History of the trigonometry test. History of trigonometry: news and development І historical note about development of trigonometry

The demand of the virgin tricycles earlier than the whole anniversary of astronomy: and during the whole hour the trigonometry developed as one of the witnesses of astronomy.

Naskіlki vіdomo: the ways of the revival of tricutniks (spherical) for the first time in writing by the wiklade by the walnut astronomer Hipparchus in the middle of the 2nd century BC With the help of the most recent advances, the walnut trigonometry of goiters is associated with the astronomer Ptolemy (2 centuries A.D.), the creators of the geocentric system of light, who panuval to Copernicus. Greek astronomers did not know sines, cosines and tangents. To replace the tables of the stink values, they used the tables: it is allowed to draw the cola chord by pulling the dusi. The arcs were measured in degrees and hilines; Hordes were also measured in degrees (one degree becoming sixty parts of a radius), chilines and seconds. Tse shistdejkova was born by the Greeks among the Babylonians.

Values ​​of up to and including trigonometry in the Indian mid-century astronomers. The replacement of chords with sines became the main achievements of the Indian astronomers, which allowed the introduction of different functions, tied with the sides and kutas of a rectangular tricycle. In such a rank, an ear of trigonometry is laid in the Indies, yak vchenya about trigonometric values.

Trigonometry is necessary for astronomical studies, as they are drawn up in the form of a table. The first table of sinuses є in "Surya-siddhanti" and in Ariabhati. Vaughn is given through 3,4,5. More reports were added as soon as possible: for example, Bhaskar, to produce a table at the sinuses in 1.

Pivdenno-Indian mathematicians in the 16th century have sought great achievements in the field of summing up indefinite numerical series. Mabut, the stinks were busy with the tsimi doslіdzhennyi, if they joked about the ways of calculating the greater exact values ​​of the number of P. Nilakant, verbally producing the rules for arranging the arctangent into a number of statues. And in the anonymous treatise "Karanapaddhati" ("Technique of calculation") there are given the rules for placing the sine and cosine in non-finite state series. It is necessary to say that in Europe, to more results in the past 17-18 centuries. So, the series for sine and cosine viviv I. Newton is close to 1666 rock, and the series of arctangent buv knowledge of J. Gregori in 1671 r and G. V. Leibnits 1673 r

Trigonometry is a mathematical discipline of vivchaє abundance between sides and kutami of a tricot. Trigonometriya is the word for walnut and literally means vimir trikutnikiv.

The verdict of trigonometry is tied to land surveying, astronomy and wake-up call. Trigonometry winikla from practical needs of people. In addition, it is possible to see up to inaccessible objects and, in the context of the sutta, to help the process of geodesic collection of objects for folding geographic maps.

First, the methods of reviving trikutniks, based on the fallow lands between the sides and kutas of the trikutnik, were known by the ancient Greek astronomers Hipparchus (2 century BC) and I Claudius Ptolemy (2 century N. Ye.). Ptolemy vivіv spіvvіdnoshennya mіzh chords in number, scho to produce up to the current formulas for the sinuses of a half cut. Trivial history is a sinus. In fact, the growth of the number of tricytes and colas is still seen in the III century BC. in the robots of the great mathematicians of the Ancient Greece Euclid, Archmed, Apolony of Perg. In the Roman period, Menelaus (I century n.e.) systematically read it to the end of the day, if they didn’t get a special name.

A special sine, for example, screwing a yak of a half-chord, a central cube spirals onto the yak, as if yak is a chord with a base of an arc. The word cosine is nabagato younger. The cosine of the fast latin viraz is completely sinus, ie "Dodatkovy sinus".

Tangensi was found in conjunction with the solutions of the tasks about the value of the dinner. The tangent (and also the cotangent) was introduced in the X century by the Arabic mathematician Abul Wafoy, which is the first table for the meaning of tangents and cotangents.

The further development of trigonometry was neglected in the ancestors of the prominent astronomers Micoli Copernicus (1473-1543), the creator of the heliocentric system of light, Tycho Brahe (1546-1601) and Johannes Kepler (1571-1630), as well as in robots by the mathematician François Bіє3 I revisited the problem about the designation of all the elements of a flat or spherical tricycle for three data. The analytical theory of trigonometric functions in the main theory was established by the prominent mathematician of the 18th century Leonard Eiler (1707-1783), a member of the Petersburg Academy of Sciences. Eyler himself first introduced the definition of trigonometric functions, becoming a view of the functions of a pre-catered kut, taking the formulas given.

In such a rank, trigonometry, the science of the development of tricycles has evolved into the science of trigonometric functions.

1.1 Stage development of trigonometry yak science

Trigonometry is one of the youngest examples of elementary mathematics, which had been rejected in the XVIII century. ., ta in.). European mathematicians have reached a high level of sophistication in the numerical tables of natural sinuses and tangents (Regiomontanus, XV century, Reticus and Pitiskus, XVI century, Ta in.).

The name itself is "trigonometry" of the walnut walk, which means "vimir trikutnikiv": (trigonon) - trikutnik, (metrain) - vimir.

The scientific development of trigonometry was studied by L. Eiler in his work "Jntroductio in analysis infinitorum" (1748). Having solved trigonometry as a science about functions, giving the first analytical viclades, having seen the whole variety of formulas from non-basic formulas. Signified sides in small letters and prototypical kutіv - in kind large letters allowed him to simplify all the formulas, to bring clarity and stringency into them. Eiler to trace the idea of ​​looking at trigonometric functions as from the same lines to the radius of the stake, ie. Yak number, whereby the radius of the stake as the "povny sine" is taken as one. Eiler, having eliminated a number of new spivvidnoshen, having set the links of trigonometric functions for the displays, giving the rule of signs in functions for all the quads, having trimmed the common formula for the given trigonometry in the

L. Eiler's TVir served as the foundation for the handlers of trigonometry. One of the first books, "The Mathematics Is Fast" by S. Rumovsky (1760), the eddil "The pockets are given a flat trigonometry" All viclades are built up to the date of trikutnikiv (the most simple ones), the calculation is carried out to finish with a folding path, about the functions of the day.

In such a rank, the trigonometry of the winicle on the geometric basis, the small geometric language and the definition of geometric problems. The development of algebraic symbolism made it possible to write down trigonometric relations in the formulas; The storing of negative numbers made it possible to see the straight kuti and arcs and broaden the understanding of trigonometric lines (singing vidrizki in number) for any kuti. At the end of the period, the basis for the creation of trigonometric functions as functions of a numerical argument was established, the basis for the analytical theory of trigonometric (circular) functions. An analytical apparatus that allows calculating the values ​​of trigonometric functions with a certain degree of accuracy, with Newton's fragmentation.

Trigonometry was taken away from the curious viglyad in the works of the great clergyman, member of the Russian Academy of Sciences L. Eiler (1707 - 1783). Eiler, having begun to look at the meaning of trigonometric functions like number - the magnitude of trigonometric lines in number, the radius of which is taken as a unit ("trigonometric number" or "single number"). Eiler gave a residual solution about the signs of trigonometric functions in small worms, in all trigonometric formulas from the basic ones, having established a number of non-domiciled before new formulas, in the same meaning. Itself in the first place there are records. There is also a display of links between trigonometric and display functions from a complex argument. At the stage of presentation by L. Yeler, there were bullets of trigonometry handcuffs, which will be victorious in the history of science.

Analytical (like to lay down in geometry) prompted the theory of trigonometric functions, published by Eiler, and completed in the works of the great Russian scientist N.I. Lobachevsky.

The current point of view on trigonometric functions yak on the function of a numerical argument is rich in what is ampled with the development of physics, mechanics, technology. These functions formed the basis of the mathematical apparatus, with the help of which various periodic processes take place: collisional arms, widening of chills, collapse of mechanics, collusion of a wicked electric string. Having shown J. Fur'є (1768 - 1830), any periodical flow with a certain degree of accuracy can be represented among the most simple sinusoidal (harmonic) kolivans. Yakshcho on the cob development of trigonometry if the fallowness was turning around the squares, prompting on the sides of the sinuous rectangular tricycle with the hypotenuse of 1

In such a rank, at the cob stages of its development, trigonometry served as a way of verifying numerical geometric problems. Її wickedly respected the enumeration of elements of the simplest geometric figures, tobto trikutniks. But in the current trigonometry of self-determination and in the context, the importance of the power of trigonometric functions is more important. During the period of development of trigonometry of production units in full swing the development of the mechanics of collapsible rucks, physics of sound, light and electromagnetics.

At the end of the period of time, the terms of trigonometry і, zokrem, vivedenі spіvіdnoshennya for, de n is a natural number, і ін. Functions that are now viewed from the list of state rows:

Mayzhe also vicladeno and assistant of V. Nikitina and P. Suvorov.
Cycle of Science Viklad of trigonometry and Acad. M. Є. Golovin at his friend "Flat and spherical trigonometry with algebraic proofs", 1789. In this book you can know all the most important trigonometry formulas in the same view, which was adopted by the Vicladati in the 19th century. (Behind the vignette of vortex trigonometric functions). The author does not know about the need for the introduction of the secant and cosecant, so as the function in the small types is stuck in practice.
In 1804, N. Fuss's assistant. The book is designated for schools. “Flat trigonometry,” the author says, “is a science, which is a subject of three tributes and numbers of images of parts of a straight tricycle to start three parts of one." Pidruchnik is stored in 4 parts. The main understanding, the solution of trikutniks, the addition of trigonometry to the practical geometry and geodesy that, nareshty, the theorem of the addition. Pidruchnik N. Fuss is seen as a spherical trigonometry.

Krok ahead to rob academician M.V. Ostrogradsky in 1851 r. Extensions of yogo on kuti, be of the greatest magnitude.

A set of editors A.G. Mordkovich, if you want to play without respect, your handlers may not be right. § 3. Methods for the study of those "Trigonometric functions" in the course of algebra and analysis In the educational trigonometric functions in schools, it is possible to see two main stages:

Scientists, school documentation, writing visnovki about the steps of mastery of this understanding. Provide information about the progress of mathematical misdirection and the process of formulating an understanding of a complex number. Description of methods. Diagnostics: I stage. The conversation was conducted with a math teacher, yak at 10? Classical viclades, algebra and geometry. Talking about the end of the day on the cob ...

Municipal budgetary legal mortgage

middle school №10

we will die in vivchennyam okremikh objects

Viconave project:

Pavlov Roman

student 10b class

kerivnik:

mathematic teacher

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m Єlets, 2012

1. Introduction.

3. Svit trigonometry.

· Trigonometry in physics.

· Trigonometry in planimetry.

· Trigonometry in mystery and architecture.

· Trigonometry in medicine and biology.

3.2 Graphical statements about the transformation of "little cycling" trigonometric functions in the original curves (behind the additional computer programs "Functions and graphs").

· Curves in polar coordinates (Rosettes).

· Curves in Cartesian coordinates (Curves Lissajous).

· Mathematical ornaments.

4. Visnovok.

5. List of literature.

Meta project - development of interest to the introduction of those "Trigonometry" in the course of algebra and cob analysis through the prism of applied meaning to the material; expansion of graphical viglyadі, scho revenge trigonometric functions; zasosuvannya trigonometry in such sciences as physics, biology. I will not stop the role of the won і in medicine, and, well, it is impossible to find it in music and architecture without it.

Ob'єkt doslіdzhennya - trigonometry

Subject Doslіdzhennya - applied straightness of trigonometry; graphs of deyakykh funktsіy, with victories of trigonometric formulas.

Zavdannya doslіdzhennya:

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

2. To show on specific butts is a practical test of trigonometry in the Russian sciences ..

3. Expand on specific butts the capabilities of the triggering of trigonometric functions, which allow "little trick" functions to re-transform into functions, graphs that may be able to reach the original viewer.

Hypothesis - prescription: The connection of trigonometry with navkolishnіm light, the meaning of trigonometry in the development of practical buildings, graphic flexibility of trigonometric functions allow "materializing" knowledge of schools. This allows, more beautifully, the intelligence of life, the need for knowledge, which is associated with the evocation of trigonometry, and the interest before the evacuation is given by those.

Doslіdzhennya method - analysis of mathematical literature from given ones; revision of specific applications to the nature of those given; computer model based on computer programs. See the mathematician "Functions and graphs" (Physicon).

1. Introduction

"Alone is overshadowed clear,

wicked and wonderful. "

N. Rubtsov

Trigonometry is the chain of mathematics, in which there are deposits between the values ​​of the cuts and the lengths of the sides of the trials, as well as the algebraic capabilities of trigonometric functions. Fortunately, it’s not just in mathematics lessons, but in our everyday life. You could not have considered about the whole, but the trigonometry was studied in such sciences as physics, biology, I will not stop the role of you in medicine and medicine, and, well, to find it more, without it, it is not necessary to navigate in music. I have a significant role to play in the development of a tool for storing theoretical knowledge on practical theoretical knowledge, taken away with the development of mathematics, and playing with practical knowledge. Skin vivchaє mathematician, tsіkavit yak і de stagnate іt іs otrimanі knowledge. Look at the power supply and given the robot.

2. The history of the development of trigonometry.

word trigonometry folded from two walnut words: τρίγονον (trіgonon-tricutnik) and і μετρειν (metrain - vimіryuvati) literally means vimir tricutnikiv.

The very tse of the work - vimir trikutnikiv, as it is now accepted, the decision of the trikutnik, i.e. the basis of practical zasosuvan trigonometry.

As a science, trigonometry evolved from human practice, in the process of revising specific practical buildings. The first step in the development of trigonometry is tightly linked with the development of astronomy. The great influx on the development of astronomy and the trigonometry tightly tied with it needed to develop the seafaring, for which it would be necessary to correctly set the course of the ship in the open sea behind the material setting of the heavenly lights. An important role in the development of trigonometry has played a role in the folding of geographic maps and is clearly linked to the need for the correct designation of great events on the earth's surface.

Fundamental for the development of trigonometry in the era of the birth of Mali, the robots of the ancient Greek astronomer Hipparchus(Middle of the II century BC). Trigonometry is like a science, in the bitterly intelligent word it wasn’t just from the Hipparchus, but from the old ones, because they didn’t miss the stink of understanding about the functions of the huts and didn’t put the three sides in the backyard vigilance of food. Along the way, the stench, creeping through the elements of the elementary geometries, violated the tidings, how to deal with trigonometry. At the same time, the main reason for rejecting the required results is the number of circular chords being counted on the left side of the same spacing between the sides of the correct three-, chotiroh-, five - and decimal points and a radius.

Gipparkh sklavs of the first table of chords, i.e., the tables, which twist the chord for the new central kutiv in the number of continuous radius. Tse buli, according to the day, tables of the sublinear sinuses of the half of the central kut. In addition, the original tables of the Hipparchus (as well as all of them were written) did not go to us, and we can tell about them by the head of the rite from the creator of "Great Pobudova" or (in Arabic translation) "Almagest" of the famous astronomer Claudia Ptolemy He is alive in the middle of the II century. e.

Ptolemy has a circumference of 360 degrees, and a diameter of 120 parts. Win vvvav radius equal to 60 parts (60 ¢¢). The skin from the parts of the wine is 60 ¢, the skin of the quil is 60 ¢¢, the second is 60 thirds (60 ¢¢¢), etc. at the viglyad there are 60 parts of the radius (60 hours), and the side of the inscribed square is either a chord of 90 ° with the number 84h51 ¢ 10². The chord at 120 ° is the side of the inscribed one-sided tricycle - the side of the inscribed single-sided tricycle is a win in the number of 103ch55 ¢ 23 ² and so on. , which is suitable for the diameter of the stake, but having written down the Pyfagorian theorems on the display: (chord a) 2 + (chord | 180-a |) 2 = (diameter) 2, which will be based on the current formula sin2a + cos2a = 1.

"Almagest" is a table of chords through pivdegrees from 0 ° to 180 °, as from our current point of view is a table of sine waves for kutiv from 0 ° to 90 ° through the skin quarter of a degree.

In the basis of all trigonometric calculation among the Greeks, Ptolemy's theorem lay in the house of Hipparchus: "A straight-up, motivation on the diagonals of a chotirikutnik inscribed in a number, down-to-earth sums of upright people, motivated on the opposite sides" (T.E. Corrosive with a theorem, the Greeks (for the additional theorem of Pythagoras) along the chords of the two kutis (or the chord of growth), the chord of the sumi (or the chord of growth), the chord of the half of the given kut, i.e. rіznitsі) two kutіv or half a kuta.

New crocs in the development of trigonometry tied with the development of the mathematical culture of peoples India, Middle Asia and Europe (V-XII).

An important crochet forward in the period from the V to the XII century of bustling with the Hindus, as on the view of the Greeks they began to see and live in the number of not the whole chord MM ¢ (div. then, which is now called the line of the sinus a- half of the central kuta.

The order of the sine of the Indusi was introduced into the trigonometry cosine, more precisely, apparently, they began to live in their numbers the cosine line. (The term cosine itself appeared signifi- cantly increase in European robots in the first place in the end of the 16th century. The so-called “sinus addition”, i.e. ) sinus complementi quickly write yak sinus co or co-sinus).

Їm buli vіdomі also spіvіdnoshennya cosa = sin (90 ° -a) і sin2a + cos2a = r2, as well as the formulas for the sine sumi і difference two kutіv.

The offensive stage at the development of the trigonometry of dressings with the edges

Middle Asia, Near Descent, Transcaucasia (VII-XV century)

Rozvivayuchis in tіsnomu zv'yazku of astronomієyu i geografієyu - serednoazіatska mathematics small yaskravo virazheny "obchislyuvalny character" i bula spryamovana on virіshennya Applied zavdan vimіryuvalnoї geometrії i trigonometrії, and trigonometry in sformuvalasya Especially ically mathematical distsiplіnu in znachnіy mіrі sama in Prace serednoazіatskih vchenih. Among the number of successful successes that they have grown, they have gone into the first place due to the introduction of all six trigonometric lines: sine, cosine, tangent, cotangent, secant and cosecant, for which the first two bullets are seen by the Greeks and the Greeks.

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

Abu al-Wafa from Khorosan, who is alive in the X century (940-998), with an analogous "table of tangents", that is, having counted the amount of tini b = a × = a × tgj, it appears to be a horizontal shock of singing (a = 60) on the vertical wall (see armchair).

Slid to mean that the very terms "tangent" (in the literal translation - "how to feel") and "cotangent" came from the Latin language and appeared in Europe meaningfully (XVI-XVII centuries). Middle Asian vcheni called different lines "tinny": cotangent - "first tinnyu", tangent - "other tinnyu".

Abu-l-Wafa gave absolutely more accurate geometric value of the tangent line in trigonometric number and adding tangent to the tangent line and cotangent of the secant line and cosecant. In the same sense (verbally) algebraic deposits and trigonometric functions are used, and for a drop, if the radius cola is of the same type. Tsey supervisingly important vipadok will be considered by European vchenyi for 300 years ago. Nareshti, Abu-l-Wafa sklav the table of sinuses through the skin 10 ¢.

Among the ancestors of the Middle Asians, trigonometry was transformed from science, as service astronomy, into a special mathematical discipline, which represents an independent interest.

Trigonometry becomes an independent science. Tse call me to mate with the names of an Azerbaijani mathematician Nasіreddіna Tusі ().

For the first time in the European scientific string, the viclades of trigonometry is given in the book "About the tricotniki of the Apostles', written Johann Müller, Bilsh see in mathematics for im'yam Regiomontana (). Winning in the new methods of visualizing rectangular tricycles and giving tables of sinuses with an accuracy of up to 0.0000001. At the same time, it is miraculous for those who have taken into account the radius of the stake equal, that is, E. Vislov having taken the value of trigonometric functions in tens of fractions, having actually changed from sixty systems and numbers to tens.

English teachings of the XIV century Bradwardin () The first in Europe is in the trigonometric calculation of the cotangent under the name "straight tini" and the tangent under the name "zorotnoy tini".

At the time of the XVII century. The development of trigonometry has a new direct analytical approach. As far as the head metric of trigonometry, the development of tricycles was important, the calculation of the elements of geometric figures and the idea of ​​trigonometric functions was based on a geometric basis, then in the XVII-XIX centuries. trigonometry gradually grows into one of the chapters of mathematical analysis. About the power of periodicity of trigonometric functions know Viet The first mathematical dosage was set before trigonometry.

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

At the first half of the XIX century. French vcheny J. Fur'є dov, how be it from time to time, the ruh can be represented in the viglyadi sumi of simple harmony kolivan.

The great importance in the history of trigonometry is the small creativity of the famous Petersburg academician Leonard Eiler (), win dodav all trigonometrii suchasny viglyad.

In his work "Introduced into Analysis" (1748), Euler broke down trigonometry as a science about trigonometric functions, giving the first analytical viclade, having taken the whole set of trigonometric formulas from the basic formulas.

Eiler to trace the residual nutritional status about the signs of trigonometric functions in all stake worms, the formulas given for the zagalnye types.

Having introduced into mathematics new functions - trigonometric, it became an assistant to put nutrition about the distribution of these functions in an unlimited number. Appear, such distribution can be:

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This number allows the folding table of trigonometric values ​​to be significantly reduced for the meaning of any degree of accuracy.

Analytical motivation of the theory of trigonometric functions, published by Eiler, bulo completed in robots , Gauss, Koshi, Fur'є and іnshikh.

"Geometric view, - writes Lobachevsky," is necessary until the ear of the trigonometry, until the stench begins to see the power of trigonometric functions ... The geometry of all trigonometric functions must be taken care of.

Our hour of trigonometry is no longer looking like a self-styled gilka of mathematics. Particularly about trigonometric functions - a part of the larger part, prompted from a single point of view, about the functions that are involved in mathematical analysis; іnsha f part - the solution of trikutnikiv - you can see it as the head of geometry.

3.The world of trigonometry.

3.1 The stagnation of trigonometry in the ancient sciences.

Trigonometric calculation is used practically in all areas of geometry, physics and engineering.

The great significance of the technique of triangulation is that it allows the visibility of the near future in astronomy, among the organizers in geography, and control over the navigation systems of the companions. Slid means the stagnation of trigonometry in the offensive areas: technology of navigation, theory of music, acoustics, optics, analysis of financial markets, electronics, theory of imaging, statistics, biology, medicine (including) numbers, seismology, meteorology, oceanology, cartography, abundant distribution of physics, topography, geodesy, architecture, phonetics, economics, electronic technology, mechanical engineering, computer graphics, crystallography.

Trigonometry in physics.

Harmonious communication.

If a point collapses along a straight line alternately in one direction, then in one direction, then it seems that the point is going kolyvannya.

One of the simplest types of kolivan is a ruch along the axis of the projection of the point M, which can be wrapped around the stake. The law of qih kolivan maє viglyad x =Rcos (https://pandia.ru/text/78/114/images/image010_59.gif "width =" 19 "height =" 41 src = ">.

Call me the frequency and watch cycle frequencyw =, showing the cut-out speed of the wrapping, rotated in radians per second. U tsikh poznachennyah maєmo: x =Rcos (wt +a). (2)

number a name cob phase kolyvannya.

Vivchennya kolivan of any kind is importantly even wanting to be able to do it with a lot of rubbish, for the praise of my sticks, often in a day of light and because of the great success of the victorious ones (sound of sickness, electromagnetism of sickness).

Mechanical equipment.

Mechanic kolyvannyi call the ruch til, which is repeated exactly (or approximately) through the same interval of an hour. The butts of simple collar systems can be used to wrap on springs or a pendulum. At the same time, for example, a weight, I pivot on the springs (div. Fig.) And move down. The kettlebell more often rolls 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 "> Graph of the room (2) go from the graph of the room (1)

on. The number a is called the cob phase.

https://pandia.ru/text/78/114/images/image020_33.gif "width =" 29 "height =" 45 src = ">), de l is the dozen of the pendulum, and j0 is the cob cutout. More than a pendulum, more than a pendulum. (This can be seen in Fig. 1-7 Appendix VIII). On fig. 8-16, supplement VIII, it can be clearly seen, like the change of cob vidhilennya flowing into the amplitude of the swing of the pendulum, the period does not change at all. During the period of the pendulum swinging of the house, it is possible to calculate the accelerated earthly burden g at the lower points of the earth's surface.

Capacitor discharge.

It is not only a lot of mechanics that are seen behind the sinusoidal law. І in the electric lancers there are sinusoids. So in the lancet, pictured in the upper right cuff of the model, the charge on the capacitor plates changes according to the law q = CU + (q0 - CU) cos ωt, de C- capacitor unit, U-voltage on the jerel strum, L inductance: / coil, https /pandia.ru/text/78/114/images/image022_30.jpg "align =" left "width =" 348 "height =" 253 src = "> Capacitor model settings, explicitly in the" Functions and graphics "program, can be set using parameters On the graphs 1-4, you can clearly see how the pressure is injected onto the change in force or the charge of the capacitor, when it is clear that, with a positive load, the charge also inflates the same values. Fig. 5-8 add-on IX shows that when the capacitor is changing (when the inductance of the coil is changed in Fig. 9-14 of Appendix IX) and saving the same parameters, the period of jitter changes, that is, the jets change. frequency of the charge of the capacitor .. (div. additions no IX).

Yak z'adnati dvі trumpets.

Put the butt on, you can shoot the enemy, so that the sinusoid is sounded only in the connection with the bells. However, it is not so. For example, sinusoidal vikoristovytsya when closing two cylindrical pipes from the cut one to one. Schob z'єdnati two trumpets in such a rank, demanding to see them navskosi.

As soon as the pipe is flared up, then it will shine with a sinusoidal shape. You can turn over, wrapping the candle with paper, turning it up and opening it up with paper. To that, you can trim the rivny spikes of the pipe, you can form the metal sheet from the top along the sinusoidal path and bend it into the pipe.

Veselka theory.

Forward, the theory of Veselka Bula is given in 1637 rotsi Rene Descartes... Vin explaining the vestka, as a manifestation, tied with images and broken light in the board drenches.

A rainbow of water comes through those that are sleepy and seemingly broken in the dots of water, which are important in the law of breaking:

de n1 = 1, n2≈1.33 are indications of breakage and water breakage, α is a breakdown rate, and β is a breakpoint rate.

Pivnichne syayvo

Penetration into the upper balls of the atmosphere of the planets by charging the particles of the sleepy wind through the interaction of the magnetic field of the planet with the sleepy window.

The force that is given to the ruch in the magnetic field is charged to the particle is called, the force Lorenz. Vona is proportional to the charge of the particle and the vector addition of the field and the liquid to the dust of the particle

Head of trigonometry with a practical wizard.

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Viznachennya kofіtsієnta rubbing.

Tilo wagi P lay on the hijacked area with the cut in the head a. Tilo from the beginning of the yogo vlasnoy vagi passed at an accelerated rate of S in t seconds. Visibility k.

The force of the grip to grab the area = kPcosa.

Force that pulls down the road F = Psina-kPcosa = P (sina-kcosa). (1)

If you just collapse on the stolen area, then accelerated a = https: //pandia.ru/text/78/114/images/image029_22.gif "width =" 20 "height =" 41 "> == gF; 2)

Z іvnostі (1) і (2) vipliv, scho g (sina-kcosa) = https: //pandia.ru/text/78/114/images/image032_21.gif "width =" 129 "height =" 48 "> = gtga-.

Trigonometry in planimetry.

Basic formulas for the definition of geometries and 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.

Spіvvіdnoshennya sides and kutіv in a rectangular tricycle:

1) The leg of the rectangular tricycle to the rear to the tangent of the protylny kut.

2) The leg of the rectus tricuspid is back to the side of the hypotenuse on the sinus of the adjacent knuckle.

3) The leg of the rectangular tricycle is ready to add the hypotenuse to the cosine of the close knot.

4) The leg of the rectangular tricycle is back to the cotangent of the adjacent leg.

Task1:On the side sides AB and CD isosceles trapezoidABCD taken points М иN in such a rank, so straightMN is parallel to the trapezium basics. Seemingly, in the skin with small trapeziumsMBCN iAMND can be entered color iR for sure. know aboutAD iBC.

given: ABCD-trapezium, AB = CD, MєAB, NєCD, ​​MN || AD, in the trapezium MBCN and AMND it is possible to enter a number with a radius r and R as an example.

know: AD and BC.

Decision:

Nekhai O1 and O2 - centers inscribed in the small trapezium of kil. Direct O1K || CD.

В Δ O1O2K cosα = O2K / O1O2 = (R-r) / (R + r).

Since ΔO2FD is rectangular, then O2DF = α / 2 => FD = R * ctg (α / 2). Since AD ​​= 2DF = 2R * ctg (α / 2),

similarly BC = 2r * tg (α / 2).

cos α = (1-tg²α / 2) / (1 + tg² (α / 2)) => (Rr) / (R + r) = (1-tg² (α / 2)) / (1 + tg² (α / 2)) => (1-r / R) / (1 + r / R) = (1-tg²α / 2) / (1 + tg² (α / 2)) => tan (α / 2) = √ (r / R) => ctg (α / 2) = √ (R / r), toodi AD = 2R * ctg (α / 2), BC = 2r * tan (α / 2), we know.

vidpovid : AD = 2R√ (R / r), BC = 2r√ (r / R).

Task2:At the tricotnik ABC on the side b, from the top to the top, from the top A. Count the area of ​​the tricycle ABC.

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

know: SΔABC.

Decision:

Come on CE = EB = x, AE = y, AED = γ. According to the cosine theorem in ΔAEC, b² = x² + y²-2xy * cosγ (1); and in ΔACE, by the cosine theorem, c² = x² + y² + 2xy * cosγ (2). From 1 equal to 2 we can accept c²-b² = 4xy * cosγ (3).

T.K. SΔABC = 2SΔACE = xy * sinγ (4), then 3 divided by 4 is acceptable: (c²-b²) / S = 4 * ctgγ, ale ctgγ = tgαb, also SΔABC = (c? -B²) / 4 * tgα.

View: (с²- b² ) / 4 * tg α .

Trigonometry in mystery and architecture.

Architecture is not a single sphere of science, in which trigonometric formulas are used. Most of the compositional decisions and motives of the little ones took place on the basis of additional geometry. Allegedly theoretical data mean little. I would like to aim the butt to induce one of the sculptures of the French master of the Golden Capital of Art.

Proportionate performance in the motivation of the statue is ideal. However, when the statue was raised on the high pedestal, she looked conniving. The sculptor didn’t know how, in perspective to the horizon, a lot of details change, and when looking from below, up the hill, the enemy of ideality doesn’t emerge. It was carried out without the help of rosrahunks, the figurine with a great height looked proportionally. In the main, the stench of boules is based on the methods of vizuvannya, to be approximate vimiruvannya, on the eye. However, the improvement in quiet proportions allowed the growth of the figure to be closer to the ideal. In this rank, I know I will bring you closer to the statue from the point of view, and from the top of the statue to the eyes of the people and the height of the statue, you can take a look at the other table (the same can be done from the lower point of view), we ourselves point of view (fig. 1)

The situation changes (Fig. 2), so as the statue is raised to the height of the AC and NS, it is possible to develop the value of the cosine of the cut C, according to the table, it is known to see it fall. In the process, it is possible to develop the AN, as well as the sinus kuta C, to allow the reconversion of the results beyond the basic trigonometric likelihood cos 2a +sin 2a = 1.

Having adjusted the analysis of the Academy of Sciences in the first and in the other type, one can know the ratio of proportions. The next step is to make an armchair, and then a sculpture, when it is visually raised, the figure will be brought closer to the ideal.

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Trigonometry in medicine and biology.

biorhythm model

The biorhythm model can be used for additional trigonometric functions. To induce the rhythm model, it is necessary to enter the date of the population, the date of the date (day, month, day) and the triviality of the forecast (number of days).

Rukh rib at water look for the law of sine or cosine, like fixing a point on the tail, and then look at the trajectory of the ruch. When floating, the ribbing takes the shape of a crooked one, like the graph of the function y = tgx.

Heart formula

As a result of a pre-session held by a student of the Iranian University Shiraz Vahidom-Rezoy Abbasi, For the first time, physicians have neglected the ability to organize information, how to communicate to the electrical activity of the heart, in other words, electrical cardiography.
The formula, which I will call Tehran, was presented by a wide scientific community at the 14th Conference of Geographic Medicine and, moreover, at the 28th Conference on Nutrition of Computer Technology in Cardiology, which is being presented in Nizhny Novgorod. This formula is a complex algebraic-trigonometric parity, which can be stored in 8 variations, 32 functions and 33 basic parameters, including a number of additional parameters for arrhythmia problems. Like doctors, the formula in the meaning of the world will lie down to the process of describing the main parameters of the heart's activity, and, by ourselves, setting the diagnosis and the ear is completely lukewarm.

Trigonometry helps our brain to start building up to objects.

American vcheni start, so that the brain is estimated to come to the objects, in the middle of the area of ​​land and area of ​​land. Strictly it seems, the idea of ​​"vimiruvannya kutiv" is not new. Also, the artists of Ancient China painted viddaleny objects in the field of view, even without the laws of perspective. Having formulated the theory of value for the assessment of the Arab teachings of the XI century Alhazen. Psychologist James Gibson reanimated the idea of ​​being reanimated in the middle of the last century. However, for that about the theory

I’ve been screwed up again.

The results of the new release, as it is possible to let it go, appear not without interest to the engineers, who are designing navigation systems for robots, as well as to the people who work over the leaves of the most realistic virtual models. Possible supplements in the field of medicine, with the rehabilitation of patients from the ear regions of the brain.

3.2 Graphical statements about the transformation of "little cycling" trigonometric functions in the original curves.

Curves in polar coordinates.

with. 16іс. 19 Sockets.

Polar coordinates have a single interval e, Pole O and polar air Oh. The position of any point M will start with the polar radius OM and the polar cut j, we will approve the exchange OM and exchange Oh. The number r, when the bend is equal to OM through e(ОМ = rе) і the numerical value of the kuta j, rotated in degrees or in radians, are called the polar coordinates of the point M.

For any point, as seen from point O, you can take 0≤j<2p и r>0.However, when prompting curves that are similar to the form r = f (j), it is natural to put pressure on changeable j (including both negative and 2p), and r can be positive or negative.

In order to know the point (j, r), carried out from the point Pro min, which will set up from the end of the Oh cut j, і put on the new one (for r> 0) or on the other side (for r> 0) on the other side (for r> 0). r ½e.

Everything is meaningfully forgiven, as soon as the coordinate grid is in front of the front, it can be stored in concentric keels with radii e, 2e, 3e, etc. ..., 340 °, 350 °; there will be some changes and at j<0°, и при j>360 °; for example, at j = 740 ° і at j = -340 ° mi, it is acceptable for a minute, for a j = 20 °.

Doslіdzhennyu danykh grafіkіv dopomagaє computer program "Functions and graphs"... Corystyuchis, with the power of the whole program, you can read the graphs of trigonometric functions.

1 .It is easy to see the curves, given by the family:r =a +sin3j

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

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

In crooked IV, the least value of r = 0.5 and the peel may have incomplete views. With a> 1, the trilis dwarf may not end with such a rank for a> 1.

2.Considering curvesat a = 0; 1/2; 1; 3/2

With a = 0 (Fig. 1), with a = 1/2 (Fig. 2), with a = 1 (Fig. 3), the peaks may end in the endings of the viewer, with a = 3/2, there will be five incomplete peaks., (Fig. 4).

3.In the zagalny vipad at crookedr = 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 pelust a" sector "is required, I translate 360 ​​°.

On pic1-4 readings, the view of the peels is at = https: //pandia.ru/text/78/114/images/image044_13.gif "width =" 16 "height =" 41 src = ">. Gif" width = "16" height = "41 src =">.

4.Equations, known by a good mathematician-naturalist Khabenikhta for geometrical shapes, which can be seen in the light of the roslin. For example, r = 4 (1 + cos3j) і r = 4 (1 + cos3j) + 4sin23j show the curves shown in Fig. 1.2.

Curves in Cartesian coordinates.

Curves to Lissajous.

A lot of curves can be created in Cartesian coordinates. Particularly tricky to see the curves, which is given in the parametric view:

De t-dopom_zhne zminne (parameter). For example, the curves of Lissajous are discernible, but they are characterized by the following:

If you take an hour for the parameter t, then the Lissage figuri will be the result of the folding of two harmonious collapsible collapses, so that it will rise at mutually perpendicular strains. In the zagalny vipadku, the curve will grow in the middle of the rectangle with the sides 2a and 2b.

Clearly visible on the offensive butts

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 and non-closed.

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

Tsikavi and civil lines, in accordance with the common type

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

З рівняння y = arcsin (sinx) next:

1) i 2) siny = sinx.

With two minds, the happy function is y = x. Graph її in the interval (-; https: //pandia.ru/text/78/114/images/image053_13.gif "width =" 77 "height =" 41 "> matimemo y = p-x, so yak sin (px ) = sinx i in the same interval

... Here the graph will appear as a type of aircraft.

So as sinx is a periodic function with a period of 2p, then lamana ABC is prompted in the interval (,) to repeat itself on the same dilenks.

Rivnyannya y = arcsin (sinkx) will be rendered laman linia from the period https://pandia.ru/text/78/114/images/image058_13.gif "width =" 79 height = 48 "height =" 48 ">

Satisfied with the coordinates of the points, which lie at once in a sinusoidal shape (for them y> sinx) and lower than the curve y = -sinx, that is, the "solution area" of the system will fold out of the areas that are farmed in Fig. 1.

2.Considering the irregularities

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

For the verification of this irregularity, the following graphs of functions will be used: y = sinx; y = -sinx.

Let's go to the farms of the region, de y> sinx і one hour y<-sinx; затем закрашиваем области, где y< sinx и одновременно y>-sinx.

A lot of irregularities will be satisfied with the region, prepared in Fig. 2

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

Let's move on to the offensive indifference:

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

For the revision of this irregularity, the list will be graphs of functions: y = ± arcsin (sinx); y = ± arcsin (sin (x + )) .

Warehouse table of available options for solutions.

Let's take a look at the far-flung version of the offensive systems.

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

2) Another multiplier is less than zero, that is, gif "width =" 17 "height =" 41 ">) |.

3) The third multiplier is less than zero, tobto | Y |<|sin(x-)|, другие множители положительны, т. е. |y|>| Sinx | i | y |> | sin (x + Basic disciplines "href =" / text / category / uchebnie_distciplini / "rel =" bookmark "> basic disciplines, technologies, in bytes.

The victorious model of the program "Functions and graphics" significantly expanded the possibilities of carrying out the presentations, made it possible to materialize knowledge when looking at additional trigonometry in physics. The directors of the program carried out laboratory computers to monitor the mechanical hands on the butt of the pendulum, open the hands in an electric lance. The victorious computer programs have given the ability to follow the cicabs of mathematical curves, to be asked for additional trigonometric rivnyans and by-the-clock graphs in polar and Cartesian coordinates. The graphical solution of trigonometric irregularities called for the look of cyclic mathematical ornaments.

5.List of vicary literature.

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

2.Vilenkin in nature and technology: Book. for pozaklasny reading of the IX-X class-M .: Enlightenment, 5s (Svit knowledge).

3. Domoryad іgri and rozvagi. Hold. ed. fiz-mat. lit. M, 9p.

4. Kozhurov trigonometry for technikum. Hold. ed. technical and theoretical lit. M., 1956

5. Book. for post-class reading in mathematics in high school. Hold. first-ped. ed. Min. Education. RF, M., p.

6., Tarakanova trigonometry. 10 cl ..- M .: Bustard, p.

7. About trigonometry and not only about her: a book for students of 9-11 grades .. -M .: Education, 1996-80s.

8. Shapiro is a practical wizard in mathematics. Book. for the teacher.-M .: Education, 1990-96s.

The trigonometry of the winicle was developed in ancient times as one of the distributions of astronomy, as a calculating apparatus; according to the practical needs of the people. Astronomy itself was the beginning of the fact that the spherical trigonometry of the winicle was earlier flat.

Acts of trigonometric views of the boule are of the ancient Babylonians and Egyptians, ale the foundations of the science of science in Ancient Greece, ancient Greek astronomers successfully watched the times of nutrition and trigonometry. However, the stench did not look at the lines of the sine, cosine and іn., But the chordi. The role of the line of sinuses of the kuta is the vikonuval of the chord, tightening the arc, rivnu 2a.

Greek astronomer Gіpparkh in the II century. BC That is, in the table of the numerical values ​​of the chords in the presence of the quantities, the arcs are pulled together by them. More opportunities from trigonometry to take place at Ptolemy's house "Almagesta".

Ptolemy has a circumference of 360 degrees, and a diameter of 120 parts. Win vvazhav radius equal to 60 parts (60H). The skin, from the parts of the wines, is 60 ", and the skin of the quill is 60", the second - 60 thirds (60 ""), etc. In these words, the speed of Shistdejkov's system of numbers, quite a bit, is suspected of being in the Babylonians. Zastosovyuchi was said, Ptolemy lingered on the side of the correct inscribed six-kutnik or a chord, embossing an arc of 60 ° in the view of 60 parts of the radius (60 H), and the side of the inscribed square, or a chord in 90 °, attributed to the number 84 H 5110 - ". of the inscribed equal-sided tricycle - win with the number 103 × 55 "23" etc.

Having stagnated with the geometrical theorem, the knowledge of the fallowness, which is equal to the next step by the day formulas for thinking:

Having bridged with a pair of spindles and bends in parts of the radius with the values ​​of chords 60 ° "and 72 °, vin virahuvov in the chord, embossing an arc at 6 °, at 3 °; 1.5 ° i, nareshti, --0.75 °. (The value of the chord is in Mr.

Broken rosary allowed Ptolemy to fold the table, like the chordi from 0 to 180 °, calculated from an accuracy of up to 1 "radius.

The table, which has been preserved until our hour, is equivalent to the table of sinuses from 0 to 90 ° with a crochet of 0.25 ° with five ten signs.

Name the line of sine and cosine for the first time introduced by the Indians. Smells have put together the first tables of sinuses, I want them to be accurate, not Ptolemy. In India and read about the day about trigonometric quantities, called more goniometry (from "gonia" - kut і "Mehr" - vimiryuyu).

Submitted development about trigonometric values ​​otrimalo in IX - XV centuries. in the lands of the Middle and Near Retreat, a number of mathematicians, who did not just become quick at that time, reached the end of the galus, but made their significant contributions to science.

Vidomy Muhammad ibn Musa al-Khorezm (IX century) Sclavs of tables of sinuses and cotangens. Al-Habash abo (Ahmed ibn Abdallah al-Marvazi) calculating tables for tangent, cotangent and cosecant.

More important in the development of trigonometry is pratsi al-Battani (bl. 850-929) and Abu-l-Wafi al-Buzjani (940-998). The rest of the theorem of sinuses in spherical trigonometry, calculating for the sinuses a table with an interval of 15 "

Abu Raikhan Muhammad ibn Ahmad-al-Berun (according to the Biruni's transcript (973-1048)) published and at the same time clarified the results, which reached him in the area of ​​trigonometry. At the pratsi "Canon Mas" ud "win viclavs all at that hour of the position of trigonometry and suttuly updating them. An important innovation, the destruction of Abu-l-Wafa, approved by al-Berun. Al-Beruni, explaining the reason for the change in his report, showing that everything is calculated with a single radius is simpler.

Nasir ad-Din Muhammad at-Tusi (1201-1274), in his "Treatise on the Four-Siders", was the first viclav of trigonometric views as an independent view of mathematics, and not an appendage to astronomy. Yogo treatise for the year after celebrating the great influx on the robots of Regiomontana (1436-1476).

In the first half of the XV century. Dzhemshid ibn Masud al-Kashi counted with great accuracy the trigonometric tables in croc. Г, which stretched out 250 rockets, were not overturned.

In Europe XII - XV centuries, for that, yak bouli was translated from Arabic and Greek language into Latin deyak classic mathematical and astronomical create, development of trigonometry is trivial. With the appearance of flat tricycles, the theorem of sinuses was widely adopted, the knowledge of the sinuses lived in Pivdenny France by Leo Gersonid (1288-1344), the trigonometry of a bull in 1342 was translated into Latin language. He was known as the European representative of the era of the era in the area of ​​trigonometry buv Regiomontan. Yogo great tables of sinuses through G with exactness up to the 7th significant digit and thi maisternally vicladenia trigonometric pratsyu "Five books about tricusters of all kinds" was a great value for a given development, trigonometry in the XVI-XVII centuries.

At the time of the XVII century. in the development of trigonometry, there is a new direct - analytic. As far as the head metric of trigonometry, the development of tricycles was important, the calculation of the elements of geometric figures and the idea of ​​trigonometric functions was based on a geometric basis, then in the XVII - XIX centuries. trigonometry gradually grows into one of the chapters of mathematical analysis. There is a wide range of stasis in mechanics, physics and technology, especially in the case of the evacuation of collisional problems and other periodic processes. About the power of periodicity of trigonometric functions know Viet, the first mathematical predictions of what were put before trigonometry. The Swiss mathematician Johann Bernoulli (1642-1727) has already fixed the symbols of trigonometric functions. If the development of algebraic symbolism, the projection of negative numbers and the direct description of the number of ideas in the expansion of the understanding of the kut and the arc, then the development of the development of the number of collisions, about the sound, light and electrical processes led to the former From the physics of vidomo, how to match the harmonious string (for example, the swing of the pendulum, the alternating electric strum)

The graphs of harmonic lines are often called sinusoidal lines in physics and technology.

At the first half of the XIX century. french teachings by Zh. Fur'є dovіv, so be it from time to time, it can be represented (with a certain degree of accuracy) in the viglyadі sumi of simple harmony coliwans.

The expansion of the phenomenon about trigonometric functions called to understand the mathematical analogous base before the formation of the trigonometric functions:

The development of the analytical theory of trigonometric functions was given by I. Newton and L. Euler. The founder of the center of the theory of slides in honor of L. Eiler. Win dodav all trigonometrii suchasny viglyad. The further development of the theory was continued in the 19th century. M. I. Lobachevsky and others. Our hour of trigonometry is no longer looking like a self-styled gilka of mathematics. Nayvazhlivisha її part - about trigonometric functions, - є partly larger, prompted from a single point of view about the functions that are used in mathematical analysis; The insha and the part - the solution of the triangles - look like the head of geometry (flat and spherical).

History of trigonometry

Trigonometry is the word walnut and literally means vimir trikutnikiv ( - trikutnik, and  - vimiryuyu).

In this particular type of trikutnikov, there is a reason for the development of trikutniks, that is, the designation of the sides, kutiv and the other elements of the trikutnik, as given by the actions of them. There is a great number of practical establishments, as well as the establishments of planimetry, stereometry, astronomy and those who are brought up to the task of developing triplets.

The verdict of trigonometry is tied to land surveying, astronomy and wake-up call.

I want the name of the science to be known recently, it’s a lot of money to get it at once before the trigonometry of the understanding, and in fact, there were two thousand rockets.

First, the methods of reviving trikutniks, based on the fallow lands between the sides and kutas of the trikutnik, were known by the ancient Greek astronomers Hipparchus (2 century BC) and I Claudius Ptolemy (2 century N. Ye.). With the increase in the number of fallow areas between the two sides of the tricycle and the thighs, they began to be called trigonometric functions.

Significant additions in the development of trigonometry were made by Arabic al-Batanov (850-929) and Abu-l-Wafa, Mohamed-bin Mohamed (940-998), which list of sinus and tangent tables after 10’ with accuracy up to 1/60 4 ... The theorem of sinuses was already known by the Indian teachings of Bkhaskar (p. 1114, the death of nevidomas) and the Azerbaijani astronomer and mathematician Nasireddin Tusi Muhamed (1201-1274). In addition, Nasireddin Tusi in his robot "A treatise on the four-sided" Viklav flat and spherical trigonometry as an independent discipline.

Trivial history is a sinus. In fact, the development of these types of tricytes and colas (and, by the way, and trigonometric functions) are developed in the sameIIIstolitti BC in the robots of the great mathematicians of Ancient Greece - Euclid, Archmed, Apolony of Perg. In the Roman period, Menelaus (Istolittya n.e.), if I didn't get a special name. A lucky sine , for example, when a yak of a half-chord is screwed into the yak, a central cube of  is spiraling onto the yak, for example, a chord is a base of the arc.

 M

A '

Small. 1

V IV- Vthe capitals have already appeared a special term in the astronomy of the great Indian scientist Ariabhati, in which names are the first Indian companion of the Earth. Vidrizok AM (Fig. 1) was named ardkhadzhiv (ardha - half, jiva - tyativa onion, yaku nagaduє chord). The name jiva is shorter when it has appeared. Arab mathematicians inIXstolіttі tse word bulo has been replaced in Arabic by the word jayb (opuklіst). When translating Arabic mathematical texts in the capital, the bulo is replaced by the Latin sine (sinus- vigin, curvature).

The word cosine is nabagato younger. Cosine - tse speedy latin virazucompletelysinus, T. E. "Dodatkovy sinus" (or іnakshe "sine of the dodatkovy arc";cos = sin(90  -  )).

Tangensi was found in conjunction with the solutions of the tasks about the value of the dinner. Tangent (and also cotangent) introduced inXcapital by the Arabic mathematician Abu-l-Wafa, which is the first table for the meaning of tangents and cotangens. However, the last hour was left unrecognizable by the European clerk, and the tangency bullyXIVstolitty by the Nimetsian mathematician, astronomer Regimontanus (1467). Winning the tangential theorem. Regiomontan sklav also reports trigonometric tables; planners and spherical trigonometry has become an independent discipline in Europe.

Named "tangent", scho to resemble Latintanger(Stosuvatisya), announced in 1583 rTangensto change yak "scho stosuєtsya" (the line of tangents is similar to the same number).

The further development of trigonometry was neglected in the ancestors of the prominent astronomers Micoli Copernicus (1473-1543) - the creator of the heliocentric system of light, Tycho Brahe (1546-1601) and Johannes Kepler (1571-1630), as well as in robots by the mathematician François Bi I will raise the problem about the designation of all the elements of a flat or spherical tricycle for three data.

For the last hour, trigonometry was of a purely geometric character, ie, facts, which at once were formulated in terms of trigonometric functions, were formulated and brought to the aid of the geometrical ones to understand and solidity. Such was the case in the middle century, I wanted some analytic methods to be used in some of the victorious methods, especially when logarithms appeared. Mabut, the greatest stimulus before the development of trigonometry was determined in conjunction with the news of the astronomy staff, was of great practical interest (for example, for the development of the design of the mission of the vessel afterwards). Astronomers tsіkavili spіvіdnoshennya mіzh sides and kutami spherical tricycles. I need to respect the fact that mathematicians have long been coping with the assignments.

Fixing XVIIin., trigonometric functions began to be set up to the date of development of tasks, tasks of mechanics, optics, electricity, radiotechnics, for describing colival processes, widening of problems, the rundown of new mechanisms, for the development of advanced functions of and they have come up with an important meaning for all mathematics.

The analytical theory of trigonometric functions in the main field was established by a prominent mathematicianXviiistolitti Leonard Eyler (1707-1783) a member of the Petersburg Academy of Sciences. Eiler's scientific decline is great, including very short results, such as mathematical analysis, geometry, number theory, mechanics and additions of mathematics. Eyler himself first introduced the definition of trigonometric functions, becoming a view of the functions of a pre-catered kut, taking the formulas given. Pislya Euler trigonometry nabula of forms of calculation: the facts began to be brought along the path of formal stasis of trigonometry formulas, the proofs became more compact and simpler,

In such a rank, trigonometry, the science of the development of tricycles has become a science of trigonometric functions.

Formerly a part of trigonometry, as a result of the power of trigonometric functions and depletion among them, they thought it was called goniometry. The term goniometriya is practically impossible to get used to for an hour.