Lessons: Trigonometry. Trigonometry made simple and clear Learning trigonometry

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1. Introduction.

Approaching the school, I hear the voices of the guys from the gym, I move on - they sing, draw... emotions and feelings are everywhere. My office, algebra lesson, tenth graders. Here is our textbook, in which the trigonometry course makes up half of its volume, and there are two bookmarks in it - these are the places where I found words that are not related to the theory of trigonometry.

Among the few are students who love mathematics, feel its beauty and do not ask why it is necessary to study trigonometry, where is the material learned applied? The majority are those who simply complete assignments so as not to get a bad grade. And we firmly believe that the applied value of mathematics is to gain knowledge sufficient to successfully pass the Unified State Exam and enter a university (enroll and forget).

The main goal of the presented lesson is to show the applied value of trigonometry in various fields of human activity. The examples given will help students see the connection between this section of mathematics and other subjects studied at school. The content of this lesson is an element of professional training for students.

Tell something new about a seemingly long-known fact. Show a logical connection between what we already know and what remains to be learned. Open the door a little and look beyond the school curriculum. Unusual tasks, connections with today's events - these are the techniques that I use to achieve my goals. After all, school mathematics as a subject contributes not so much to learning as to the development of the individual, his thinking, and culture.

2. Lesson summary on algebra and principles of analysis (grade 10).

Organizing time: Arrange six tables in a semicircle (protractor model), worksheets for students on the tables (Annex 1).

Announcing the topic of the lesson: “Trigonometry is simple and clear.”

In the course of algebra and elementary analysis, we begin to study trigonometry; I would like to talk about the applied significance of this section of mathematics.

Lesson thesis:

“The great book of nature can only be read by those who know the language in which it is written, and that language is mathematics.”
(G. Galileo).

At the end of the lesson, we will think together whether we were able to look into this book and understand the language in which it was written.

Trigonometry of an acute angle.

Trigonometry is a Greek word and translated means “measurement of triangles.” The emergence of trigonometry is associated with measurements on earth, construction, and astronomy. And your first acquaintance with it happened when you picked up a protractor. Have you noticed how the tables are positioned? Think about it in your mind: if we take one table as a chord, then what is the degree measure of the arc that it subtends?

Let's remember the measure of angles: 1 ° = 1/360 part of a circle (“degree” - from the Latin grad - step). Do you know why the circle was divided into 360 parts, why not divided into 10, 100 or 1000 parts, as happens, for example, when measuring lengths? I'll tell you one of the versions.

Previously, people believed that the Earth is the center of the Universe and it is motionless, and the Sun makes one revolution around the Earth per day, the geocentric system of the world, “geo” - Earth ( Figure No. 1). Babylonian priests who carried out astronomical observations discovered that on the day of the equinox the Sun, from sunrise to sunset, describes a semicircle in the vault of heaven, in which the visible diameter (diameter) of the Sun fits exactly 180 times, 1 ° - trace of the Sun. ( Figure No. 2).

For a long time, trigonometry was purely geometric in nature. In you continue your introduction to trigonometry by solving right triangles. You learn that the sine of an acute angle of a right triangle is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, tangent is the ratio of the opposite side to the adjacent side and cotangent is the ratio of the adjacent side to the opposite. And remember that in a right triangle having a given angle, the ratio of the sides does not depend on the size of the triangle. Learn the sine and cosine theorems for solving arbitrary triangles.

In 2010, the Moscow metro turned 75 years old. Every day we go down to the subway and don’t notice that...

Task No. 1. The inclination angle of all escalators in the Moscow metro is 30 degrees. Knowing this, the number of lamps on the escalator and the approximate distance between the lamps, you can calculate the approximate depth of the station. There are 15 lamps on the escalator at the Tsvetnoy Boulevard station, and 2 lamps at the Prazhskaya station. Calculate the depth of these stations if the distances between the lamps, from the escalator entrance to the first lamp and from the last lamp to the escalator exit, are 6 m ( Figure No. 3). Answer: 48 m and 9 m

Homework. The deepest station of the Moscow metro is Victory Park. What is its depth? I suggest you independently find the missing data to solve your homework problem.

I have a laser pointer in my hands, which is also a range finder. Let's measure, for example, the distance to the board.

Chinese designer Huan Qiaokun guessed to combine two laser rangefinders and a protractor into one device and obtained a tool that allows you to determine the distance between two points on a plane ( Figure No. 4). What theorem do you think solves this problem? Remember the formulation of the cosine theorem. Do you agree with me that your knowledge is already sufficient to make such an invention? Solve geometry problems and make small discoveries every day!

Spherical trigonometry.

In addition to the flat geometry of Euclid (planimetry), there may be other geometries in which the properties of figures are considered not on a plane, but on other surfaces, for example, on the surface of a ball ( Figure No. 5). The first mathematician who laid the foundation for the development of non-Euclidean geometries was N.I. Lobachevsky – “Copernicus of Geometry”. From 1827 for 19 years he was the rector of Kazan University.

Spherical trigonometry, which is part of spherical geometry, considers the relationships between the sides and angles of triangles on a sphere formed by arcs of great circles on a sphere ( Figure No. 6).

Historically, spherical trigonometry and geometry arose from the needs of astronomy, geodesy, navigation, and cartography. Think about which of these areas has received such rapid development in recent years that its results are already being used in modern communicators. ... A modern application of navigation is a satellite navigation system, which allows you to determine the location and speed of an object from a signal from its receiver.

Global Navigation System (GPS). To determine the latitude and longitude of the receiver, it is necessary to receive signals from at least three satellites. Receiving a signal from the fourth satellite makes it possible to determine the height of the object above the surface ( Figure No. 7).

The receiver computer solves four equations in four unknowns until a solution is found that draws all the circles through one point ( Figure No. 8).

Knowledge of acute angle trigonometry turned out to be insufficient for solving more complex practical problems. When studying rotational and circular movements, the value of the angle and circular arc are not limited. The need arose to move to the trigonometry of a generalized argument.

Trigonometry of a generalized argument.

The circle ( Figure No. 9). Positive angles are plotted counterclockwise, negative angles are plotted clockwise. Are you familiar with the history of such an agreement?

As you know, mechanical and sun watches are designed in such a way that their hands rotate “along the sun,” i.e. in the same direction in which we see the apparent movement of the Sun around the Earth. (Remember the beginning of the lesson - the geocentric system of the world). But with the discovery by Copernicus of the true (positive) motion of the Earth around the Sun, the motion of the Sun around the Earth that we see (i.e., apparent) is fictitious (negative). Heliocentric system of the world (helio - Sun) ( Figure No. 10).

Warm-up.

Extend your right arm in front of you, parallel to the surface of the table, and perform a circular rotation of 720 degrees.
Extend your left arm in front of you, parallel to the surface of the table, and perform a circular rotation of (–1080) degrees.
Place your hands on your shoulders and make 4 circular movements back and forth. What is the sum of the rotation angles?

In 2010, the Winter Olympic Games were held in Vancouver; we learn the criteria for grading a skater’s exercise performed by solving the problem.

Task No. 2. If a skater makes a 10,800-degree turn while performing the “screw” exercise in 12 seconds, then he receives an “excellent” rating. Determine how many revolutions the skater will make during this time and the speed of his rotation (revolutions per second). Answer: 2.5 revolutions/sec.

Homework. At what angle does the skater turn, who received an “unsatisfactory” rating, if at the same rotation time his speed was 2 revolutions per second.

The most convenient measure of arcs and angles associated with rotational movements turned out to be the radian (radius) measure, as a larger unit of measurement of an angle or arc ( Figure No. 11). This measure of measuring angles entered science through the remarkable works of Leonhard Euler. Swiss by birth, he lived in Russia for 30 years and was a member of the St. Petersburg Academy of Sciences. It is to him that we owe the “analytical” interpretation of all trigonometry, he derived the formulas that you are now studying, introduced uniform signs: sin x,cos x, tg x,ctg x.

If until the 17th century the development of the doctrine of trigonometric functions was built on a geometric basis, then, starting from the 17th century, trigonometric functions began to be applied to solving problems in mechanics, optics, electricity, to describe oscillatory processes and wave propagation. Wherever we have to deal with periodic processes and oscillations, trigonometric functions have found application. Functions expressing the laws of periodic processes have a special property inherent only to them: they repeat their values through the same interval of change in argument. Changes in any function are most clearly conveyed on its graph ( Figure No. 12).

We have already turned to our body for help when solving problems involving rotation. Let's listen to our heartbeat. The heart is an independent organ. The brain controls any of our muscles except the heart. It has its own control center - the sinus node. With each contraction of the heart, an electric current spreads throughout the body - starting from the sinus node (the size of a millet grain). It can be recorded using an electrocardiograph. He draws an electrocardiogram (sinusoid) ( Figure No. 13).

Now let's talk about music. Mathematics is music, it is a union of intelligence and beauty.
Music is mathematics in calculation, algebra in abstraction, trigonometry in beauty. Harmonic oscillation (harmonic) is a sinusoidal oscillation. The graph shows how the air pressure on the listener's eardrum changes: up and down in an arc, periodically. The air presses, now stronger, now weaker. The force of impact is very small and vibrations occur very quickly: hundreds and thousands of shocks every second. We perceive such periodic vibrations as sound. The addition of two different harmonics gives a vibration of a more complex shape. The sum of three harmonics is even more complex, and natural sounds and sounds of musical instruments are made up of a large number of harmonics. ( Figure No. 14.)

Each harmonic is characterized by three parameters: amplitude, frequency and phase. The oscillation frequency shows how many shocks of air pressure occur in one second. High frequencies are perceived as “high”, “thin” sounds. Above 10 KHz – squeak, whistle. Small frequencies are perceived as “low”, “bass” sounds, rumble. Amplitude is the range of vibrations. The larger the scope, the greater the impact on the eardrum, and the louder the sound we hear ( Figure No. 15). Phase is the displacement of oscillations in time. Phase can be measured in degrees or radians. Depending on the phase, the zero point on the graph shifts. To set a harmonic, it is enough to specify the phase from –180 to +180 degrees, since at large values the oscillation is repeated. Two sinusoidal signals with the same amplitude and frequency, but different phases, are added algebraically ( Figure No. 16).

Lesson summary. Do you think we were able to read a few pages from the Great Book of Nature? Having learned about the applied significance of trigonometry, did its role in various spheres of human activity become clearer to you, did you understand the material presented? Then remember and list the areas of application of trigonometry that you met today or knew before. I hope that each of you found something new and interesting in today's lesson. Perhaps this new thing will tell you the way in choosing a future profession, but no matter who you become, your mathematical education will help you become a professional and an intellectually developed person.

Homework. Read the lesson summary (

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Usually, when they want to scare someone with SCARY MATHEMATICS, they cite all sorts of sines and cosines as an example, as something very complex and disgusting. But in fact, this is a beautiful and interesting section that can be understood and solved.
The topic begins in 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 rushing headlong into formulas, you need to understand from geometry what sine, cosine, etc. are.
Sine of angle- the ratio of the opposite (angle) side to the hypotenuse.
Cosine- the ratio of adjacent to hypotenuse.
Tangent- opposite side to 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)
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Thin red lines are the perpendicular from the intersection point of the circle and the right angle on the ox and oy axis. The red x and y are the value of the x and y coordinate 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 calculated from the positive direction of the ox axis counterclockwise.
Let's find the sine, cosine, etc. for it.
sin a: opposite side is equal to y, hypotenuse is equal to 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 look at the triangles.
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AF = AE = 1 - radius of the circle.
Therefore AB = 1 as the radius. AB - hypotenuse.
BD = CA = y - as the value for oh.
AD = CB = x - as the value according to oh.
sin a = BD / AB = y / 1 = y
Next is the cosine:
cos a: adjacent side - AD = x
cos a = AD / AB = x / 1 = x

We also output tangent and cotangent.
tg a = y / x = sin a / cos a
cot a = x / y = cos a / sin a
Suddenly we have derived the formula for tangent and cotangent.

Well, let's take a concrete look at how this is solved.
For example, a = 45 degrees.
We get a right triangle with one angle of 45 degrees. It’s immediately clear to some that this is an equilateral triangle, but I’ll describe it anyway.
Let's find the third angle of the triangle (the first is 90, the second is 5): b = 180 - 90 - 45 = 45
If two angles are equal, then their sides are equal, that’s what 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 = 1. By the Pythagorean theorem, we know that the diagonal of a square with side a is equal to a roots of two.
Now we think. If 1 (the hypotenuse aka diagonal) is equal to the side of the square times the root of two, then the side of the square should be equal to 1/sqrt(2), and if we multiply the numerator and denominator of this fraction by the root of two, 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
You need to work with the other angle values in the same way. Only the triangles will not be isosceles, but the sides can be found just as easily using the Pythagorean theorem.
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 compose it yourself without any hassle: Draw a table like this and write the numbers 1 2 3 in the boxes.
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Now from these 1 2 3 you take 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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The tangent is just as easy to derive - you need to divide the value of the sine line by the value of the cosine line:
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The cotangent value is the inverted value of the tangent. As a result, we get something like this:
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note that tangent does not exist in P/2, for example. Think about why. (You cannot divide by zero.)

What you need to remember here: sine is the y value, cosine is the x value. Tangent is the ratio of y to x, and cotangent is the opposite. so, to determine the values of sines/cosines, it is enough to draw the table that I described above and a circle with coordinate axes (it is convenient to look at the values at angles of 0, 90, 180, 360).
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Well, I hope that you can distinguish 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 in the second and third quarters x is negative, and y is negative in the third and fourth. Nothing scary or scary.

I think it wouldn’t be amiss to mention reduction formulas ala ghosts, as everyone hears, which has a grain of truth. There are no formulas as such, as they are unnecessary. The very meaning of this whole action: We easily find the angle values only for the first quarter (30 degrees, 45, 60). Trigonometric functions are periodic, so we can drag any large angle into the first quarter. Then we will immediately find its meaning. But simply dragging is not enough - you need to remember about the sign. This is what reduction formulas are for.
So, we have a large angle, or rather more than 90 degrees: a = 120. And we need to find its sine and cosine. To do this, we will decompose 120 into the following 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 there is positive, therefore the + sign in front of the sine is preserved.
To get rid of 90 degrees, we change the sine to cosine. Well, this is a rule you need to remember:
sin (90 + 30) = cos 30 = sqrt(3) / 2
Or you can imagine it 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 the 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 one, 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 to transfer angles to the first quarter:
- decompose the angle into digestible terms;
-take into account which quarter the angle is in and put the appropriate sign if the function in this quarter is negative or positive;
-get rid of unnecessary things:
*if you need to get rid of 90, 270, 450 and the remaining 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’s difficult to explain in words, but oh well).
That's all. I don’t think it’s 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 also compile cumbersome tables, then we know:
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Basic equations of trigonometry: you need to know them very, very well, by heart.
Fundamental trigonometric identity(equality):
sin^2(a) + cos^2(a) = 1
If you don't believe it, it's better to check it yourself and see for yourself. Substitute the values of different angles.
This formula is very, very useful, always remember it. using it you can express sine through cosine and vice versa, which is sometimes very useful. But, like any other formula, you need to know how to handle it. Always remember that the sign of the trigonometric function depends on the quadrant in which the angle is located. That's why when extracting the root you need to know the quarter.

Tangent and cotangent: We already derived these formulas at the very beginning.
tg a = sin a / cos a
cot 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 are cancelled.

As you can see, all formulas are a game and a combination.
Here are two more, obtained from 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 limitation on the value of angle a, since you cannot divide by zero.

Addition formulas: are proven using vector algebra.
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Rarely used, but accurately. There are formulas in the scan, but they 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 the betta with an alpha.
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The two subsequent formulas are derived from the first substitution sin^2(a) = 1 - cos^2(a) and cos^2(a) = 1 - sin^2(a).
The sine of a double angle 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 tan a = sin a / cos a, etc.
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For the above mentioned persons Triple angle formulas: they are derived by adding angles 2a and a, since we already know the formulas for double angles.
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Half angle formulas:
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I don’t know how they are derived, or more accurately, how to explain it... If we write out these formulas, substituting the main trigonometric identity with a/2, then the answer will converge.

Formulas for addition and subtraction of trigonometric functions:
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They are obtained from addition formulas, but no one cares. They don't happen often.

As you understand, there are still a bunch of formulas, listing which is simply pointless, 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 previous existing formulas. Everything is terribly logical and precise. I'll just tell you lastly about the auxiliary angle method:
Converting the expression a cosx + b sinx to the form Acos(x+) or Asin(x+) is called the method of introducing an auxiliary angle (or an additional argument). The method is used when solving trigonometric equations, when estimating the values of functions, in extremum problems, and it is important to note that some problems cannot be solved without introducing an auxiliary angle.
No matter how you tried to explain this method, nothing came of it, so you’ll have to do it yourself:
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A scary thing, but useful. If you solve the problems, it should work out.
From here, for example: mschool.kubsu.ru/cdo/shabitur/kniga/trigonom/metod/metod2/met2/met2.htm

Next in the course are graphs of trigonometric functions. But that's enough for one lesson. Considering that at school they teach this for six months.

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

Sine, cosine, tangent - when pronouncing these words in the presence of high school students, you can be sure that two thirds of them will lose interest in further conversation. The reason lies in the fact that the basics of trigonometry at school are taught in complete isolation from reality, and therefore students do not see the point in studying formulas and theorems.

In fact, upon closer examination, this area of knowledge turns out to be very interesting, as well as applied - trigonometry is used in astronomy, construction, physics, music and many other fields.

Let's get acquainted with the basic concepts and name several reasons to study this branch of mathematical science.

Story

It is unknown at what point in time humanity began to create the future trigonometry from scratch. However, it is documented that already in the second millennium BC, the Egyptians were familiar with the basics of this science: archaeologists found a papyrus with a task in which it was required to find the angle of inclination of the pyramid on two known sides.

The scientists of Ancient Babylon achieved more serious successes. Over the centuries, studying astronomy, they mastered a number of theorems, introduced special methods for measuring angles, which, by the way, we use today: degrees, minutes and seconds were borrowed by European science in the Greco-Roman culture, into which these units came from the Babylonians.

It is assumed that the famous Pythagorean theorem, relating to the basics of trigonometry, was known to the Babylonians almost four thousand years ago.

Name

Literally, the term “trigonometry” can be translated as “measurement of triangles.” The main object of study within this section of science for many centuries was the right triangle, or more precisely, the relationship between the magnitudes of the angles and the lengths of its sides (today, the study of trigonometry from scratch begins with this section). There are often situations in life when it is practically impossible to measure all the required parameters of an object (or the distance to the object), and then it becomes necessary to obtain the missing data through calculations.

For example, in the past, people could not measure the distance to space objects, but attempts to calculate these distances occurred long before the advent of our era. Trigonometry also played a crucial role in navigation: with some knowledge, the captain could always navigate by the stars at night and adjust the course.

Basic Concepts

Mastering trigonometry from scratch requires understanding and remembering several basic terms.

The sine of a certain angle is the ratio of the opposite side to the hypotenuse. Let us clarify that the opposite leg is the side lying opposite the angle we are considering. Thus, if an angle is 30 degrees, the sine of this angle will always, for any size of the triangle, be equal to ½. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.

Tangent is the ratio of the opposite side to the adjacent side (or, which is the same, the ratio of sine to cosine). Cotangent is the unit divided by the tangent.

It is worth mentioning the famous number Pi (3.14...), which is half the length of a circle with a radius of one unit.

Popular mistakes

People learning trigonometry from scratch make a number of mistakes - mostly due to inattention.

First, when solving geometry problems, you must remember that the use of sines and cosines is only possible in a right triangle. It happens that a student “automatically” takes the longest side of a triangle as the hypotenuse and gets incorrect calculation results.

Secondly, at first it is easy to confuse the values of sine and cosine for the selected angle: recall that the sine of 30 degrees is numerically equal to the cosine of 60, and vice versa. If you substitute an incorrect number, all further calculations will be incorrect.

Thirdly, until the problem is completely solved, you should not round any values, extract roots, or write a common fraction as a decimal. Often students strive to get a “beautiful” number in a trigonometry problem and immediately extract the root of three, although after exactly one action this root can be reduced.

Etymology of the word "sine"

The history of the word “sine” is truly unusual. The fact is that the literal translation of this word from Latin means “hollow.” This is because the correct understanding of the word was lost during translation from one language to another.

The names of the basic trigonometric functions originate from India, where the concept of sine was denoted by the word “string” in Sanskrit - the fact is that the segment, together with the arc of the circle on which it rested, looked like a bow. During the heyday of Arab civilization, Indian achievements in the field of trigonometry were borrowed, and the term passed into Arabic as a transcription. It so happened that this language already had a similar word denoting a depression, and if the Arabs understood the phonetic difference between the native and borrowed word, then the Europeans, translating scientific treatises into Latin, mistakenly literally translated the Arabic word, which had nothing to do with the concept of sine . We still use it to this day.

Tables of values

There are tables that contain numerical values for sines, cosines and tangents of all possible angles. Below we present data for angles of 0, 30, 45, 60 and 90 degrees, which must be learned as a mandatory section of trigonometry for “dummies”; fortunately, they are quite easy to remember.

If it happens that the numerical value of the sine or cosine of an angle “got out of your head,” there is a way to derive it yourself.

Geometric representation

Let's draw a circle and draw the abscissa and ordinate axes through its center. The abscissa axis is horizontal, the ordinate axis is vertical. They are usually signed as "X" and "Y" respectively. Now we will draw a straight line from the center of the circle so that the angle we need is obtained between it and the X axis. Finally, from the point where the straight line intersects the circle, we drop a perpendicular to the X axis. The length of the resulting segment will be equal to the numerical value of the sine of our angle.

This method is very relevant if you forgot the required value, for example, during an exam, and you don’t have a trigonometry textbook at hand. You won’t get an exact number this way, but you will definitely see the difference between ½ and 1.73/2 (sine and cosine of an angle of 30 degrees).

Application

Some of the first experts to use trigonometry were sailors who had no other reference point on the open sea except the sky above their heads. Today, captains of ships (airplanes and other modes of transport) do not look for the shortest path using the stars, but actively resort to GPS navigation, which would be impossible without the use of trigonometry.

In almost every section of physics, you will find calculations using sines and cosines: be it the application of force in mechanics, calculations of the path of objects in kinematics, vibrations, wave propagation, refraction of light - you simply cannot do without basic trigonometry in the formulas.

Another profession that is unthinkable without trigonometry is a surveyor. Using a theodolite and a level or a more complex device - a tachometer, these people measure the difference in height between different points on the earth's surface.

Repeatability

Trigonometry deals not only with the angles and sides of a triangle, although this is where it began its existence. In all areas where cyclicity is present (biology, medicine, physics, music, etc.) you will encounter a graph whose name is probably familiar to you - this is a sine wave.

Such a graph is a circle unfolded along the time axis and looks like a wave. If you've ever worked with an oscilloscope in physics class, you know what we're talking about. Both the music equalizer and the heart rate monitor use trigonometry formulas in their work.

Finally

When thinking about how to learn trigonometry, most middle and high school students begin to consider it a difficult and impractical science, since they only get acquainted with boring information from a textbook.

As for impracticality, we have already seen that, to one degree or another, the ability to handle sines and tangents is required in almost any field of activity. As for the complexity... Think: if people used this knowledge more than two thousand years ago, when an adult had less knowledge than today's high school student, is it realistic for you personally to study this field of science at a basic level? A few hours of thoughtful practice solving problems - and you will achieve your goal by studying the basic course, the so-called trigonometry for dummies.

Back in 1905, Russian readers could read in William James’s book “Psychology” his reasoning about “why is rote learning such a bad way of learning?”

“Knowledge acquired through simple rote learning is almost inevitably forgotten completely without a trace. On the contrary, mental material, acquired by memory gradually, day after day, in connection with various contexts, associated associatively with other external events and repeatedly subjected to discussion, forms such a system, enters into such a connection with the other aspects of our intellect, is easily restored in memory by a mass of external occasions, which remains a durable acquisition for a long time.”

More than 100 years have passed since then, and these words remain amazingly topical. You become convinced of this every day when working with schoolchildren. The massive gaps in knowledge are so great that it can be argued: the school mathematics course in didactic and psychological terms is not a system, but a kind of device that encourages short-term memory and does not care at all about long-term memory.

Knowing the school mathematics course means mastering the material of each area of mathematics and being able to update any of them at any time. To achieve this, you need to systematically contact each of them, which is sometimes not always possible due to the heavy workload in the lesson.

There is another way of long-term memorization of facts and formulas - these are reference signals.

Trigonometry is one of the large sections of school mathematics, studied in the course of geometry in grades 8 and 9 and in the course of algebra in grade 9, algebra and elementary analysis in grade 10.

The largest volume of material studied in trigonometry falls on the 10th grade. Most of this trigonometry material can be learned and memorized on trigonometric circle(a circle of unit radius with its center at the origin of the rectangular coordinate system). Appendix1.ppt

These are the following trigonometry concepts:

definitions of sine, cosine, tangent and cotangent of an angle;
radian angle measurement;
domain of definition and range of values of trigonometric functions
values of trigonometric functions for some values of the numerical and angular argument;
periodicity of trigonometric functions;
evenness and oddity of trigonometric functions;
increasing and decreasing trigonometric functions;
reduction formulas;
values of inverse trigonometric functions;
solving simple trigonometric equations;
solving simple inequalities;
basic formulas of trigonometry.

Let's consider studying these concepts on the trigonometric circle.

1) Definition of sine, cosine, tangent and cotangent.

After introducing the concept of a trigonometric circle (a circle of unit radius with a center at the origin), the initial radius (the radius of the circle in the direction of the Ox axis), and the angle of rotation, students independently obtain definitions for sine, cosine, tangent and cotangent on a trigonometric circle, using the definitions from the course geometry, that is, considering a right triangle with a hypotenuse equal to 1.

The cosine of an angle is the abscissa of a point on a circle when the initial radius is rotated by a given angle.

The sine of an angle is the ordinate of a point on a circle when the initial radius is rotated by a given angle.

2) Radian measurement of angles on a trigonometric circle.

After introducing the radian measure of an angle (1 radian is the central angle, which corresponds to the length of the arc equal to the length of the radius of the circle), students conclude that the radian measurement of the angle is the numerical value of the angle of rotation on the circle, equal to the length of the corresponding arc when the initial radius is rotated by given angle. .

The trigonometric circle is divided into 12 equal parts by the diameters of the circle. Knowing that the angle is in radians, you can determine the radian measurement for angles that are multiples of .

And radian measurements of angles, multiples, are obtained similarly:

3) Domain of definition and range of values of trigonometric functions.

Will the correspondence between rotation angles and coordinate values of a point on a circle be a function?

Each angle of rotation corresponds to a single point on the circle, which means this correspondence is a function.

Getting the functions

On the trigonometric circle you can see that the domain of definition of functions is the set of all real numbers, and the range of values is .

Let us introduce the concepts of lines of tangents and cotangents on a trigonometric circle.

1) Let Let us introduce an auxiliary straight line parallel to the Oy axis, on which tangents are determined for any numerical argument.

2) Similarly, we obtain a line of cotangents. Let y=1, then . This means that the cotangent values are determined on a straight line parallel to the Ox axis.

On a trigonometric circle you can easily determine the domain of definition and range of values of trigonometric functions:

for tangent -

for cotangent -

4) Values of trigonometric functions on a trigonometric circle.

The leg opposite the angle in is equal to half the hypotenuse, that is, the other leg according to the Pythagorean theorem:

This means that by defining sine, cosine, tangent, cotangent, you can determine values for angles that are multiples or radians. The sine values are determined along the Oy axis, the cosine along the Ox axis, and the tangent and cotangent values can be determined using additional axes parallel to the Oy and Ox axes, respectively.

The tabulated values of sine and cosine are located on the corresponding axes as follows:

Table values of tangent and cotangent -

5) Periodicity of trigonometric functions.

On the trigonometric circle you can see that the values of sine and cosine are repeated every radian, and tangent and cotangent - every radian.

6) Evenness and oddness of trigonometric functions.

This property can be obtained by comparing the values of positive and opposite angles of rotation of trigonometric functions. We get that

This means that cosine is an even function, all other functions are odd.

7) Increasing and decreasing trigonometric functions.

The trigonometric circle shows that the sine function increases and decreases

Reasoning similarly, we obtain the intervals of increasing and decreasing functions of cosine, tangent and cotangent.

8) Reduction formulas.

For the angle we take the smaller value of the angle on the trigonometric circle. All formulas are obtained by comparing the values of trigonometric functions on the legs of selected right triangles.

Algorithm for applying reduction formulas:

1) Determine the sign of the function when rotating through a given angle.

When turning a corner the function is preserved, when rotated by an angle - an integer, odd number, the cofunction (

9) Values of inverse trigonometric functions.

Let us introduce inverse functions for trigonometric functions using the definition of a function.

Each value of sine, cosine, tangent and cotangent on the trigonometric circle corresponds to only one value of the angle of rotation. This means that for a function the domain of definition is , the range of values is - For the function the domain of definition is , the range of values is . Similarly, we obtain the domain of definition and range of values of the inverse functions for cosine and cotangent.

Algorithm for finding the values of inverse trigonometric functions:

1) finding the value of the argument of the inverse trigonometric function on the corresponding axis;

2) finding the angle of rotation of the initial radius, taking into account the range of values of the inverse trigonometric function.

For example:

10) Solving simple equations on a trigonometric circle.

To solve an equation of the form , we find points on the circle whose ordinates are equal and write down the corresponding angles, taking into account the period of the function.

For the equation, we find points on the circle whose abscissas are equal and write down the corresponding angles, taking into account the period of the function.

Similarly for equations of the form The values are determined on the lines of tangents and cotangents and the corresponding angles of rotation are recorded.

All concepts and formulas of trigonometry are learned by the students themselves under the clear guidance of the teacher using a trigonometric circle. In the future, this “circle” will serve as a reference signal or an external factor for them to reproduce in memory the concepts and formulas of trigonometry.

Studying trigonometry on a trigonometric circle helps:

choosing the optimal communication style for a given lesson, organizing educational cooperation;
lesson targets become personally significant for each student;
new material is based on the student’s personal experience of action, thinking, and feeling;
the lesson includes various forms of work and ways of obtaining and assimilating knowledge; there are elements of mutual and self-learning; self- and mutual control;
there is a quick response to misunderstanding and error (joint discussion, support tips, mutual consultations).