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

The concept has a long historysinus. In fact, various ratios of the segments of a triangle and a circle (and, in essence, trigonometric functions) are already found in the 3rd century. BC e. in the works of great mathematicians Ancient Greece- Euclid, Archimedes, Apollonius of Perga. In the Roman period, these relations were already quite systematically studied by Menelaus (1st century AD), although they did not acquire a special name.

In the subsequent period, mathematics for a long time most actively developed by Indian and Arab scientists. In the IV-V centuries. appeared, in particular, special term in the works on astronomy of the great Indian scientist Aryabhata (476 - ca. 550), after whom the first Indian satellite of the Earth is named. The segment he called ardhajiva.

Later, more short title jiva. Arab mathematicians in the ninth century. the word jiva (or jiba) was replaced by the Arabic word jaib (bulge). When translating Arabic mathematical texts in the XII century. this word has been replaced by the Latinsinus (sinus - bend, curvature).

The word cosine is much younger.Cosine is an abbreviation latin expression complementy sinus, i.e. “additional sine” (or otherwise “sine of an additional arc”; remember cos a \u003d sin (90 ° - a)).

Tangents arose in connection with the solution of the problem of determining the length of the shadow. Tangent (as well as cotangent, secant and cosecant) was introduced in the 10th century. Arab mathematician Abul-Vafa, who also compiled the first tables for finding tangents and cotangents. However, these discoveries remained unknown to European scientists for a long time, and tangents were rediscovered in the 14th century. first by the English scientist T. Braverdin, and later by the German mathematician, astronomer Regiomontanus (1467).

The name "tangent", which comes from the Latin tanger (to touch), appeared in 1583. Tangens is translated as "touching" (the line of tangents is a tangent to unit circle).

Modern designationsarcsin and arctg appear in 1772 in the works of the Viennese mathematician Scherfer and the famous French scientist Lagrange, although J. Bernoulli, who used a different symbolism, had already considered them a little earlier. But these symbols became generally accepted only at the end XVIII century. The prefix "arc" comes from the Latin arcus(bow, arc), which is quite consistent with the meaning of the concept: arcsin x, for example, is an angle (or, one might say, an arc), the sine of which is x.

For a long time, trigonometry developed as part of geometry. Perhaps the greatest incentives for the development of trigonometry arose in connection with the solution of astronomical problems, which was of great practical interest (for example, for solving problems of determining the location of a ship, predicting eclipses, etc.).

Astronomers were interested in the relationship between the sides and angles of spherical triangles made up of great circles lying on a sphere.

In any case, in geometric shape many formulas of trigonometry were discovered and rediscovered by ancient Greek, Indian, Arab mathematicians. (True, the difference formulas trigonometric functions became known only in the 17th century - they were brought out by the English mathematician Napier to simplify calculations with trigonometric functions. And the first drawing of a sinusoid appeared in 1634.)

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

The modern form of trigonometry was given by the largest mathematician of the 18th centuryL . Euler(1707-1783), Swiss by birth, long years who worked in Russia and was a member of the St. Petersburg Academy of Sciences. It was Euler who first introduced the well-known definitions of trigonometric functions, began to consider functions of an arbitrary angle, and obtained reduction formulas. All this is a small fraction of what long life Euler managed to do in mathematics: he left over 800 papers, proved many theorems that have become classical, related to the most diverse areas of mathematics. (Despite the fact that in 1776 Euler lost his sight, he last days continued to dictate more and more new works.)

After Euler, trigonometry took on the form of calculus: various facts began to be proved by the formal application of trigonometry formulas, the proofs became much more compact, simpler.

The scope of trigonometry covers the most different areas mathematics, some branches of natural science and technology.

Trigonometry has several varieties:

Spherical trigonometry deals with the study of spherical triangles.


Rectilinear or plane trigonometry usually studies triangles.

Ancient Greek and Hellenistic scientists developed trigonometry significantly. However, in the works of Euclid and Archimedes, trigonometry is presented in geometric form. Theorems about the length of chords are applied in the laws of sines. And Archimedes' theorem for the division of chords corresponds to the formulas for the sines of the sum and difference of angles.

At present, mathematicians use new record well-known theorems, for example, sin α/ sin β< α/β < tan α/ tan β, где 0° < β < α < 90°, тем самым, компенсируют недостатки таблиц хорд, времен Аристарха Самосского.

Presumably the first trigonometric tables were compiled Hipparchus of Nicaea, who is rightly considered the "father of trigonometry". He is credited with creating a summary table of arcs and chords for a series of angles. Moreover, it was Hipparchus of Nicaea who first began to use 360 ​​° circles.

Claudius Ptolemy significantly developed and expanded the teachings of Hipparchus. Ptolemy's theorem states that the sum of the products of opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. The consequence of Ptolemy's theorem was the understanding of the equivalence of the four sum and difference formulas for sine and cosine. In addition, Ptolemy derived the formula for the half angle. Ptolemy used all his results in compiling trigonometric tables. Unfortunately, not a single authentic trigonometric table of Hipparchus and Ptolemy has survived to this day.

Trigonometric calculations have found their application in almost all areas of geometry, physics and engineering.
With the help of trigonometry (triangulation technique), you can measure the distances between stars, between landmarks in geography, and control satellite navigation systems.

Trigonometry is successfully applied in navigation technique, music theory, acoustics, optics, analysis financial markets, electronics, probability theory, statistics, biology and medicine, chemistry and number theory (cryptography), seismology, meteorology, oceanology, cartography, topography and geodesy, architecture and phonetics, mechanical engineering and computer graphics e .

TRIGONOMETRY- (from the Greek. trigwnon - triangle and metrew - measure) - a mathematical discipline that studies the relationship between the angles and sides of triangles and trigonometric functions.

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

In trigonometry, three types of relationships are distinguished: 1) between the trigonometric functions themselves; 2) between elements of a flat triangle (trigonometry on a plane); 3) between the elements of a spherical triangle, i.e. a figure carved on a sphere by three planes passing through its center. Trigonometry began precisely with the most complex, spherical part. It arose primarily from practical needs. The ancients watched the movement of heavenly bodies. Scientists processed measurement data in order to keep a calendar and correctly determine the start time of sowing and harvesting, the dates of religious holidays. The stars were used to calculate the location of a ship at sea or the direction of a caravan in the desert. Observations for starry sky from time immemorial, astrologers have also led.

Naturally, all measurements related to the location of the luminaries in the sky are indirect measurements. Straight lines could only be drawn on the surface of the Earth, but even here it was far from always possible to directly determine the distance between some points, and then they again resorted to indirect measurements. For example, they calculated the height of a tree by comparing the length of its shadow with the length of the shadow from some pole, the height of which was known. The size of the island in the sea was calculated in a similar way. Similar tasks are reduced to the analysis of a triangle in which some of its elements are expressed through others. This is what trigonometry does. And since the stars and planets were represented by ancient points on celestial sphere, it was spherical trigonometry that first began to develop. It was considered a branch of astronomy.

And it all started a very long time ago. The first fragmentary information on trigonometry was preserved on the cuneiform tablets of Ancient Babylon. The astronomers of Mesopotamia learned to predict the position of the Earth and the Sun, and it was from them that the system for measuring angles in degrees, minutes and seconds came to us, because the Babylonians adopted a sexagesimal number system.

However, the first truly important achievements belong to ancient Greek scientists. For example, the 12th and 13th theorems of the second book Began Euclid (late 4th-3rd century BC) essentially express the cosine theorem. In the 2nd century BC. the astronomer Hipparchus of Nicaea (180–125 BC) compiled a table to determine the ratios between the elements of triangles. Such tables are needed because the values ​​of trigonometric functions cannot be calculated from the arguments using arithmetic operations. Trigonometric functions had to be calculated in advance and stored in tables. Hipparchus calculated in a circle of a given radius the length of the chords corresponding to all angles from 0 to 180 °, a multiple of 7.5 °. Essentially, this is a table of sines. The works of Hipparchus have not come down to us, but many of the information from them is included in Almagest(II century) - a famous essay in 13 books by the Greek astronomer and mathematician Claudius Ptolemy (d. ca. 160 AD). The ancient Greeks did not know sines, cosines and tangents; instead of tables of these quantities, they used tables that made it possible to find the chord of a circle along a contracted arc. IN Almagest the author gives a table of chord lengths of a circle with a radius of 60 units, calculated in 0.5° steps with an accuracy of 1/3600 of a unit, and explains how this table was compiled. Ptolemy's work served as an introduction to trigonometry for astronomers for several centuries.

To understand how ancient scientists compiled trigonometric tables, one must become familiar with Ptolemy's method. The method is based on the theorem that the product of the diagonals of a quadrangle inscribed in a circle is equal to the sum of the products of its opposite sides.

Let ABCD- inscribed quadrilateral , AD - diameter of a circle, and a point O is its center (Fig. 1). If you know how to calculate the chords subtracting the angles DOC= a and DOB = b, i.e. side CD and diagonal b, then, by the Pythagorean theorem, from right triangles ADB And ADC can be found AB and AC and then, according to Ptolemy's theorem, BC = (AU· BD - AB· CD) /AD, i.e. chord that subtends an angle WOS= b – a. Some chords, such as the sides of a square, a regular hexagon, and an octagon, corresponding to angles of 90, 60, and 45°, are easy to determine. The side of a regular pentagon is also known, which subtends the arc to 72°. The above rule allows you to calculate chords for the differences of these angles, for example for 12° = 72° - 60°. In addition, you can find the chords of half angles, but this is not enough to calculate what the chord of an arc is 1 °, if only because all the angles named are multiples of 3 °. For the chord 1°, Ptolemy found an estimate, showing that it is greater than 2/3 of the chord (3/2)° and less than 4/3 of the chord (3/4)°, two numbers that coincide with sufficient accuracy for his tables.

If the Greeks calculated chords in the corners, then Indian astronomers in the writings of the 4th-5th centuries. we switched to half-chords of a double arc, i.e. exactly to the sine lines (Fig. 2). They also used the lines of the cosine - or rather, not of its own, but of the “inverted” sine, which later received the name “sine-versus” in Europe, now this function is equal to 1 - cos a is no longer in use. Subsequently, the same approach led to the definition of trigonometric functions in terms of the ratios of the sides of a right triangle.

Per unit of segments MP,OP,PA minute of arc was taken. So, the sine line of the arc AB= 90° yes OB is the radius of the circle; arc AL, equal to the radius, contains (rounded) 57°18" = 3438".

come down to us indian tables sinuses (the oldest was compiled in the 4th-5th century AD) are not as accurate as the Ptolemaic ones; they are drawn through 3 ° 45 "(i.e. through 1/24 of the arc of the quadrant).

The terms "sine" and "cosine" came from the Indians, not without a curious misunderstanding. The Indians called the semi-chord "Ardhajiva" (translated from Sanskrit - "half of the bowstring"), and then reduced this word to "jiva". Muslim astronomers and mathematicians, who received knowledge of trigonometry from the Indians, perceived it as "jiba", and then it turned into "jib", which in Arabic means "bulge", "bosom". Finally, in the 7th c. "jaib" was literally translated into Latin with the word "sinus" , which had nothing to do with the concept it denoted. The Sanskrit “kotijiva” is the sine of the remainder (up to 90 °), and in Latin it is sinus complementi, i.e. sine complement, in the 17th century. abbreviated to the word "cosine". The names "tangent" and "secant" (translated from Latin meaning "tangent" and "secant") were introduced in 1583 by the German scientist Fink.

A great contribution to the development of trigonometry was made by Arab scientists, for example, Al-Battani (c. 900 AD). In the 10th century the Baghdad scholar Muhammad from Bujan, known as Abu-l-Vefa (940–997), added lines of tangents, cotangents, secants and cosecants to the lines of sines and cosines. He gives them the same definitions that are found in our textbooks. Abu-l-Vefa also establishes the basic relationships between these lines.

So, by the end of the 10th c. scientists of the Islamic world have already operated, along with sine and cosine, four other functions - tangent, cotangent, secant and cosecant; discovered and proved several important theorems of plane and spherical trigonometry; used a circle of unit radius (which made it possible to interpret trigonometric functions in modern sense); came up with the polar triangle of a spherical triangle. Arab mathematicians compiled exact tables, for example, tables of sines and tangents with a step of 1 "and an accuracy of 1/700,000,000. A very important applied task was this: to learn how to determine the direction to Mecca for five daily prayers, wherever a Muslim is.

A particularly important influence on the development of trigonometry was Treatise on the Complete Quadrilateral astronomer Nasir-ed-Din from Tus (1201-1274), also known as at-Tusi. It was the first work in the world in which trigonometry was treated as an independent area of ​​mathematics.

In the 12th century was transferred from Arabic on the Latin series of astronomical works, for the first time Europeans got acquainted with trigonometry.

Nasir-ed-Din's treatise made a great impression on the German astronomer and mathematician Johann Müller (1436–1476). Contemporaries knew him better under the name of Regiomontanus (as it is translated into latin name his hometown Koenigsberg, now Kaliningrad). Regiomontan compiled extensive tables of sines (after 1 minute, accurate to the seventh significant digit). For the first time, he deviated from the sexagesimal division of the radius and took one ten-millionth of the radius as the unit of measurement for the sine line. Thus, the sines were expressed as whole numbers, not sexagesimal fractions. Before introduction decimal fractions there was only one step left, but it took more than 100 years. Labor of Regiomontanus Five books on triangles of all kinds played the same role in European mathematics as the work of Nasir al-Din in the science of Muslim countries.

The tables of Regiomontanus were followed by a number of others, even more detailed. A friend of Copernicus, Rheticus (1514-1576), together with several assistants, worked for 30 years on the tables, completed and published in 1596 by his student Otto. The angles went through 10 "", and the radius was divided into 1,000,000,000,000,000 parts, so the sines had 15 correct digits.

The further development of trigonometry followed the path of accumulation and systematization of formulas, clarification of basic concepts, formation of terminology and notation. Many European mathematicians worked in the field of trigonometry. Among them are such great scientists as Nicolaus Copernicus (1473-1543), Tycho Brahe (1546-1601) and Johannes Kepler (1571-1630). François Viet (1540–1603) supplemented and systematized various cases of solving plane and spherical triangles, discovered the “flat” cosine theorem and formulas for trigonometric functions of multiple angles. Isaac Newton (1643–1727) expanded these functions into series and opened the way for their use in calculus. Leonhard Euler (1707-1783) introduced both the very concept of a function and the symbolism accepted today. sin values x, cos x etc. he considered as functions of a number x is the radian measure of the corresponding angle. Euler gave the number x all possible values: positive, negative and even complex. He also discovered a connection between trigonometric functions and the exponent of a complex argument, which made it possible to turn numerous and often very intricate trigonometric formulas into simple corollaries from the rules of addition and multiplication of complex numbers. He also introduced inverse trigonometric functions.

By the end of the 18th century trigonometry as a science has already taken shape. Trigonometric functions have found application in mathematical analysis, physics, chemistry, technology - wherever you have to deal with periodic processes and oscillations - be it acoustics, optics or the swing of a pendulum.

The solution of any triangles, ultimately, comes down to the solution of right-angled triangles (i.e. those in which one of the corners is right). Since all right triangles with a given acute angle are similar to each other, the ratios of their respective sides are the same. For example, in a right triangle ABC the ratio of its two sides, for example, the leg A to the hypotenuse With, depends on the size of one of the acute angles, for example A. The ratios of different pairs of sides of a right triangle are called trigonometric functions its acute angle. There are six such relations in the triangle, and they correspond to six trigonometric functions (the designations of the sides and angles of the triangle in Fig. 3).

Because A + IN= 90°, then

sin A= cos B= cos (90° – A),

A=ctg B= ctg (90° – A).

Several equalities follow from the definitions, connecting the trigonometric functions of the same angle with each other:

Taking into account the Pythagorean theorem a 2 + b 2 = c 2 it is possible to express all six functions through any one. For example, sine and cosine are related by the basic trigonometric identity

sin 2 A+ cos2 A = 1.

Some relationships between functions:

These formulas are also valid for the trigonometric functions of any angle, but they must be used with care, since the right and left sides may have different domains of definition.

There are only two right triangles that have both "good" angles (expressed as an integer or rational number of degrees) and at least one of the ratios of the sides is rational. It is an isosceles triangle (with angles of 45, 45 and 90°) and half equilateral triangle(with angles of 30, 60, 90°) - just those two cases when the values ​​of trigonometric functions can be calculated directly by definition. These values ​​are shown in the table

Dictionary Ushakov

Trigonometry

trigonometry, trigonometry, pl. No, female(from Greek trigonos - triangle and metreo - measure) ( mat.). Department of geometry on the relationship between the sides and angles of a triangle.

encyclopedic Dictionary

Trigonometry

(from the Greek. trigonon - a triangle and ... metrics), a branch of mathematics that studies trigonometric functions and their applications to geometry.

Ozhegov's dictionary

TRIGON E TRIA, And, and. The branch of mathematics that studies the relationship between the sides and angles of a triangle.

| adj. trigonometric, oh, oh.

Dictionary of Efremova

Trigonometry

and.
Branch of mathematics that studies trigonometric functions and their application to
problem solving.

Encyclopedia of Brockhaus and Efron

Trigonometry

The ratios between the sides and angles of triangles (see) are expressed using a special kind of functions, called. trigonometric. These functions are given special names: sine, cosine, tangent, cotangent, secant And cosecant.

Let us assume that, taking the point ABOUT behind the center, radius OA describe the arc AB. Dot A called start arcs AB, a dot IN - end arcs AB. Imagine an angle AOW, whose vertex is at the point ABOUT, and the sides pass through the points A And IN. When changing the radius OA arc AB, bounded by the sides of a given angle, changes, but the ratio AB/OA remains unchanged. This relationship serves measure given angle. Since equal angles can be plotted along different sides straight oa, then, in order to distinguish one angle from another, they agreed to express one of the angles as a positive number, and the other as a negative number. If arcs AB And AB", described by radius OA are equal, then the angle AOB equal to the angle AOW". If eg. AB/OA = 1/3 , then we agree to say that the angle AOB equals 1/3 and that angle AOB" equals ( - 1/3) . Thus every abstract number (positive or negative) corresponds to a well-defined angle. If we are from the end of the arc IN let's drop the perpendiculars VR And BQ directly OA and directly OS, perpendicular to OA, then we get segments OR And OQ(Fig. 2), which are called. projections 0V on OA and on OS. Let's assume that the angle AOB does not change, but the radius changes OA; in this case the relationship OP/OA And OQ/OA remain unchanged.

The following special cases are possible here. Projection 0V on O A can be directed in the same direction as the segment OA or in the opposite direction (Fig. 3).

Just like a projection 0V on OS may have a direction OS or the opposite direction (Fig. 4).

Direction OS chosen so that the direct

corner A OS was positive. If the angle AOB equals α , That sinus α (Sin α) name attitude OQ/OA if OQ has the same direction as OS. If OQ directed oppositely OS, That

Sinα = -OQ/OA

Attitude OP/OA name cosine α, (Cosα) if OR the same direction as O.A. If OR has the opposite direction from oa, That

Cosα = -OP/OA

In textbooks T. you can find proof of the following formulas:

Sin ( - α) = -Sinα, Cos ( -α) = Cos α,

Sin (π /2 - α) = Cos α, Cos (π /2 -α) = Sin α,

Sin(π- α) = Sin α, Cos (π - α) = -Cosα,

Sin (π + α) = - Sinα, Cos(π + a) = -Cosα,

Sin(2π- α) = -Sin α, Cos (2 π -α) = Cos α,

Sin (2 π + α) = Sin α, Cos (2 π + α) - Cosα.

Using these formulas, the calculation of Sinα and Cosα is reduced to the case when α is a positive number not exceeding π /4

From formulas

Sin (α + β) = Sin α Cosß + Cos α Sinß,

Cos (α + ß) = Cos α Cosß - Sin α Sinß

Sina + Sinb = 2Sin[(a + b)/2] Cos[(a -b)/2],

Sina- Sinb = 2Sin[(a -b)/2] Cos[(a + b)/2],

Cosa + Cosb = 2Cos[(a + b)/2] Cos[(a - b)/2],

cosa- Cosb = 2Sin[(a + b)/2] Sin[(a -b)/2].

Functions Sin2α And Cos2α expressed through Sinα And Cosα in the following way:

Sin2α = 2Sinα Cosα,

Cos2α = Cos2α - Sin2α.

Due to the ratio

Cos 2 α + Sin 2 α = 1

the last formula takes the following forms;

Cos2a = 1 -2Sin2α or Cos2a = SCos 2α - 1.

Here for brevity it is written Sin2α And Cos 2 a instead of (Sinα) 2 And (Cosα) 2. Trigonometric functions tangent (tg), cotangent (ctg), secant (sec) And cosecant (cosec) are defined as follows:

tan α = Sin α / Cos α, ctg α = Cos α / Sin α,

sec α = 1/Cos α, cosec α = 1/Sin α

We note some properties of the tangent.

tg(α + β) = (tg α + tg β)/(1 -tan α tan β)

tg2 α = (2tg α)/(1 - tg2α)

tg α /2 = Sin α /(1 + Cos α) = (1 - Cos α)/Sin α

Functions inverse to trigonometric called. circular: arcsine (arc Sin), arccosine (arc Cos), arctangent (arc tg), arccotangent (arc ctg), arcsecant (arc sec) and arccosecant (arc cosec). If eg. tgα = a, That α = arctga. Because given number a corresponds to many different α , then, for greater certainty, we agreed under arc-tga understand the number lying in the interval (- π /2, π /2). In this interval, the tangent can have any value. Similarly, it is assumed that the numbers arc Sina, arc ctga And arc coseca lie between - π /2 And π /2, and the numbers arc cosa And arc seca between ABOUT And π . Trigonometric functions are very important: they are found in so many questions of analysis and geometry. Since calculations are facilitated with the help of logarithms, not the most trigonometric functions are placed in the tables, but their logarithms (see). The angles in the tables are not expressed in numbers, but in degrees. If this angle is α , then it contains 180 α / π degrees; 60th part of a degree minute and the 60th part of a minute is second. Trigonometric tables are calculated using series (see).

The ratios between the sides and angles of a rectilinear triangle (see) are expressed by the following formulas. If we denote the angles of a triangle by A, IN And WITH, and opposite sides through a, b And With, then we get

A + B + C = π,

SinA/a = SmB/b = SinC/c

a 2 \u003d b 2 + c 2 - 2bс.CosA,

a = b.CosC + c.CosB,

tg[(A - Β)/2] = [(a - b)/(a + b)]Ctg(C/2)

If the perimeter of the triangle, i.e. a + b + c denote for brevity by 2p, then we get

In these formulas, the square root has a positive value. If s denotes the area of ​​a triangle, then s = 1/2(ab).Sinc or s = √.

If R the radius of a circle circumscribed about a triangle, and r is the radius of the inscribed circle, then

R = a/(2SinA) = (abc)/(4s) And r = s/p.

From the above formulas, others can be deduced by rearranging the letters. For example, from the formula

A 2 = b 2 + c 2 - 2bс.CosA

b 2 \u003d a 2 + c 2 - 2ac. Cosb.

With the help of these formulas, according to the given parts of the triangle, the remaining parts of it are calculated. A similar task called solving triangles, occurs in many practical issues: in geodetic surveys, in determining heights, in finding the distance between impregnable points, etc.

We now turn to spherical triangles. The solution of these triangles is the subject spherical trigonometry. Suppose that on the surface of a ball of radius R a triangle is drawn, the vertices of which are A, B And WITH. Connecting the center of the ball ABOUT with dots A, B And WITH, we get a trihedral angle containing three plane angles and three dihedral angles. Quantities dihedral angles, whose edges are OA, OV And OS, denote by A, B And WITH, and the values ​​of the opposite plane angles through a, b And With. We will assume that six numbers A, B, C, a, b, c expressed in degrees, and that none of them exceeds 180°. Between these numbers, the following main relationships take place:

Cosa = Cosb.Cosс + Sinb. Sins. CosA,

SinA/Sina = SinB/Sinb = SinC/Sinc

Cosa.Sinb- Sina.Cosb.CosC = Sinc.CosA,

Cosa.SinB- Cosb.CosC.SinA = CosA.Sin C,

Ctga. Sinb- CtgA.SinC = Cosb.CosC,

CosA = - CosB.CosC + SinB.SinC.Cosa.

If a + b + c = 2p, That

The sum of the angles of a spherical triangle contains more than 180°. Number A + B + C -180° called spherical excess of this triangle and is denoted by the letter ε . To determine the number of degrees contained in one of the sides of a spherical triangle, the angles of which are given, the formulas are

The area of ​​a spherical triangle is (π /180) ε.R 2, Where R ball radius.

Luillier's formula (l "Huillier) makes it possible to calculate the spherical excess on the sides of a triangle.

Let us also point out the Delambre formulas:

Sin[(A + B)/2]:Cos = Cos[(a -b)/2]:Cos

Sin[(A - B)/2]:Cos = Sin[(a -b)/2]:Sin

Cos[(A + B)/2]:Sin = Cos[(a + b)/2]:Cos

Cos[(A - B)/2]:Sin = Sin[(a + b)/2]:Sin

and on Napier's formulas:

tg[(A + B)/2] = (ctg)(Cos[(a -b)/2]/Cos[(a + b)/2])

tg[(A - B)/2] = (ctg)(Sin[(a -b)/2]/Sin[(a + b)/2])

tg[(a + b)/2] = (tg)(Cos[(A - B)/2]/Cos[(A + B)/2])

tg[(a - b)/2] = (tg)(Sin[(A -B)/2]/Sin[(A + B)/2]) From the above formulas, we obtain new ones by rearranging the letters.

The formulas for spherical thermometers are very often used in astronomy.

Without listing trigonometry textbooks, we point to J. A. Serret, "Trait é de Trigonomé trie". Information on the history of T. can be found in the work: Moritz Cantor, "Vorlesungen ü ber Geschichte der Mathematik", brought up to 1759 (before the year of Lagrange's birth). In addition, in 1900 the first part of the work appeared: A. von Braunm ühl, "Vorlesungen ü ber Geschichte der Trigonometrie", in which the history of T. was brought to half of XVII table. (before the invention of logarithms).

D.S.

Russian language dictionaries

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Usually, when they want to scare someone with TERRIBLE MATH, all sorts of sines and cosines are cited as an example, as something very complex and nasty. But really, it's beautiful and interesting section which can be understood and solved.
The topic begins to take place in the 9th grade and everything is not always clear the first time, there are many subtleties and tricks. I tried to say something on the topic.

Introduction to the world of trigonometry:
Before throwing headlong into formulas, you need to understand from geometry what sine, cosine, etc. are.
Sine of an angle- the ratio of the opposite (angle) side to the hypotenuse.
Cosine is the ratio of the adjacent to the hypotenuse.
Tangent- opposite side in adjacent side
Cotangent- adjacent to the opposite.

Now consider a circle of unit radius on the coordinate plane and mark some angle alpha on it: (pictures are clickable, at least some of them)
-
-
Thin red lines are the perpendicular from the point of intersection of the circle and the right angle on the x and y axes. The red x and y are the value of the x and y coordinates on the axes (the gray x and y are just to indicate that these are coordinate axes and not just lines).
It should be noted that the angles are counted from the positive direction of the x-axis counterclockwise.
We find for it the sine, cosine, and so on.
sin a: opposite side is y, hypotenuse is 1.
sin a = y / 1 = y
To make it completely clear where I get y and 1 from, for clarity, let's arrange the letters and consider triangles.
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AF = AE = 1 - radius of the circle.
Therefore, AB = 1, as a radius. AB is the hypotenuse.
BD = CA = y - as value for oh.
AD \u003d CB \u003d x - as a value for oh.
sin a = BD / AB = y / 1 = y
Further cosine:
cos a: adjacent side - AD = x
cos a = AD / AB = x / 1 = x

We also deduce tangent and cotangent.
tg a = y / x = sin a / cos a
ctg a = x / y = cos a / sin a
Already suddenly we have derived the formula of tangent and cotangent.

Well, let's take a look at how it is solved with specific angles.
For example, a = 45 degrees.
We get right triangle at one 45 degree angle. It is immediately clear to someone that this is a triangle with different sides, but I will sign it anyway.
Find the third corner of the triangle (first 90, second 5): b = 180 - 90 - 45 = 45
If two angles are equal, then the sides are equal, as it sounded like.
So, it turns out that if we add two such triangles on top of each other, we get a square with a diagonal equal to radius \u003d 1. By the Pythagorean theorem, we know that the diagonal of a square with side a is equal to the roots of two.
Now we think. If 1 (the hypotenuse aka the diagonal) is equal to the side of the square times the square root of 2, then the side of the square must equal 1/sqrt(2), and if we multiply the numerator and denominator of that fraction by the root of 2, we get sqrt(2)/2 . And since the triangle is isosceles, then AD = AC => x = y
Finding our trigonometric functions:
sin 45 = sqrt(2)/2 / 1 = sqrt(2)/2
cos 45 = sqrt(2)/2 / 1 = sqrt(2)/2
tg 45 = sqrt(2)/2 / sqrt(2)/2 = 1
ctg 45 = sqrt(2)/2 / sqrt(2)/2 = 1
With the rest of the angles, you need to work in the same way. Only the triangles will not be isosceles, but the sides are just as easy to find using the Pythagorean theorem.
In this way, we get a table of values ​​​​of trigonometric functions from different angles:
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Moreover, this table is cheating and very convenient.
How to make it yourself without any hassle: you draw such a table and write the numbers 1 2 3 in the cells.
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Now from these 1 2 3 you extract the root and divide by 2. It turns out like this:
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Now we cross out the sine and write the cosine. Its values ​​are the mirrored sine:
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It is just as easy to derive the tangent - you need to divide the value of the sine line by the value of the cosine line:
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The value of the cotangent is the inverted value of the tangent. As a result, we get something like this:
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note that the tangent does not exist in P/2, for example. Think why. (You can't divide by zero.)

What to remember here: sine is the y value, cosine is the x value. The tangent is the ratio of y to x, and the cotangent is the other way around. so, in order to determine the values ​​of sines / cosines, it is enough to draw a plate, which I described above and a circle with coordinate axes (it is convenient to look at the values ​​\u200b\u200bat angles 0, 90, 180, 360).
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Well, I hope you can tell quarters:
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The sign of its sine, cosine, etc. depends on which quarter the angle is in. Although, absolutely primitive logical thinking will lead you to the correct answer if you take into account that x is negative in the second and third quarters, and y is negative in the third and fourth. Nothing terrible or frightening.

I think it would not be superfluous to mention reduction formulas ala ghosts, as everyone hears, which has a grain of truth. There are no formulas as such, for uselessness. The very meaning of all this action: We easily find the values ​​of the angles only for the first quarter (30 degrees, 45, 60). Trigonometric functions are periodic, so we can drag any large angle to the first quadrant. Then we will immediately find its meaning. But just dragging is not enough - you need to remember about the sign. That's what casting formulas are for.
So, we have a large angle, or rather more than 90 degrees: a \u003d 120. And you need to find its sine and cosine. To do this, we decompose 120 into such angles that we can work with:
sin a = sin 120 = sin (90 + 30)
We see that this angle lies in the second quarter, the sine is positive there, therefore the + sign in front of the sine is preserved.
To get rid of 90 degrees, we change the sine to cosine. Well, here's a rule to remember:
sin (90 + 30) = cos 30 = sqrt(3) / 2
And you can imagine it in another way:
sin 120 = sin (180 - 60)
To get rid of 180 degrees, we do not change the function.
sin (180 - 60) = sin 60 = sqrt(3) / 2
We got the same value, so everything is correct. Now cosine:
cos 120 = cos (90 + 30)
The cosine in the second quarter is negative, so we put a minus sign. And we change the function to the opposite, since we need to remove 90 degrees.
cos (90 + 30) = - sin 30 = - 1 / 2
Or:
cos 120 = cos (180 - 60) = - cos 60 = - 1 / 2

What you need to know, be able to do and do in order to translate corners in the first quarter:
-decompose the angle into digestible terms;
- take into account in which quarter the angle is located, and put the appropriate sign if the function in this quarter is negative or positive;
-get rid of excess
*if you need to get rid of 90, 270, 450 and the rest 90+180n, where n is any integer, then the function is reversed (sine to cosine, tangent to cotangent and vice versa);
*if you need to get rid of 180 and the remaining 180+180n, where n is any integer, then the function does not change. (There is one feature here, but it is difficult to explain it in words, well, okay).
That's all. I do not consider it necessary to memorize the formulas themselves, when you can remember a couple of rules and use them easily. By the way, these formulas are very easy to prove:
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And they make up bulky tables, then we know:
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Basic trigonometry equations: they need to be known very, very well, by heart.
Basic trigonometric identity(equality):
sin^2(a) + cos^2(a) = 1
If you don't believe me, check it out yourself and see for yourself. Substitute the values ​​of the different angles.
This formula is very, very useful, always remember it. with it, you can express the sine through the cosine and vice versa, which is sometimes very useful. But, like with any other formula, you need to be able to handle it. Always remember that the sign of the trigonometric function depends on the quarter in which the angle is located. That's why when extracting the root, you need to know a quarter.

Tangent and cotangent: we have already derived these formulas at the very beginning.
tg a = sin a / cos a
ctg a = cos a / sin a

Product of tangent and cotangent:
tg a * ctg a = 1
Because:
tg a * ctg a = (sin a / cos a) * (cos a / sin a) = 1 - fractions cancel.

As you can see, all formulas are a game and a combination.
Here are two more, obtained by dividing by the cosine square and sine square of the first formula:
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Please note that the last two formulas can be used with a restriction on the value of the angle a, since you cannot divide by zero.

Addition formulas: are proved using vector algebra.
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They are used rarely, but aptly. There are formulas on the scan, but it may be illegible or the digital form is easier to perceive:
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Double angle formulas:
They are obtained based on addition formulas, for example: the cosine of a double angle is cos 2a = cos (a + a) - does it remind you of anything? They just replaced beta with alpha.
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The two following formulas are derived from the first substitution sin^2(a) = 1 - cos^2(a) and cos^2(a) = 1 - sin^2(a).
With the sine of a double angle, it is simpler and is used much more often:
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And special perverts can derive the tangent and cotangent of a double angle, given that tg a \u003d sin a / cos a, and so on.
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For the above persons Triple angle formulas: they are derived by adding the angles 2a and a, since we already know the formulas for the double angle.
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Half angle formulas:
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I don’t know how they are derived, or rather how to explain it ... If you write these formulas, substituting the basic trigonometric identity with a / 2, then the answer will converge.

Formulas for adding and subtracting trigonometric functions:
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They are obtained from addition formulas, but no one cares. Meet not often.

As you understand, there are still a bunch of formulas, listing which is simply meaningless, because I won’t be able to write something adequate about them, and dry formulas can be found anywhere, and they are a game with the previous existing formulas. Everything is terribly logical and accurate. I'll just tell you last about the auxiliary angle method:
The transformation of the expression a cosx + b sinx to the form Acos(x+) or Asin(x+) is called the method of introducing an auxiliary angle (or additional argument). The method is applied when solving trigonometric equations, when estimating the values ​​of functions, in extremum problems, and what is important to note, some problems cannot be solved without introducing an auxiliary angle.
As you, I did not try to explain this method, nothing came of it, so you have to do it yourself:
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It's scary, but useful. If you solve problems, it should work.
From here for example: mschool.kubsu.ru/cdo/shabitur/kniga/trigonom/metod/metod2/met2/met2.htm

Next on the course are graphs of trigonometric functions. But one lesson is enough. Considering that this is taught at school for six months.

Write your questions, solve problems, ask for scans of some tasks, figure it out, try it.
Always yours, Dan Faraday.

This note is methodical in nature and is intended to remind (or teach :)) what is random walk and what is its role in stock trading. Random walk (or Brownian motion or random walk) is a process with independent increments, with each increment having zero mean. An example of such a process: we take a coin and throw it. If heads, then the next increment is +1, if tails, the next increment is -1. We throw many times and summarize the cumulative total. In general, you can't imagine it easier.
Despite the simplicity of such a construction, it has an extremely important role to understand the dynamics of prices on the stock exchange. Let's take a look at the random walk graph:

This picture is quite typical. As you can see, there are many of the favorite attributes of technical analysis here - levels, patterns, trends, etc. And in general, the picture is clearly similar to the real prices. So the random walk is clearly not a bad market model.

Since we found such a successful mathematical model real life, then it would be nice to discuss the properties of the model. The main properties are:
1) You can't make money on a random walk. No methods, including money management and risk management. This is due to the fact that this process has no memory - each next increment is in no way connected with the previous one.
2) A random walk with a probability tending to 1 will reach any predetermined level, even a million, even a billion. This, on average, occurs in a time proportional to the square of the magnitude of the level.
Already from property 1) it follows that fans of indiscriminate use of technical analysis do not understand what they are doing. And even if they earn, they do not know why something is bad. I am not against technical analysis, but the reasons why it sometimes works are quite non-trivial.
It follows from property 2) that the market can go a hell of a lot for no reason at all - hello option sellers and traders without stops.
Now let's answer the question - why is the market so similar to a random walk? There are two reasons:
1) Just a continuous stream of limit and market orders, each unrelated to any other, will cause the price to randomly wander.
2) Traders, as a rule, look for patterns in price (that is, price deviations from a random walk). And if they find it, they start trading near this pattern. Then a non-trivial evolution takes place, which I will not explain here, but as a result of this evolution, sooner or later, the regularity will cease to exist. That is why successful traders do not like to just share their trading systems.
Finally, let's discuss the philosophical aspects of the model. The random walk model is just a mathematical model. And the real market is a set of people. And, of course, if we knew everything about all traders, then we would not need any random walk model at all - for us, each price movement would not be random, but completely understandable. But you can’t know everything about everyone, but something about some is easy. And any good trading system is, first of all, knowledge of a certain peculiarity of the behavior of some traders in the market.

Application: Random Walk Generation in Excel
To generate a random walk in Excel, you can use, for example, the following code:

Option Explicit
Sub Rand_Walk()
Dim x As Single, s As Single
Dim i As Integer, imax As Integer
imax = 10000
s = 0
For i = 1 To imax
Randomize
x = Rnd()
x = 2 * x - 1
s = s + x
Cells(i, 1) = i
Cells(i, 2) = s
Next i
end sub

It must be copied into the code of any Excel sheet. Run and build a graph on the first two columns of the sheet. After that, you can admire quasi-stock quotes.