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.

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In this lesson we will talk about how the need to introduce trigonometric functions arises and why they are studied, what you need to understand in this topic, and where you just need to get better at it (what is a technique). Note that technique and understanding are two different things. Agree, there is a difference: learning to ride a bicycle, that is, understanding how to do it, or becoming a professional cyclist. We will talk specifically about understanding, about why trigonometric functions are needed.

There are four trigonometric functions, but they can all be expressed in terms of one using identities (equalities that relate them).

Formal definitions of trigonometric functions for acute angles in right triangles (Fig. 1).

Sinus The acute angle of a right triangle is the ratio of the opposite side to the hypotenuse.

Cosine The acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse.

Tangent The acute angle of a right triangle is the ratio of the opposite side to the adjacent side.

Cotangent The acute angle of a right triangle is the ratio of the adjacent side to the opposite side.

Rice. 1. Determination of trigonometric functions of an acute angle of a right triangle

These definitions are formal. It is more correct to say that there is only one function, for example, sine. If they were not so needed (not so often used) in technology, so many different trigonometric functions would not be introduced.

For example, the cosine of an angle is equal to the sine of the same angle with the addition of (). In addition, the cosine of an angle can always be expressed through the sine of the same angle up to sign, using the basic trigonometric identity (). The tangent of an angle is the ratio of sine to cosine or an inverted cotangent (Fig. 2). Some don't use cotangent at all, replacing it with . Therefore, it is important to understand and be able to work with one trigonometric function.

Rice. 2. Relationship between various trigonometric functions

But why were such functions needed at all? What practical problems are they used to solve? Let's look at a few examples.

Two people ( A And IN) push the car out of the puddle (Fig. 3). Human IN can push the car sideways, but it is unlikely to help A. On the other hand, the direction of his efforts can gradually shift (Fig. 4).

Rice. 3. IN pushes the car sideways

Rice. 4. IN begins to change the direction of his efforts

It is clear that their efforts will be most effective when they push the car in one direction (Fig. 5).

Rice. 5. The most effective joint direction of effort

How much IN helps push the machine to the extent that the direction of its force is close to the direction of the force with which it acts A, is a function of the angle and is expressed through its cosine (Fig. 6).

Rice. 6. Cosine as a characteristic of effort efficiency IN

If we multiply the magnitude of the force with which IN, on the cosine of the angle, we obtain the projection of its force onto the direction of the force with which it acts A. The closer the angle between the directions of forces is to , the more effective the result of joint actions will be. A And IN(Fig. 7). If they push the car with the same force in opposite directions, the car will stay in place (Fig. 8).

Rice. 7. Effectiveness of joint efforts A And IN

Rice. 8. Opposite direction of forces A And IN

It's important to understand why we can replace an angle (its contribution to the final result) with a cosine (or other trigonometric function of an angle). In fact, this follows from this property of similar triangles. Since in fact we are saying the following: the angle can be replaced by the ratio of two numbers (side-hypotenuse or side-side). This would be impossible if, for example, for the same angle of different right triangles these ratios were different (Fig. 9).

Rice. 9. Equal side ratios in similar triangles

For example, if the ratio and ratio were different, then we would not be able to introduce the tangent function, since for the same angle in different right triangles the tangent would be different. But due to the fact that the ratios of the lengths of the legs of similar right triangles are the same, the value of the function will not depend on the triangle, which means that the acute angle and the values ​​of its trigonometric functions are one-to-one.

Suppose we know the height of a certain tree (Fig. 10). How to measure the height of a nearby building?

Rice. 10. Illustration of the condition of example 2

We find a point such that a line drawn through this point and the top of the house will pass through the top of the tree (Fig. 11).

Rice. 11. Illustration of the solution to the problem of example 2

We can measure the distance from this point to the tree, the distance from it to the house, and we know the height of the tree. From the proportion you can find the height of the house: .

Proportion is the equality of the ratio of two numbers. In this case, the equality of the ratio of the lengths of the legs of similar right triangles. Moreover, these ratios are equal to a certain measure of the angle, which is expressed through a trigonometric function (by definition, this is a tangent). We find that for each acute angle the value of its trigonometric function is unique. That is, sine, cosine, tangent, cotangent are really functions, since each acute angle corresponds to exactly one value of each of them. Consequently, they can be further explored and their properties used. The values ​​of trigonometric functions for all angles have already been calculated and can be used (they can be found from the Bradis tables or using any engineering calculator). But we cannot always solve the inverse problem (for example, using the value of the sine to restore the measure of the angle that corresponds to it).

Let the sine of some angle be equal to or approximately (Fig. 12). What angle will correspond to this sine value? Of course, we can again use the Bradis table and find some value, but it turns out that it will not be the only one (Fig. 13).

Rice. 12. Finding an angle by the value of its sine

Rice. 13. Polysemy of inverse trigonometric functions

Consequently, when reconstructing the value of the trigonometric function of an angle, the multivalued nature of the inverse trigonometric functions arises. This may seem difficult, but in reality we face similar situations every day.

If you curtain the windows and don’t know whether it’s light or dark outside, or if you find yourself in a cave, then when you wake up, it’s difficult to say whether it’s one o’clock in the afternoon, at night, or the next day (Fig. 14). In fact, if you ask us “What time is it?”, we must answer honestly: “Hour plus multiplied by where”

Rice. 14. Illustration of polysemy using the example of a clock

We can conclude that this is a period (the interval after which the clock will show the same time as now). Trigonometric functions also have periods: sine, cosine, etc. That is, their values ​​are repeated after some change in the argument.

If there was no change of day and night or change of seasons on the planet, then we could not use periodic time. After all, we only number the years in ascending order, but the days have hours, and every new day the counting begins anew. The situation is the same with months: if it is January now, then in a few months January will come again, etc. External reference points help us use periodic counting of time (hours, months), for example, the rotation of the Earth around its axis and the change in the position of the Sun and Moon in the sky. If the Sun always hung in the same position, then to calculate time we would count the number of seconds (minutes) from the moment this very calculation began. The date and time might then read like this: a billion seconds.

Conclusion: there are no difficulties in terms of polysemy of inverse functions. Indeed, there may be options when for the same sine there are different angle values ​​(Fig. 15).

Rice. 15. Restoring an angle from the value of its sine

Usually, when solving practical problems, we always work in the standard range from to . In this range, for each value of the trigonometric function there are only two corresponding values ​​of the angle measure.

Consider a moving belt and a pendulum in the form of a bucket with a hole from which sand pours out. The pendulum swings, the tape moves (Fig. 16). As a result, the sand will leave a trace in the form of a graph of the sine (or cosine) function, which is called a sine wave.

In fact, the graphs of sine and cosine differ from each other only in the reference point (if you draw one of them and then erase the coordinate axes, you will not be able to determine which graph was drawn). Therefore, there is no point in calling the cosine graph a graph (why come up with a separate name for the same graph)?

Rice. 16. Illustration of the problem statement in example 4

The graph of a function can also help you understand why inverse functions will have many values. If the value of the sine is fixed, i.e. draw a straight line parallel to the abscissa axis, then at the intersection we get all the points at which the sine of the angle is equal to the given one. It is clear that there will be an infinite number of such points. As in the example with the clock, where the time value differed by , only here the angle value will differ by the amount (Fig. 17).

Rice. 17. Illustration of polysemy for sine

If we consider the example of a clock, then the point (clockwise end) moves around the circle. Trigonometric functions can be defined in the same way - consider not the angles in a right triangle, but the angle between the radius of the circle and the positive direction of the axis. The number of circles that the point will go through (we agreed to count the movement clockwise with a minus sign, and counterclockwise with a plus sign), this is a period (Fig. 18).

Rice. 18. The value of sine on a circle

So, the inverse function is uniquely defined on a certain interval. For this interval, we can calculate its values, and get all the rest from the found values ​​by adding and subtracting the period of the function.

Let's look at another example of a period. The car is moving along the road. Let's imagine that her wheel has driven into paint or a puddle. Occasional marks from paint or puddles on the road may be seen (Figure 19).

Rice. 19. Period illustration

There are quite a lot of trigonometric formulas in the school course, but by and large it is enough to remember just one (Fig. 20).

Rice. 20. Trigonometric formulas

The double angle formula can also be easily derived from the sine of the sum by substituting (similarly for the cosine). You can also derive product formulas.

In fact, you need to remember very little, since with solving problems these formulas themselves will be remembered. Of course, someone will be too lazy to decide much, but then he will not need this technique, and therefore the formulas themselves.

And since the formulas are not needed, then there is no need to memorize them. You just need to understand the idea that trigonometric functions are functions that are used to calculate, for example, bridges. Almost no mechanism can do without their use and calculation.

1. The question often arises whether wires can be absolutely parallel to the ground. Answer: no, they cannot, since one force acts downward and the others act in parallel - they will never balance (Fig. 21).

2. A swan, a crayfish and a pike pull a cart in the same plane. The swan flies in one direction, the crayfish pulls in the other, and the pike in the third (Fig. 22). Their powers can be balanced. This balancing can be calculated using trigonometric functions.

3. Cable-stayed bridge (Fig. 23). Trigonometric functions help calculate the number of cables, how they should be directed and tensioned.

Rice. 23. Cable-stayed bridge

Rice. 24. “String Bridge”

Rice. 25. Bolshoi Obukhovsky Bridge

Links to site ma-te-ri-a-lyInternetUrok

Mathematics 6th grade:

Geometry 8th grade:

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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 point of intersection 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 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 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.

The video course “Get an A” includes all the topics necessary to successfully pass the Unified State Exam in mathematics with 60-65 points. Completely all tasks 1-13 of the Profile Unified State Exam in 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 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.