the threshold of the 17th century in the development of trigonometry, a new direction begins - analytical.

Trigonometry provides the necessary method for the development of many concepts and methods for solving real problems that arise in physics, mechanics, astronomy, geodosy, cartography and other sciences. In addition, trigonometry is a great help in solving stereometric problems.

IV. Work on computers with a presentation:

“Basic formulas of trigonometry” (Appendix 1)

Pre-remind safety precautions in the computer science classroom.

Basic trigonometric identities.
Addition formulas.
Cast formulas
Formulas for the sum and difference of sines (cosines).
Double argument formulas.
Half argument formulas.

V. Application of trigonometric formulas to the transformation of expressions.

a) One student completes the task on the back of the board, the rest from the place check and raise the signal cards (correct - “+”, incorrect - “-“) from the place.

Choose an answer.

Simplify the expression 7 cos - 5.

a) 1+cos; b) 2; at 12; d) 12

Simplify expression 5 – 4 si n

a) 1; b) 9; c) 1+8sin; d) 1+cos.

study, the beginning of which resembles a small wave, after which there is a systolic rise. A small wave usually shows atrial contraction. The beginning of the rise coincides with the beginning of the ejection of blood into the aorta. On the same tape, you can see another maximum peak, which signals the closing of the semilunar valves. The shape of this segment of the maximum rise can be quite diverse, which leads to different results of this study. After the maximum rise follows the descent of the curve, which continues to the very end. This segment of the apical cardiogram is accompanied by the opening of the mitral valve. After that, a slight rise in the wave. It indicates the fast filling time. The rest of the curve is referred to as passive ventricular filling time. Such a study of the right ventricle can indicate possible pathological abnormalities.

Trigonometry in medicine and biology

Borhythm model 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).

Heart formula. As a result of a study conducted by a student at the Iranian Shiraz University, Wahid-Reza Abbasi, for the first time, doctors were able to streamline information related to the electrical activity of the heart, or, in other words, electrocardiography. The formula is a complex algebraic-trigonometric equation, consisting of 8 expressions, 32 coefficients and 33 main parameters, including several additional ones for calculations in cases of arrhythmia. According to doctors, this formula greatly facilitates the process of describing the main parameters of the activity of the heart, thereby speeding up the diagnosis and the start of the actual treatment.

Trigonometry also helps our brain determine the distances to objects.

1) Trigonometry helps our brain to 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. Even the artists of Ancient China painted 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 the psychologist James

2)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=tg(x)
5. Conclusion

As a result of the research work:

· I got acquainted with the history of trigonometry.

· Systematized methods for solving trigonometric equations.

· Learned about the applications of trigonometry in architecture, biology, medicine.