Trigonometry in life. Report on trigonometry in real life

(1561-1613), and science itself was used in ancient times for calculations in astronomy, geodesy and architecture.

Trigonometric calculations are used in virtually all areas of geometry, physics, and engineering. Of great importance is the technique of triangulation, which allows one to measure distances to nearby stars in astronomy, between landmarks in geography, and to control satellite navigation systems. Also notable are the applications of trigonometry in fields such as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound (US) and computed tomography), pharmaceuticals, chemistry, number theory ( and, as a consequence, cryptography), seismology, meteorology, oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography.

In the USSR School it had the status of an academic subject.

Definition of trigonometric functions

Originally, trigonometric functions were related to aspect ratios in a right triangle. Their only argument is an angle (one of the acute angles of this triangle).

Sine is the ratio of the opposite side to the hypotenuse.
Cosine is the ratio of the adjacent leg to the hypotenuse.
Tangent is the ratio of the opposite side to the adjacent side.
Cotangent is the ratio of the adjacent side to the opposite side.
Secant is the ratio of the hypotenuse to the adjacent leg.
Cosecant is the ratio of the hypotenuse to the opposite side.

These definitions allow you to calculate function values for acute angles, that is, from 0° to 90° (from 0 to radians). In the 18th century, Leonhard Euler gave modern, more general definitions, expanding the scope of definition of these functions to the entire number line. Let us consider a circle of unit radius in a rectangular coordinate system (see figure) and plot an angle from the horizontal axis (if the angle is positive, then we plot it counterclockwise, otherwise clockwise). Let us denote the point of intersection of the constructed side of the angle with the circle A. Then:

For acute angles, the new definitions coincide with the previous ones.

A purely analytical definition of these functions is also possible, which is not related to geometry and represents each function by its expansion into an infinite series.

Story

Ancient Greece

Ancient Greek mathematicians used the chord technique in their constructions related to the measurement of arcs of a circle. A perpendicular to the chord, lowered from the center of the circle, bisects the arc and the chord resting on it. Half a chord bisected is the sine of half an angle, and so the sine function is also known as "half a chord." Because of this relationship, a significant number of trigonometric identities and theorems known today were also known to ancient Greek mathematicians, but in equivalent chordal form.

Although the works of Euclid and Archimedes do not contain trigonometry in the strict sense of the word, their theorems are presented in geometric form, equivalent to specific trigonometric formulas. Archimedes' theorem for dividing chords is equivalent to the formulas for the sines of the sum and difference of angles. To compensate for the lack of a table of chords, mathematicians from the time of Aristarchus sometimes used a well-known theorem, in modern notation - sin α/ sin β< α/β < tan α/ tan β, где 0° < β < α < 90°, совместно с другими теоремами.

Ptolemy's theorem entails the equivalence of the four sum and difference formulas for sine and cosine. Ptolemy later developed the half-angle formula. Ptolemy used these results to create his trigonometric tables, although these tables may have been derived from the work of Hipparchus. Neither the tables of Hipparchus nor Ptolemy have survived to this day, although the evidence of other ancient authors removes doubts about their existence.

Medieval India

Other sources report that it was the replacement of chords with sinuses that became the main achievement of Medieval India. This replacement made it possible to introduce various functions related to the sides and angles of a right triangle. Thus, in India, the beginning of trigonometry was laid as the study of trigonometric quantities.

Indian scientists used various trigonometric relations, including those that are expressed in modern form as

The Indians also knew formulas for multiple angles, , where .

Trigonometry is necessary for astronomical calculations, which are presented in the form of tables. The first table of sines is found in the Surya Siddhanta and Aryabhata. Later, scientists compiled more detailed tables: for example, Bhaskara gives a table of sines every 1°.

South Indian mathematicians in the 16th century made great strides in the field of summing infinite number series. Apparently, they were doing this research while they were looking for ways to calculate more accurate values for the number π. Nilakanta verbally gives the rules for expanding the arctangent into an infinite power series. And in the anonymous treatise “Karanapaddhati” (“Computation Technique”), rules are given for the expansion of sine and cosine into infinite power series. It must be said that in Europe similar results were achieved only in the 17th and 18th centuries. Thus, the series for sine and cosine were derived by Isaac Newton around 1666, and the arctangent series was found by J. Gregory in 1671 and G. W. Leibniz in 1673.

In the 8th century. Scientists from the countries of the Near and Middle East became acquainted with the works of Indian mathematicians and astronomers and translated them into Arabic. In the middle of the 9th century, the Central Asian scientist al-Khwarizmi wrote an essay “On Indian Accounting”. After the Arabic treatises were translated into Latin, many ideas of Indian mathematicians became the property of European and then world science.

Notes

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See what "Trigonometry" is in other dictionaries:

Trigonometry... Spelling dictionary-reference book

- (Greek, from tri, gonia angle, and metron measure). The part of mathematics concerned with the measurement of triangles. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. TRIGONOMETRY Greek, from trigonon, triangle, and metreo, I measure.… … Dictionary of foreign words of the Russian language

Modern encyclopedia

Trigonometry- (from the Greek trigonon triangle and... geometry), a branch of mathematics in which trigonometric functions and their applications to geometry are studied. Certain problems of trigonometry were solved by astronomers of Ancient Greece (3rd century BC);... ... Illustrated Encyclopedic Dictionary

- (from the Greek trigonon triangle and... geometry) a branch of mathematics in which trigonometric functions and their applications to geometry are studied... Big Encyclopedic Dictionary

TRIGONOMETRY IN OUR LIFE

Many people ask: why is trigonometry needed? How is it used in our world? What can trigonometry be related to? And here are the answers to these questions. Trigonometry or trigonometric functions are used in astronomy (especially for calculating the positions of celestial objects) when spherical trigonometry is required, in sea and air navigation, in music theory, in acoustics, in optics, in financial market analysis, in electronics, in probability theory, in statistics, biology, medical imaging such as computed tomography and ultrasound, pharmacy, chemistry, number theory, seismology, meteorology, oceanography, many physical sciences, land surveying and surveying, architecture, phonetics , in economics, in electrical engineering, in mechanical engineering, in civil engineering, in computer graphics, in cartography, in crystallography, in game development and many other fields.

Geodesy

Surveyors often have to deal with sines and cosines. They have special tools to accurately measure angles. Using sines and cosines, angles can be converted into lengths or coordinates of points on the earth's surface.

Ancient astronomy

The beginnings of trigonometry can be found in mathematical manuscripts of Ancient Egypt, Babylon and Ancient China. The 56th problem from the Rhinda papyrus (2nd millennium BC) suggests finding the inclination of a pyramid whose height is 250 cubits and the length of the base side is 360 cubits.

The further development of trigonometry is associated with the name of the astronomer Aristarchus Samos (III century BC). His treatise “On the magnitudes and distances of the Sun and Moon” posed the problem of determining the distances to celestial bodies; this problem required calculating the ratio of the sides of a right trianglefor a known value of one of the angles. Aristarchus considered the right triangle formed by the Sun, Moon and Earth during a quadrature. He needed to calculate the value of the hypotenuse (the distance from the Earth to the Sun) through the leg (the distance from the Earth to the Moon) with a known value of the adjacent angle (87°), which is equivalent to calculating the valuesin of angle 3. According to Aristarchus, this value lies in the range from 1/20 to 1/18, that is, the distance to the Sun is 20 times greater than to the Moon; in fact, the Sun is almost 400 times further away than the Moon, an error caused by an inaccuracy in the measurement of the angle.

Several decades later Claudius Ptolemy in his works “Geography”, “Analemma” and “Planispherium” he gives a detailed presentation of trigonometric applications to cartography, astronomy and mechanics. Among other things, it is describedstereographic projection, several practical problems have been studied, for example: determining altitude and azimuthheavenly body according to him declination and hour angle. In terms of trigonometry, this means that you need to find the side of a spherical triangle from the other two sides and the opposite angle.

In general, we can say that trigonometry was used for:

· accurately determining the time of day;

· calculations of the future location of celestial bodies, the moments of their sunrise and sunset, solar eclipses and the Moon;

· finding the geographic coordinates of the current location;

· calculating the distance between cities with known geographical coordinates.

Gnomon is the oldest astronomical instrument, a vertical object (stele, column, pole),

allowing for the least

The length of its shadow (at noon) determines the angular height of the sun.

Thus, cotangent was understood as the length of the shadow from a vertical gnomon with a height of 12 (sometimes 7) units; initially these concepts were used to calculate sundials. The tangent was the shadow of a horizontal gnomon. The cosecant and secant were the hypotenuses of the corresponding right triangles (segments AO in the figure on the left)

Architecture

Trigonometry is widely used in construction, and especially in architecture. Most compositional solutions and constructions

The drawings were made precisely with the help of geometry. But theoretical data means little. I would like to give an example of the construction of one sculpture by a French master of the Golden Age of art.

a lot of calculations to make the figure look proportional from a great height. They were mainly based on the method of sighting, that is, approximate measurement by eye. However, the difference coefficient of certain proportions made it possible to make the figure closer to the ideal. Thus, knowing the approximate distance from the statue to the point of view, namely from the top of the statue to the person’s eyes and the height of the statue, we can calculate the sine of the angle of incidence of the view using a table (we can do the same with the lower point of view), thereby finding the point vision

The situation changes as the statue is raised to a height, so the distance from the top of the statue to the person’s eyes increases, and therefore the sine of the angle of incidence increases. By comparing changes in the distance from the top of the statue to the ground in the first and second cases, we can find the coefficient of proportionality. Subsequently, we will receive a drawing, and then a sculpture, when lifted, the figure will be visually closer to the ideal

Medicine and biology.

Bohrhythm model can be constructed using trigonometric functions. To build a biorhythm model, you need to enter the person’s date of birth, reference date (day, month, year) and forecast duration (number of days).

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

Trigonometry also helps our brain determine distances to objects.

American scientists claim that the brain estimates the distance to objects by measuring the angle between the plane of the earth and the plane of vision. 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. The theory of determining distance by estimating angles was formulated by the 11th century Arab scientist Alhazen. After a long period of oblivion in the middle of the last century, the idea was revived by psychologist James

Gibson (James Gibson), who based his conclusions on the basis of his experience working with military aviation pilots. However, after that about the theory

forgotten again.

Movement of fish in water occurs according to the law of sine or cosine, if you fix a point on the tail and then consider the trajectory of movement. When swimming, the fish's body takes shape

a curve that resembles the graph of the function y=tgx.

Measuring work

Municipal budgetary educational institution

secondary school No. 10

with in-depth study of individual subjects

Project completed:

Pavlov Roman

10b grade student

Supervisor:

mathematic teacher

Boldyreva N. A

Yelets, 2012

1. Introduction.

3. The world of trigonometry.

· Trigonometry in physics.

· Trigonometry in planimetry.

· Trigonometry in art and architecture.

· Trigonometry in medicine and biology.

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

· Curves in polar coordinates (Rosettes).

· Curves in Cartesian coordinates (Lissajous Curves).

· Mathematical ornaments.

4. Conclusion.

5. List of references.

Objective of the project - development of interest in studying 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 graphical representations containing trigonometric functions; the use of trigonometry in sciences such as physics and biology. It also plays an important role in medicine, and, what is most interesting, even music and architecture cannot do without it.

Object of study - trigonometry

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

Research objectives:

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

2. Show practical applications of trigonometry in various sciences using specific examples.

3. Using specific examples, reveal the possibilities of using trigonometric functions, which allow turning “little interesting” functions into functions whose graphs have a very original appearance.

Hypothesis - assumptions: The connection of trigonometry with the outside world, the importance of trigonometry in solving many practical problems, and the graphical capabilities of trigonometric functions make it possible to “materialize” the knowledge of schoolchildren. This allows you to better understand the vital necessity of the knowledge acquired through the study of trigonometry, and increases interest in the study of this topic.

Research methods - analysis of mathematical literature on this topic; selection of specific applied tasks on this topic; computer modeling based on a computer program. Open mathematics “Functions and graphs” (Physikon).

1. Introduction

“One thing remains clear: the world is structured

menacing and beautiful."

N. Rubtsov

Trigonometry is a branch of mathematics that studies the relationships between angles and side lengths of triangles, as well as algebraic identities of trigonometric functions. It’s hard to imagine, but we encounter this science not only in mathematics lessons, but also in our everyday life. You might not have suspected it, but trigonometry is found in such sciences as physics, biology, it plays an important role in medicine, and, most interestingly, even music and architecture cannot do without it. Problems with practical content play a significant role in the development of skills in applying theoretical knowledge acquired 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 the answer to this question.

2. History of the development of trigonometry.

Word trigonometry was made up of two Greek words: τρίγονον (trigonon-triangle) and and μετρειν (metrein - to measure) in literal translation means measuring triangles.

It is precisely this task - measuring triangles or, as they say now, solving triangles, i.e. determining all the sides and angles of a triangle from its three known elements (a side and two angles, two sides and an angle, or three sides) - since ancient times it has been the basis of practical applications of trigonometry.

Like any other science, trigonometry grew out of human practice, in the process of solving specific practical problems. The first stages of the development of trigonometry are closely related to the development of astronomy. The development of astronomy and closely related trigonometry was greatly influenced by the needs of developing navigation, which required the ability to correctly determine the course of a ship on the open sea by the position of celestial bodies. A significant role in the development of trigonometry was played by the need to compile geographical maps and the closely related need to correctly determine large distances on the earth's surface.

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

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

Ptolemy divided the circle into 360 degrees and the diameter into 120 parts. He considered the radius to be 60 parts (60¢¢). He divided each part into 60¢, each minute into 60¢¢, a second into 60 thirds (60¢¢¢), etc., using the indicated division, Ptolemy expressed the side of a regular inscribed hexagon or a chord subtending an arc of 60° in the form of 60 parts of a radius (60h), and the side of an inscribed square or a chord of 90° was equated to the number 84h51¢10². The chord of 120° - the side of an inscribed equilateral triangle - he expressed the number 103h55¢23², etc. For a right triangle with a hypotenuse , equal to the diameter of the circle, he wrote down on the basis of the Pythagorean theorem: (chord a)2+(chord|180-a|)2=(diameter)2, which corresponds to the modern formula sin2a+cos2a=1.

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

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

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

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

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

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

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

Central Asia, Middle East, Transcaucasia(VII-XV century)

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

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

Abu-l-Wafa from Khorosan, who lived in the 10th century (940-998), compiled a similar “tangent table”, that is, he calculated the length of the shadow b=a×=a×tgj cast by a horizontal pole of a certain length (a=60) on a vertical wall ( see drawing).

It should be noted that the terms “tangent” (literally translated as “touching”) and “cotangent” themselves originate from the Latin language and appeared in Europe much later (XVI-XVII centuries). Central Asian scientists called the corresponding lines “shadows”: cotangent - “first shadow”, tangent - “second shadow”.

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

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

Trigonometry is separated from astronomy and becomes an independent science. This department is usually associated with the name of the Azerbaijani mathematician Nasireddin Tusi ().

For the first time in European science, a harmonious presentation of trigonometry was given in the book “On Triangles of Different Kinds,” written by Johann Muller, better known in mathematics as regiomontana(). He generalizes in it methods for solving right triangles and gives tables of sines with an accuracy of 0.0000001. What is remarkable is that he assumed the radius of a circle to be equal to miles, that is, he expressed the values of trigonometric functions in decimal fractions, actually moving from the sexagesimal number system to the decimal one.

14th century English scientist Bradwardin () was the first in Europe to introduce into trigonometric calculations the cotangent called the “direct shadow” and the tangent called the “reverse shadow”.

On the threshold of the 17th century. A new direction is emerging in the development of trigonometry - analytical. If before this the main goal of trigonometry was considered to be the solution of triangles, the calculation of the elements of geometric figures and the doctrine of trigonometric functions were built on a geometric basis, then in the 17th-19th centuries. trigonometry is gradually becoming one of the chapters of mathematical analysis. I also knew about the periodicity properties of trigonometric functions Viet, whose first mathematical studies related to trigonometry.

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

In the first half of the 19th century. French scientist J. Fourier proved that any periodic motion can be represented as a sum of simple harmonic oscillations.

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

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

Euler was responsible for the final solution to the question of the signs of trigonometric functions in all quarters of the circle and the derivation of reduction formulas for general cases.

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

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

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

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

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

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

3. The world of trigonometry.

3.1 Application of trigonometry in various sciences.

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

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

Trigonometry in physics.

Harmonic vibrations.

When a point moves in a straight line alternately in one direction or the other, the point is said to make fluctuations.

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

Usually, instead of this frequency, we consider cyclic frequencyw=, showing the angular velocity of rotation expressed in radians per second. In this notation we have: x=Rcos(wt+a). (2)

Number a called initial phase of oscillation.

The study of vibrations of all kinds is important simply because we encounter oscillatory movements or waves very often in the world around us and use them with great success (sound waves, electromagnetic waves).

Mechanical vibrations.

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

on . The number a is called the initial phase.

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

Capacitor discharge.

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

How to connect two pipes.

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

If you unfold a pipe cut obliquely, it will turn out to be bounded at the top by a sinusoid. You can verify this by wrapping the candle in paper, cutting it diagonally and unfolding the paper. Therefore, in order to get an even cut of the pipe, you can first cut the metal sheet from above along a sinusoid and roll it into a pipe.

Rainbow theory.

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

A rainbow occurs because sunlight is refracted by water droplets suspended in the air according to the law of refraction:

where n1=1, n2≈1.33 are the refractive indices of air and water, respectively, α is the angle of incidence, and β is the angle of refraction of light.

Northern lights

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

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

Trigonometry problems with practical content.

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

Determination of friction coefficient.

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

The force of body pressure on an inclined plane =kPcosa.

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

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

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

Trigonometry in planimetry.

Basic formulas for solving geometry problems using trigonometry:

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

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

Ratio of sides and angles in a right triangle:

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

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

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

4) A leg of a right triangle is equal to the product of another leg and the cotangent of the adjacent angle.

Task 1:On the sides AB and CD of an isosceles trapezoidABCD points M andN in such a way that the straight lineMN is parallel to the bases of the trapezoid. It is known that in each of the formed small trapezoidsMBCN andAMND we can inscribe a circle, and the radii of these circles are equalr andR accordingly. Find reasonsAD andB.C.

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

Find: AD and BC.

Solution:

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

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

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

similarly BC = 2r* tan(α/2).

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

Answer : AD=2R√(R/r), BC=2r√(r/R).

Problem 2:In a triangle ABC known parties b, c and the angle between the median and the height coming from the vertex A. Calculate the area of the triangle ABC.

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

Find: S∆ABC.

Solution:

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

T.K. S∆ABC=2S∆ACE=xy*sinγ(4), then dividing the 3 equality by 4 we get: (c²-b²)/S=4*ctgγ, but ctgγ=tgαb, therefore S∆ABC= ( с²-b²)/4*tgα.

Answer: (s²- b² )/4*tg α .

Trigonometry in art and architecture.

Architecture is not the only field of science in which trigonometric formulas are used. Most of the compositional decisions and construction of drawings took place precisely with the help of geometry. But theoretical data means little. I would like to give an example of the construction of one sculpture by a French master of the Golden Age of art.

The proportional relationship in the construction of the statue was ideal. However, when the statue was raised on a high pedestal, it looked ugly. The sculptor did not take into account that in perspective, towards the horizon, many details are reduced and when looking from the bottom up, the impression of its ideality is no longer created. Many calculations were made to ensure that the figure from a great height looked proportional. They were mainly based on the method of sighting, that is, approximate measurement by eye. However, the difference coefficient of certain proportions made it possible to make the figure closer to the ideal. Thus, knowing the approximate distance from the statue to the point of view, namely from the top of the statue to the person’s eyes and the height of the statue, we can calculate the sine of the angle of incidence of the view using a table (we can do the same with the lower point of view), thereby finding the point vision (Fig. 1)

The situation changes (Fig. 2), since the statue is raised to a height AC and NS increases, we can calculate the values of the cosine of angle C, and from the table we will find the angle of incidence of the gaze. In the process, you can calculate AN, as well as the sine of the angle C, which will allow you to check the results using the basic trigonometric identity cos 2a+sin 2a = 1.

By comparing the AN measurements in the first and second cases, one can find the proportionality coefficient. Subsequently, we will receive a drawing, and then a sculpture, when raised, the figure will be visually closer to the ideal.

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Trigonometry in medicine and biology.

Biorhythm model

A model of biorhythms can be built using trigonometric functions. To build a biorhythm model, you need to enter the person’s date of birth, reference date (day, month, year) and forecast duration (number of days).

Movement of fish in water occurs according to the law of sine or cosine, if you fix a point on the tail and then consider the trajectory of movement. When swimming, the fish's body takes the shape of a curve that resembles the graph of the function y=tgx.

Heart formula

As a result of a study conducted by an Iranian university student Shiraz by Vahid-Reza Abbasi, For the first time, doctors were able to organize information related to the electrical activity of the heart or, in other words, electrocardiography.
The formula, called Tehran, was presented to the general scientific community at the 14th conference of geographical medicine and then at the 28th conference on the use of computer technology in cardiology, held in the Netherlands. This formula is a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 main parameters, including several additional ones for calculations in cases of arrhythmia. According to doctors, this formula greatly facilitates the process of describing the main parameters of heart activity, thereby speeding up the diagnosis and the start of treatment itself.

Trigonometry helps our brain determine distances to objects.

American scientists claim that the brain estimates the distance to objects by measuring the angle between the plane of the earth and the plane of vision. 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. The theory of determining distance by estimating angles was formulated by the 11th century Arab scientist Alhazen. After a long period of oblivion, the idea was revived in the middle of the last century by psychologist James Gibson, who based his conclusions on the basis of his experience working with military aviation pilots. However, after that about the theory

forgotten again.

The results of the new study, as one might assume, will be of interest to engineers who design navigation systems for robots, as well as specialists who work on creating the most realistic virtual models. Applications in the field of medicine are also possible, in the rehabilitation of patients with damage to certain areas of the brain.

3.2 Graphic representations of the transformation of “little interesting” trigonometric functions into original curves.

Curves in polar coordinates.

With. 16is. 19 Sockets.

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

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

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

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

Researching graph data helps computer program "Functions and graphs". Using the capabilities of this program, we will explore some interesting graphs of trigonometric functions.

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

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

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

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

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

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

3. In general, the curver=https://pandia.ru/text/78/114/images/image042_15.gif" width="45 height=41" height="41">), because in this sector 0°≤≤180 °..gif" width="20" height="41">.gif" width="16" height="41"> for one petal you will need a “sector” exceeding 360°.

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

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

Curves in Cartesian coordinates.

Lissajous curves.

Many interesting curves can be constructed in Cartesian coordinates. Curves whose equations are given in parametric form look especially interesting:

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

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

Let's look at this using the following examples

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

II. x=sin 3t; y=cos 5t (Fig. 2)

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

Curves can be closed or open.

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

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

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

From the equation y=arcsin(sinx) it follows:

1) and 2) siny=sinx.

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

. Here the graph is depicted by the segment BC.

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

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

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

2. Consider the inequalities

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

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

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

This inequality will be satisfied by the areas shaded in Fig. 2

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

Let's move on to the following inequality:

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

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

Let's make a table of possible solutions.

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

Then we consider and shade the solutions of the following systems.

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

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

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

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

5. List of used literature.

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

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

3. Housekeeping games and entertainment. State ed. physics and mathematics lit. M, 9 pages.

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

5. Book. for extracurricular reading in mathematics in high school. State educational pedagogical ed. Min. Enlightenment RF, M., p.

6. ,Tarakanova trigonometry. 10th grade..-M.: Bustard, p.

7. About trigonometry and not only about it: a manual for students of grades 9-11. -M.: Education, 1996-80p.

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

a study, the beginning of which resembles a small wave, after which a systolic rise is observed. A small wave usually indicates atrial contraction. The beginning of the ascent coincides with the beginning of the expulsion 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 a given segment of maximum rise can be quite diverse, which leads to different results of this study. After the maximum rise, there is a descent of the curve, which continues until the very end. This segment of the apical cardiogram is accompanied by the opening of the mitral valve. After this there is a slight rise in the wave. It indicates the fast filling time. The entire remaining segment of the curve is designated as the time of passive ventricular filling. Such an examination of the right ventricle can indicate possible pathological abnormalities.

Trigonometry in medicine and biology

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

Trigonometry also helps our brain determine distances to objects.

1) Trigonometry helps our brain determine distances to objects.

2)Movement of fish in water occurs according to the law of sine or cosine, if you fix a point on the tail and then consider the trajectory of movement. When swimming, the fish's body takes the shape of a curve that resembles the graph of the function y=tg(x)
5.Conclusion

As a result of the research work:

· I became acquainted with the history of trigonometry.

· Systematized methods for solving trigonometric equations.

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