The most necessary trigonometric formulas. Trigonometry made simple and clear

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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 looks like 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 remaining 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 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 to solve trigonometric equations, when estimating the values ​​of functions, in extremum problems, and what is important to note is 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.

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When performing trigonometric conversions, follow these tips:

  1. Don't try to immediately come up with a solution to the example from start to finish.
  2. Don't try to convert the entire example at once. Take small steps forward.
  3. Remember that in addition to trigonometric formulas in trigonometry, you can still apply all fair algebraic transformations(bracketing, reducing fractions, abbreviated multiplication formulas, and so on).
  4. Believe that everything will be fine.

Basic trigonometric formulas

Most formulas in trigonometry are often used both from right to left and from left to right, so you need to learn these formulas so well that you can easily apply some formula in both directions. Let us first write down the definitions of trigonometric functions. Let there be a right triangle:

Then, the definition of sine:

Definition of cosine:

Tangent definition:

Definition of cotangent:

Basic trigonometric identity:

The simplest corollaries from the basic trigonometric identity:

Double angle formulas. Sine of double angle:

Cosine of double angle:

Tangent of double angle:

Cotangent of double angle:

Additional trigonometric formulas

Trigonometric addition formulas. Sine of the sum:

Sine of the difference:

Cosine of the sum:

Cosine of the difference:

Tangent of the sum:

Tangent of difference:

Cotangent of the amount:

Cotangent of the difference:

Trigonometric formulas for converting a sum into a product. Sum of sines:

Sine difference:

Sum of cosines:

Difference of cosines:

Sum of tangents:

Tangent difference:

Sum of cotangents:

Cotangent difference:

Trigonometric formulas for converting a product into a sum. Product of sines:

Product of sine and cosine:

Product of cosines:

Degree reduction formulas.

Half angle formulas.

Trigonometric reduction formulas

The cosine function is called cofunction sine functions and vice versa. Similarly, the tangent and cotangent functions are cofunctions. Reduction formulas can be formulated as the following rule:

  • If in the reduction formula an angle is subtracted (added) from 90 degrees or 270 degrees, then the reduced function changes to a cofunction;
  • If in the reduction formula the angle is subtracted (added) from 180 degrees or 360 degrees, then the name of the reduced function is retained;
  • In this case, the sign that the reduced (i.e., original) function has in the corresponding quadrant is placed in front of the reduced function, if we consider the subtracted (added) angle to be acute.

Reduction formulas are given in table form:

By trigonometric circle easy to determine tabular values ​​of trigonometric functions:

Trigonometric equations

To solve a certain trigonometric equation, it must be reduced to one of the simplest trigonometric equations, which will be discussed below. For this:

  • You can use the trigonometric formulas given above. At the same time, you don’t need to try to transform the entire example at once, but you need to move forward in small steps.
  • We must not forget about the possibility of transforming some expression using algebraic methods, i.e. for example, take something out of brackets or, conversely, open brackets, reduce a fraction, apply an abbreviated multiplication formula, bring fractions to a common denominator, and so on.
  • When solving trigonometric equations you can use grouping method. It must be remembered that in order for the product of several factors to be equal to zero, it is sufficient that any of them be equal to zero, and the rest existed.
  • Applying variable replacement method, as usual, the equation after introducing the replacement should become simpler and not contain the original variable. You also need to remember to perform a reverse replacement.
  • Remember that homogeneous equations often appear in trigonometry.
  • When opening modules or solving irrational equations with trigonometric functions, you need to remember and take into account all the subtleties of solving the corresponding equations with ordinary functions.
  • Remember about ODZ (in trigonometric equations, restrictions on ODZ mainly come down to the fact that you cannot divide by zero, but do not forget about other restrictions, especially about the positivity of expressions in rational powers and under the roots of even powers). Also remember that the values ​​of sine and cosine can only lie in the range from minus one to plus one, inclusive.

The main thing is, if you don’t know what to do, do at least something, and the main thing is to use trigonometric formulas correctly. If what you get gets better and better, then continue the solution, and if it gets worse, then go back to the beginning and try to apply other formulas, do this until you come across the correct solution.

Formulas for solutions of the simplest trigonometric equations. For sine there are two equivalent forms of writing the solution:

For other trigonometric functions, the notation is unambiguous. For cosine:

For tangent:

For cotangent:

Solving trigonometric equations in some special cases:

  • Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; there are only about 200 necessary formulas in physics, and even a little less in mathematics. Each of these subjects has about a dozen standard methods for solving problems basic level difficulties that can also be learned, and thus solved completely automatically and without difficulty right moment most of the DH. After this, you will only have to think about the most difficult tasks.
  • Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

    • Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.

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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... there are emotions and feelings 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 obtain knowledge sufficient for successful passing the Unified State Exam and admission to university (enter and forget).

    The main goal of the presented lesson is to show the applied value of trigonometry in various fields 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 what seems like a long time ago known fact. Show a logical connection between what we already know and what remains to be learned. Open the door a little and look outside 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 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 you take one table as a chord, then what is the degree measure the arc that it contracts?

    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 apparent 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 right triangle- this 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 directions last years has received such rapid development that its result is already 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 world (helio – Sun) ( Figure No. 10).

    Warm-up.

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

    Winter Games took place in 2010 Olympic Games in Vancouver, we learn the criteria for grading a skater’s completed exercise 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 before 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 used to solve 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 only inherent property: They repeat their values ​​after the same argument change interval. 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, it spreads throughout the entire body - starting from the sinus node (the size of a millet grain). electricity. 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, natural sounds and sounds musical instruments consist 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 show you the way to choose 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 (

    The “Get an A” video course includes all the topics you need to successful completion Unified State Examination in mathematics for 60-65 points. Completely all problems 1-13 Profile Unified State Examination mathematics. Also suitable for passing the Basic Unified State Examination in mathematics. If you want to pass the Unified State Exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!

    Preparation course for the Unified State Exam for grades 10-11, as well as for teachers. Everything you need to solve Part 1 of the Unified State Exam in mathematics (the first 12 problems) and Problem 13 (trigonometry). And this is more than 70 points on the Unified State Exam, and neither a 100-point student nor a humanities student can do without them.

    All the necessary theory. Quick ways solutions, pitfalls and secrets of the Unified State Exam. All current tasks of part 1 from the FIPI Task Bank have been analyzed. The course fully complies with the requirements of the Unified State Exam 2018.

    The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.

    Hundreds of Unified State Exam tasks. Word problems and probability theory. Simple and easy to remember algorithms for solving problems. Geometry. Theory, reference material, analysis of all types of Unified State Examination tasks. Stereometry. Tricky solutions, useful cheat sheets, development of spatial imagination. Trigonometry from scratch to problem 13. Understanding instead of cramming. Clear explanations of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. A basis for solving complex problems of Part 2 of the Unified State Exam.



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