Trigonometry Chemistry. The connection of trigonometry with real life

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Trigonometry- a microsection of mathematics that studies the relationship between the angles and the lengths of the sides of triangles, as well as the algebraic identities of trigonometric functions.
There are many areas where trigonometry and trigonometric functions are applied. Trigonometry or trigonometric functions are used in astronomy, marine and air navigation, acoustics, optics, electronics, architecture and other fields.

The history of the creation of trigonometry

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

Early centuries

From Babylonian mathematics, we are accustomed to measuring angles in degrees, minutes and seconds (the introduction of these units into ancient Greek mathematics is usually attributed to the 2nd century BC).

The main achievement of this period was the ratio of the legs and hypotenuse in a right triangle, later called the Pythagorean theorem.

Ancient Greece

A general and logically coherent presentation of trigonometric relations appeared in ancient Greek geometry. Greek mathematicians did not yet single out trigonometry as a separate science, for them it was part of astronomy.
The main achievement of the ancient trigonometric theory was the solution in a general form of the problem of "solving triangles", that is, finding the unknown elements of a triangle, based on three of its given elements (of which at least one is a side).
Applied trigonometric problems are very diverse - for example, measurable results of operations on the listed quantities (for example, the sum of angles or the ratio of side lengths) can be set.
In parallel with the development of plane trigonometry, the Greeks, under the influence of astronomy, advanced spherical trigonometry far. In Euclid's "Principles" on this topic, there is only a theorem on the ratio of the volumes of balls of different diameters, but the needs of astronomy and cartography caused the rapid development of spherical trigonometry and related areas - the celestial coordinate system, the theory of cartographic projections, and the technology of astronomical instruments.

Middle Ages

The first specialized treatise on trigonometry was the work of the Central Asian scientist (X-XI century) "The Book of the Keys of the Science of Astronomy" (995-996). The whole course of trigonometry contained the main work of Al-Biruni - "The Canon of Mas'ud" (Book III). In addition to the tables of sines (with a step of 15 "), Al-Biruni gave tables of tangents (with a step of 1 °).

After the Arabic treatises were translated into Latin in the XII-XIII centuries, many ideas of Indian and Persian mathematicians became the property of European science. Apparently, the first acquaintance of Europeans with trigonometry took place thanks to the zij, two translations of which were made in the 12th century.

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

new time

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

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

18th century

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

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

Application of trigonometry

Those who say that trigonometry is not needed in real life are right in their own way. Well, what are its usual applied tasks? Measure the distance between inaccessible objects.
Of great importance is the triangulation technique, which makes it possible to measure distances to nearby stars in astronomy, between landmarks in geography, and to control satellite navigation systems. Also of note is the application of trigonometry in such areas as navigation technology, music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound and computed tomography), pharmaceuticals, chemistry, number theory (and, as a result, cryptography), seismology, meteorology, oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography, etc.
Conclusion: trigonometry is a huge helper in our daily life.

Trigonometry in 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.

MBOU Tselinnaya secondary school

Report Trigonometry in real life

Prepared and conducted

mathematic teacher

qualification category

Ilyina V.P.

Tselinny March 2014

Table of contents.

1. Introduction .

2. The history of the creation of trigonometry:

Early centuries.

Ancient Greece.

Middle Ages.

New time.

From the history of the development of spherical geometry.

3. Trigonometry and real life:

Application of trigonometry in navigation.

Trigonometry in algebra.

Trigonometry in physics.

Trigonometry in medicine and biology.

Trigonometry in music.

Trigonometry in computer science

Trigonometry in construction and geodesy.

4. Conclusion .

5. List of references.

Introduction

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

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

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

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

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

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

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

Research objectives:

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

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

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

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

N. Rubtsov

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

The history of the creation of trigonometry

Early centuries

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

Ancient Greece

Middle Ages

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

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

new time

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

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

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

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

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

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

Tangents arose in connection with the solution of the problem of determining the length of the shadow. Tangent (as well as cotangent, secant and cosecant) is introduced inXin. Arab mathematician Abu-l-Wafa, who also compiled the first tables for finding tangents and cotangents. However, these discoveries remained unknown to European scientists for a long time, and tangents were rediscovered inXIVin. first by the English scientist T. Braverdin, and later by the German mathematician, astronomer Regiomontanus (1467). The name "tangent" comes from the Latintanger(to touch), appeared in 1583Tangentstranslated as "touching" (remember: the line of tangents is tangent to the unit circle)

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

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

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

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

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

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

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

From the history of the development of spherical geometry .

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

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

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

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

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

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

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

courses.

Trigonometry and real life

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

Application of trigonometry in navigation

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

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

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

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

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

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

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

Trigonometry in navigation 2.

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

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

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

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

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

Trigonometry in algebra.

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

Given the equation

Let be , we get

;

where: or

subject to restrictions, we get:

Trigonometry in physics

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

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

Oscillation phase.

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

sin α / sin β =n 1 /n 2

where:

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

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

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

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

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

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

Decision:

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

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

X : ma = 0 + mg sin24.5 0

Y: 0 = N - mg cos24.5 0

N = mg cos 24,5 0

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

Answer: 819 N

Trigonometry in medicine and biology

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

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

Basic earth rhythm- daily.

The model of biorhythms can be built using trigonometric functions.

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

Even some parts of the brain are called sinuses.

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

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

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

functions y= tgx.

Trigonometry in music

We listen to musicmp3.

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

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

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

Trigonometry in computer science

Trigonometric functions can be used for precise calculations.

Using trigonometric functions, you can approximate any

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

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

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

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

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

Trigonometry in construction and geodesy

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

2ab

= =

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

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

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

Conclusion

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

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

Trigonometry is reflected in our lives, and the areas in which it plays an important role will expand, so knowledge of its laws is necessary for everyone.

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

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

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

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

Bibliography:

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 sections 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.

Portal for the student. Self-training