If you are already familiar with trigonometric circle , and you just want to refresh your memory of certain elements, or you are completely impatient, then here it is:
Here we will analyze everything in detail step by step.
The trigonometric circle is not a luxury, but a necessity
Trigonometry
Many people associate it with an impenetrable thicket. Suddenly there are so many meanings trigonometric functions, so many formulas... But it didn’t work out at first, and... off and on... complete misunderstanding...
It is very important not to give up values of trigonometric functions, - they say, you can always look at the spur with a table of values.
If you constantly look at a table with values trigonometric formulas, let's get rid of this habit!
He will help us out! You will work with it several times, and then it will pop up in your head. How is it better than a table? Yes, in the table you will find a limited number of values, but on the circle - EVERYTHING!
For example, say while looking at standard table of values of trigonometric formulas , why equal to sine, say 300 degrees, or -45.
No way?.. you can, of course, connect reduction formulas... And looking at the trigonometric circle, you can easily answer such questions. And you will soon know how!
And when deciding trigonometric equations and inequalities without the trigonometric circle - nowhere at all.
Introduction to the trigonometric circle
Let's go in order.
First, let's write out this series of numbers:
And now this:
And finally this one:
Of course, it is clear that, in fact, in first place is , in second place is , and in last place is . That is, we will be more interested in the chain.
But how beautiful it turned out! If something happens, we will restore this “miracle ladder.”
And why do we need it?
This chain is the main values of sine and cosine in the first quarter.
Let us draw a circle of unit radius in a rectangular coordinate system (that is, we take any radius in length, and declare its length to be unit).
From the “0-Start” beam we lay down the corners in the direction of the arrow (see figure).
We get the corresponding points on the circle. So, if we project the points onto each of the axes, then we will get exactly the values from the above chain.
Why is this, you ask?
Let's not analyze everything. Let's consider principle, which will allow you to cope with other, similar situations.
Triangle AOB is rectangular and contains . And we know that opposite the angle b lies a leg half the size of the hypotenuse (we have the hypotenuse = the radius of the circle, that is, 1).
This means AB= (and therefore OM=). And according to the Pythagorean theorem
I hope something is already becoming clear?
So point B will correspond to the value, and point M will correspond to the value
Same with the other values of the first quarter.
As you understand, the familiar axis (ox) will be cosine axis, and the axis (oy) – axis of sines . Later.
To the left of zero along the cosine axis (below zero along the sine axis) there will, of course, be negative values.
So, here it is, the ALMIGHTY, without whom there is nowhere in trigonometry.
But we’ll talk about how to use the trigonometric circle in.
Counting angles on a trigonometric circle.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
It is almost the same as in the previous lesson. There are axes, a circle, an angle, everything is in order. Added quarter numbers (in the corners of the large square) - from the first to the fourth. What if someone doesn’t know? As you can see, quarters (they are also called a beautiful word"quadrants") are numbered counterclockwise. Added angle values on axes. Everything is clear, no problems.
And a green arrow is added. With a plus. What does it mean? Let me remind you that the fixed side of the angle Always nailed to the positive semi-axis OX. So, if we rotate the movable side of the angle along the arrow with a plus, i.e. in ascending order of quarter numbers, the angle will be considered positive. As an example, the picture shows a positive angle of +60°.
If we put aside the corners in the opposite direction, clockwise, the angle will be considered negative. Hover your cursor over the picture (or touch the picture on your tablet), you will see a blue arrow with a minus sign. This is the direction of negative angle reading. For example, a negative angle (- 60°) is shown. And you will also see how the numbers on the axes have changed... I also converted them to negative angles. The numbering of the quadrants does not change.
This is where the first misunderstandings usually begin. How so!? What if a negative angle on a circle coincides with a positive one!? And in general, it turns out that the same position of the moving side (or point on the number circle) can be called both a negative angle and a positive one!?
Yes. That's right. Let's say a positive angle of 90 degrees takes on a circle exactly the same position as a negative angle of minus 270 degrees. A positive angle, for example, +110° degrees takes exactly the same position as negative angle -250°.
No question. Anything is correct.) The choice of positive or negative angle calculation depends on the conditions of the task. If the condition says nothing in clear text about the sign of the angle, (like "determine the smallest positive angle", etc.), then we work with values that are convenient for us.
The exception (how could we live without them?!) are trigonometric inequalities, but there we will master this trick.
And now a question for you. How did I know that the position of the 110° angle is the same as the position of the -250° angle?
Let me hint that this is connected with a complete revolution. In 360°... Not clear? Then we draw a circle. We draw it ourselves, on paper. Marking the corner approximately 110°. AND we think, how much time remains until a full revolution. Just 250° will remain...
Got it? And now - attention! If angles 110° and -250° occupy a circle same thing
situation, then what? Yes, the angles are 110° and -250° exactly the same
sine, cosine, tangent and cotangent!
Those. sin110° = sin(-250°), ctg110° = ctg(-250°) and so on. Now this is really important! And in itself, there are a lot of tasks where you need to simplify expressions, and as a basis for the subsequent mastery of reduction formulas and other intricacies of trigonometry.
Of course, I took 110° and -250° at random, purely as an example. All these equalities work for any angles occupying the same position on the circle. 60° and -300°, -75° and 285°, and so on. Let me note right away that the angles in these pairs are different. But they have trigonometric functions - identical.
I think you understand what negative angles are. It's quite simple. Counterclockwise - positive counting. Along the way - negative. Consider the angle positive or negative depends on us. From our desire. Well, and also from the task, of course... I hope you understand how to move from negative angles to positive angles and back in trigonometric functions. Draw a circle, an approximate angle, and see how much is missing to complete a full revolution, i.e. up to 360°.
Angles greater than 360°.
Let's deal with angles that are greater than 360°. Are there such things? There are, of course. How to draw them on a circle? No problem! Let's say we need to understand which quarter an angle of 1000° will fall into? Easily! We make one full turn counterclockwise (the angle we were given is positive!). We rewinded 360°. Well, let's move on! One more turn - it’s already 720°. How many are left? 280°. It’s not enough for a full turn... But the angle is more than 270° - and this is the border between the third and fourth quarter. Therefore, our angle of 1000° falls into the fourth quarter. All.
As you can see, it's quite simple. Let me remind you once again that the angle of 1000° and the angle of 280°, which we obtained by discarding the “extra” full revolutions, are, strictly speaking, different corners. But the trigonometric functions of these angles exactly the same! Those. sin1000° = sin280°, cos1000° = cos280°, etc. If I were a sine, I wouldn't notice the difference between these two angles...
Why is all this needed? Why do we need to convert angles from one to another? Yes, all for the same thing.) In order to simplify expressions. Simplification of expressions, in fact, main task school mathematics. Well, and, along the way, the head is trained.)
Well, let's practice?)
We answer questions. Simple ones first.
1. Which quarter does the -325° angle fall into?
2. Which quarter does the 3000° angle fall into?
3. Which quarter does the angle -3000° fall into?
Any problems? Or uncertainty? Let's go to Section 555, Practical work with the trigonometric circle. There, in the first lesson of this very " Practical work..." all in detail... In such questions of uncertainty to be shouldn't!
4. What sign does sin555° have?
5. What sign does tg555° have?
Have you determined? Great! Do you have any doubts? You need to go to Section 555... By the way, there you will learn to draw tangent and cotangent on a trigonometric circle. A very useful thing.
And now the questions are more sophisticated.
6. Reduce the expression sin777° to the sine of the smallest positive angle.
7. Reduce the expression cos777° to the cosine of the largest negative angle.
8. Reduce the expression cos(-777°) to the cosine of the smallest positive angle.
9. Reduce the expression sin777° to the sine of the largest negative angle.
What, questions 6-9 puzzled you? Get used to it, on the Unified State Exam you don’t find such formulations... So be it, I’ll translate it. Just for you!
The words "bring an expression to..." mean to transform the expression so that its meaning hasn't changed A appearance changed according to the assignment. So, in tasks 6 and 9 we must get a sine, inside of which there is smallest positive angle. Everything else doesn't matter.
I will give out the answers in order (in violation of our rules). But what to do, there are only two signs, and there are only four quarters... You won’t be spoiled for choice.
6. sin57°.
7. cos(-57°).
8. cos57°.
9. -sin(-57°)
I assume that the answers to questions 6-9 confused some people. Especially -sin(-57°), really?) Indeed, in the elementary rules for calculating angles there is room for errors... That is why I had to do a lesson: “How to determine the signs of functions and give angles on a trigonometric circle?” In Section 555. Tasks 4 - 9 are covered there. Well sorted, with all the pitfalls. And they are here.)
In the next lesson we will deal with the mysterious radians and the number "Pi". Let's learn how to easily and correctly convert degrees to radians and vice versa. And we will be surprised to discover that this basic information on the site enough already to solve some custom trigonometry problems!
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
The sign of the trigonometric function depends solely on the coordinate quadrant in which the numerical argument is located. Last time we learned to convert arguments from a radian measure to a degree measure (see lesson “ Radian and degree measure of an angle”), and then determine this same coordinate quarter. Now let's actually determine the sign of sine, cosine and tangent.
The sine of angle α is the ordinate (y coordinate) of a point on trigonometric circle, which occurs when the radius is rotated by an angle α.
The cosine of angle α is the abscissa (x coordinate) of a point on a trigonometric circle, which occurs when the radius is rotated by angle α.
The tangent of the angle α is the ratio of sine to cosine. Or, which is the same thing, the ratio of the y coordinate to the x coordinate.
Notation: sin α = y ; cos α = x ; tg α = y : x .
All these definitions are familiar to you from high school algebra. However, we are not interested in the definitions themselves, but in the consequences that arise on the trigonometric circle. Take a look:
Blue color indicates the positive direction of the OY axis (ordinate axis), red indicates the positive direction of the OX axis (abscissa axis). On this "radar" the signs of trigonometric functions become obvious. In particular:
- sin α > 0 if angle α lies in the I or II coordinate quadrant. This is because, by definition, sine is an ordinate (y coordinate). And the y coordinate will be positive precisely in the I and II coordinate quarters;
- cos α > 0, if angle α lies in the 1st or 4th coordinate quadrant. Because only there the x coordinate (aka abscissa) will be greater than zero;
- tan α > 0 if angle α lies in the I or III coordinate quadrant. This follows from the definition: after all, tan α = y : x, therefore it is positive only where the signs of x and y coincide. This happens in the first coordinate quarter (here x > 0, y > 0) and the third coordinate quarter (x< 0, y < 0).
For clarity, let us note the signs of each trigonometric function - sine, cosine and tangent - on separate “radars”. We get the following picture:
Please note: in my discussions I never spoke about the fourth trigonometric function - cotangent. The fact is that the cotangent signs coincide with the tangent signs - there are no special rules there.
Now I propose to consider examples similar to problems B11 from the test Unified State Examination in mathematics, which took place on September 27, 2011. After all, best way understanding theory is practice. It is advisable to have a lot of practice. Of course, the conditions of the tasks were slightly changed.
Task. Determine the signs of trigonometric functions and expressions (the values of the functions themselves do not need to be calculated):
- sin(3π/4);
- cos(7π/6);
- tg(5π/3);
- sin (3π/4) cos (5π/6);
- cos (2π/3) tg (π/4);
- sin (5π/6) cos (7π/4);
- tan (3π/4) cos (5π/3);
- ctg (4π/3) tg (π/6).
The action plan is as follows: first we convert all angles from radian measures to degrees (π → 180°), and then look at which coordinate quarter the resulting number lies in. Knowing the quarters, we can easily find the signs - according to the rules just described. We have:
- sin (3π/4) = sin (3 · 180°/4) = sin 135°. Since 135° ∈ , this is an angle from the II coordinate quadrant. But the sine in the second quarter is positive, so sin (3π/4) > 0;
- cos (7π/6) = cos (7 · 180°/6) = cos 210°. Because 210° ∈ , this is the angle from the third coordinate quadrant, in which all cosines are negative. Therefore cos(7π/6)< 0;
- tg (5π/3) = tg (5 · 180°/3) = tg 300°. Since 300° ∈ , we are in the IV quarter, where the tangent takes negative values. Therefore tan (5π/3)< 0;
- sin (3π/4) cos (5π/6) = sin (3 180°/4) cos (5 180°/6) = sin 135° cos 150°. Let's deal with the sine: because 135° ∈ , this is the second quarter in which the sines are positive, i.e. sin (3π/4) > 0. Now we work with cosine: 150° ∈ - again the second quarter, the cosines there are negative. Therefore cos(5π/6)< 0. Наконец, следуя правилу «плюс на минус дает знак минус», получаем: sin (3π/4) · cos (5π/6) < 0;
- cos (2π/3) tg (π/4) = cos (2 180°/3) tg (180°/4) = cos 120° tg 45°. We look at the cosine: 120° ∈ is the II coordinate quarter, so cos (2π/3)< 0. Смотрим на тангенс: 45° ∈ — это I четверть (самый regular angle in trigonometry). The tangent there is positive, so tan (π/4) > 0. Again we get a product in which the factors have different signs. Since “minus by plus gives minus”, we have: cos (2π/3) tg (π/4)< 0;
- sin (5π/6) cos (7π/4) = sin (5 180°/6) cos (7 180°/4) = sin 150° cos 315°. We work with the sine: since 150° ∈ , we are talking about the II coordinate quarter, where the sines are positive. Therefore, sin (5π/6) > 0. Similarly, 315° ∈ is the IV coordinate quarter, the cosines there are positive. Therefore cos (7π/4) > 0. We have obtained the product of two positive numbers - such an expression is always positive. We conclude: sin (5π/6) cos (7π/4) > 0;
- tg (3π/4) cos (5π/3) = tg (3 180°/4) cos (5 180°/3) = tg 135° cos 300°. But the angle 135° ∈ is the second quarter, i.e. tg(3π/4)< 0. Аналогично, угол 300° ∈ — это IV четверть, т.е. cos (5π/3) >0. Since “minus by plus gives a minus sign,” we have: tg (3π/4) cos (5π/3)< 0;
- ctg (4π/3) tg (π/6) = ctg (4 180°/3) tg (180°/6) = ctg 240° tg 30°. We look at the cotangent argument: 240° ∈ is the III coordinate quarter, therefore ctg (4π/3) > 0. Similarly, for the tangent we have: 30° ∈ is the I coordinate quarter, i.e. the simplest angle. Therefore tan (π/6) > 0. Again we have two positive expressions - their product will also be positive. Therefore cot (4π/3) tg (π/6) > 0.
In conclusion, let's look at some more complex tasks. In addition to figuring out the sign of the trigonometric function, you will have to do a little math here - exactly as it is done in real problems B11. In principle, these are almost real problems that actually appear in the Unified State Examination in mathematics.
Task. Find sin α if sin 2 α = 0.64 and α ∈ [π/2; π].
Since sin 2 α = 0.64, we have: sin α = ±0.8. All that remains is to decide: plus or minus? By condition, angle α ∈ [π/2; π] is the II coordinate quarter, where all sines are positive. Therefore, sin α = 0.8 - the uncertainty with signs is eliminated.
Task. Find cos α if cos 2 α = 0.04 and α ∈ [π; 3π/2].
We act similarly, i.e. extract square root: cos 2 α = 0.04 ⇒ cos α = ±0.2. By condition, angle α ∈ [π; 3π/2], i.e. We are talking about the third coordinate quarter. All cosines there are negative, so cos α = −0.2.
Task. Find sin α if sin 2 α = 0.25 and α ∈ .
We have: sin 2 α = 0.25 ⇒ sin α = ±0.5. We look at the angle again: α ∈ is the IV coordinate quarter, in which, as we know, the sine will be negative. Thus, we conclude: sin α = −0.5.
Task. Find tan α if tan 2 α = 9 and α ∈ .
Everything is the same, only for the tangent. Extract the square root: tan 2 α = 9 ⇒ tan α = ±3. But according to the condition, the angle α ∈ is the I coordinate quarter. All trigonometric functions, incl. tangent, there are positive, so tan α = 3. That's it!
Reference data for tangent (tg x) and cotangent (ctg x). Geometric definition, properties, graphs, formulas. Table of tangents and cotangents, derivatives, integrals, series expansions. Expressions through complex variables. Connection with hyperbolic functions.
Geometric definition
|BD| - length of the arc of a circle with center at point A.
α is the angle expressed in radians.
Tangent ( tan α) is a trigonometric function depending on the angle α between the hypotenuse and the leg right triangle, equal to the ratio of the length of the opposite side |BC| to the length of the adjacent leg |AB| .
Cotangent ( ctg α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| .
Tangent
Where n- whole.
In Western literature, tangent is denoted as follows:
.
;
;
.
Graph of the tangent function, y = tan x
Cotangent
Where n- whole.
In Western literature, cotangent is denoted as follows:
.
The following notations are also accepted:
;
;
.
Graph of the cotangent function, y = ctg x
Properties of tangent and cotangent
Periodicity
Functions y = tg x and y = ctg x are periodic with period π.
Parity
The tangent and cotangent functions are odd.
Areas of definition and values, increasing, decreasing
The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table ( n- whole).
| y= tg x | y= ctg x | |
| Scope and continuity | ||
| Range of values | -∞ < y < +∞ | -∞ < y < +∞ |
| Increasing | - | |
| Descending | - | |
| Extremes | - | - |
| Zeros, y = 0 | ||
| Intercept points with the ordinate axis, x = 0 | y= 0 | - |
Formulas
Expressions using sine and cosine
;
;
;
;
;
Formulas for tangent and cotangent from sum and difference
The remaining formulas are easy to obtain, for example
Product of tangents
Formula for the sum and difference of tangents
This table presents the values of tangents and cotangents for certain values of the argument. 
Expressions using complex numbers
Expressions through hyperbolic functions
;
;
Derivatives
; .
.
Derivative of the nth order with respect to the variable x of the function:
.
Deriving formulas for tangent > > > ; for cotangent > > >
Integrals
Series expansions
To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in power series for functions sin x And cos x and divide these polynomials by each other, . This produces the following formulas.
At .
at .
Where Bn- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
Where .
Or according to Laplace's formula: 
Inverse functions
Inverse functions to tangent and cotangent are arctangent and arccotangent, respectively.
Arctangent, arctg
, Where n- whole.
Arccotangent, arcctg
, Where n- whole.
Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.
Allows you to establish a number of characteristic results - properties of sine, cosine, tangent and cotangent. In this article we will look at three main properties. The first of them indicates the signs of the sine, cosine, tangent and cotangent of the angle α depending on the angle of which coordinate quarter is α. Next we will consider the property of periodicity, which establishes the invariance of the values of sine, cosine, tangent and cotangent of the angle α when this angle changes by an integer number of revolutions. The third property expresses the relationship between the values of sine, cosine, tangent and cotangent of opposite angles α and −α.
If you are interested in the properties of the functions sine, cosine, tangent and cotangent, then you can study them in the corresponding section of the article.
Page navigation.
Signs of sine, cosine, tangent and cotangent by quarters
Below in this paragraph the phrase “angle of I, II, III and IV coordinate quarter” will appear. Let's explain what these angles are.
Let's take a unit circle, mark the starting point A(1, 0) on it, and rotate it around the point O by an angle α, and we will assume that we will get to the point A 1 (x, y).
They say that angle α is the angle of the I, II, III, IV coordinate quadrant, if point A 1 lies in the I, II, III, IV quarters, respectively; if the angle α is such that point A 1 lies on any of the coordinate lines Ox or Oy, then this angle does not belong to any of the four quarters.
For clarity, here is a graphic illustration. The drawings below show rotation angles of 30, −210, 585, and −45 degrees, which are the angles of the I, II, III, and IV coordinate quarters, respectively.
Angles 0, ±90, ±180, ±270, ±360, … degrees do not belong to any of the coordinate quarters.
Now let's figure out what signs have the values of sine, cosine, tangent and cotangent of the angle of rotation α, depending on which quarter angle α is.
For sine and cosine this is easy to do.
By definition, the sine of angle α is the ordinate of point A 1. Obviously, in the I and II coordinate quarters it is positive, and in the III and IV quarters it is negative. Thus, the sine of angle α has a plus sign in the 1st and 2nd quarters, and a minus sign in the 3rd and 6th quarters.
In turn, the cosine of the angle α is the abscissa of point A 1. In the I and IV quarters it is positive, and in the II and III quarters it is negative. Consequently, the values of the cosine of the angle α in the I and IV quarters are positive, and in the II and III quarters they are negative.
To determine the signs by quarters of tangent and cotangent, you need to remember their definitions: tangent is the ratio of the ordinate of point A 1 to the abscissa, and cotangent is the ratio of the abscissa of point A 1 to the ordinate. Then from rules for dividing numbers with the same and different signs it follows that tangent and cotangent have a plus sign when the abscissa and ordinate signs of point A 1 are the same, and have a minus sign when the abscissa and ordinate signs of point A 1 are different. Consequently, the tangent and cotangent of the angle have a + sign in the I and III coordinate quarters, and a minus sign in the II and IV quarters.
Indeed, for example, in the first quarter both the abscissa x and the ordinate y of point A 1 are positive, then both the quotient x/y and the quotient y/x are positive, therefore, tangent and cotangent have + signs. And in the second quarter, the abscissa x is negative, and the ordinate y is positive, therefore both x/y and y/x are negative, hence the tangent and cotangent have a minus sign.
Let's move on to the next property of sine, cosine, tangent and cotangent.
Periodicity property
Now we will look at perhaps the most obvious property of sine, cosine, tangent and cotangent of an angle. It is as follows: when the angle changes by an integer number of full revolutions, the values of the sine, cosine, tangent and cotangent of this angle do not change.
This is understandable: when the angle changes by an integer number of revolutions, we will always get from the starting point A to point A 1 on the unit circle, therefore, the values of sine, cosine, tangent and cotangent remain unchanged, since the coordinates of point A 1 are unchanged.
Using formulas, the considered property of sine, cosine, tangent and cotangent can be written as follows: sin(α+2·π·z)=sinα, cos(α+2·π·z)=cosα, tan(α+2·π· z)=tgα, ctg(α+2·π·z)=ctgα, where α is the angle of rotation in radians, z is any, the absolute value of which indicates the number of full revolutions by which the angle α changes, and the sign of the number z indicates the direction turn.
If the rotation angle α is specified in degrees, then the indicated formulas will be rewritten as sin(α+360° z)=sinα , cos(α+360° z)=cosα , tg(α+360° z)=tgα , ctg(α+360°·z)=ctgα .
Let's give examples of using this property. For example, 
, because 
, A 
. Here's another example: or .
This property, together with reduction formulas, is very often used when calculating the values of sine, cosine, tangent and cotangent of “large” angles.
The considered property of sine, cosine, tangent and cotangent is sometimes called the property of periodicity.
Properties of sines, cosines, tangents and cotangents of opposite angles
Let A 1 be the point obtained by rotating the initial point A(1, 0) around point O by an angle α, and point A 2 be the result of rotating point A by an angle −α, opposite to angle α.
The property of sines, cosines, tangents and cotangents of opposite angles is based on a fairly obvious fact: the points A 1 and A 2 mentioned above either coincide (at) or are located symmetrically relative to the Ox axis. That is, if point A 1 has coordinates (x, y), then point A 2 will have coordinates (x, −y). From here, using the definitions of sine, cosine, tangent and cotangent, we write the equalities and .
Comparing them, we come to relationships between sines, cosines, tangents and cotangents of opposite angles α and −α of the form.
This is the property under consideration in the form of formulas.
Let's give examples of using this property. For example, the equalities and 
.
It remains only to note that the property of sines, cosines, tangents and cotangents of opposite angles, like the previous property, is often used when calculating the values of sine, cosine, tangent and cotangent, and allows you to completely avoid negative angles.
References.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.