Arcsine (y = arcsin x)
is the inverse function of sine (x = siny -1 ≤ x ≤ 1 and the set of values -π /2 ≤ y ≤ π/2.
sin(arcsin x) = x
arcsin(sin x) = x
Arcsine is sometimes denoted as follows:
.
Graph of arcsine function
Graph of the function y = arcsin x
The arcsine graph is obtained from the sine graph if the abscissa and ordinate axes are swapped. To eliminate ambiguity, the range of values is limited to the interval over which the function is monotonic. This definition is called the principal value of the arcsine.
Arccosine, arccos
Arc cosine (y = arccos x)
is the inverse function of cosine (x = cos y). It has a scope -1 ≤ x ≤ 1 and many meanings 0 ≤ y ≤ π.
cos(arccos x) = x
arccos(cos x) = x
Arccosine is sometimes denoted as follows:
.
Graph of arc cosine function
Graph of the function y = arccos x
The arc cosine graph is obtained from the cosine graph if the abscissa and ordinate axes are swapped. To eliminate ambiguity, the range of values is limited to the interval over which the function is monotonic. This definition is called the principal value of the arc cosine.
Parity
The arcsine function is odd:
arcsin(- x) = arcsin(-sin arcsin x) = arcsin(sin(-arcsin x)) = - arcsin x
The arc cosine function is not even or odd:
arccos(- x) = arccos(-cos arccos x) = arccos(cos(π-arccos x)) = π - arccos x ≠ ± arccos x
Properties - extrema, increase, decrease
The functions arcsine and arccosine are continuous in their domain of definition (see proof of continuity). The main properties of arcsine and arccosine are presented in the table.
| y= arcsin x | y= arccos x | |
| Scope and continuity | - 1 ≤ x ≤ 1 | - 1 ≤ x ≤ 1 |
| Range of values | ||
| Ascending, descending | monotonically increases | monotonically decreases |
| Highs | ||
| Minimums | ||
| Zeros, y = 0 | x = 0 | x = 1 |
| Intercept points with the ordinate axis, x = 0 | y= 0 | y = π/ 2 |
Table of arcsines and arccosines
This table presents the values of arcsines and arccosines, in degrees and radians, for certain values of the argument.
| x | arcsin x | arccos x | ||
| hail | glad. | hail | glad. | |
| - 1 | - 90° | - | 180° | π |
| - | - 60° | - | 150° | |
| - | - 45° | - | 135° | |
| - | - 30° | - | 120° | |
| 0 | 0° | 0 | 90° | |
| 30° | 60° | |||
| 45° | 45° | |||
| 60° | 30° | |||
| 1 | 90° | 0° | 0 | |
≈ 0,7071067811865476
≈ 0,8660254037844386
Formulas
Sum and difference formulas
at or
at and
at and
at or
at and
at and
at
at
at
at
Expressions through logarithms, complex numbers
Expressions through hyperbolic functions
Derivatives
;
.
See Derivation of arcsine and arccosine derivatives > > >
Higher order derivatives:
,
where is a polynomial of degree . It is determined by the formulas:
;
;
.
See Derivation of higher order derivatives of arcsine and arccosine > > >
Integrals
We make the substitution x = sint. We integrate by parts, taking into account that -π/ 2 ≤ t ≤ π/2,
cos t ≥ 0:
.
Let's express arc cosine through arc sine:
.
Series expansion
When |x|< 1
the following decomposition takes place:
;
.
Inverse functions
The inverses of arcsine and arccosine are sine and cosine, respectively.
The following formulas are valid throughout the entire domain of definition:
sin(arcsin x) = x
cos(arccos x) = x .
The following formulas are valid only on the set of arcsine and arccosine values:
arcsin(sin x) = x at
arccos(cos x) = x at .
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
What is arcsine, arccosine? What is arctangent, arccotangent?
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
To concepts arcsine, arccosine, arctangent, arccotangent The student population is wary. He does not understand these terms and, therefore, does not trust this nice family.) But in vain. These are very simple concepts. Which, by the way, make life enormously easier for a knowledgeable person when solving trigonometric equations!
Doubts about simplicity? In vain.) Right here and now you will see this.
Of course, for understanding, it would be nice to know what sine, cosine, tangent and cotangent are. Yes, their tabular values for some angles... At least in the most general terms. Then there will be no problems here either.
So, we are surprised, but remember: arcsine, arccosine, arctangent and arccotangent are just some angles. No more, no less. There is an angle, say 30°. And there is a corner arcsin0.4. Or arctg(-1.3). There are all kinds of angles.) You can simply write down angles in different ways. You can write the angle in degrees or radians. Or you can - through its sine, cosine, tangent and cotangent...
What does the expression mean
arcsin 0.4 ?
This is an angle whose sine is 0.4! Yes Yes. This is the meaning of arcsine. I will specifically repeat: arcsin 0.4 is an angle whose sine is equal to 0.4.
That's all.
To keep this simple thought in your head for a long time, I will even give a breakdown of this terrible term - arcsine:
arc sin 0,4
corner, the sine of which equal to 0.4
As it is written, so it is heard.) Almost. Console arc means arc(word arch do you know?), because ancient people used arcs instead of angles, but this does not change the essence of the matter. Remember this elementary decoding of a mathematical term! Moreover, for arccosine, arctangent and arccotangent, the decoding differs only in the name of the function.
What is arccos 0.8?
This is an angle whose cosine is 0.8.
What is arctg(-1,3) ?
This is an angle whose tangent is -1.3.
What is arcctg 12?
This is an angle whose cotangent is 12.
Such elementary decoding allows, by the way, to avoid epic blunders.) For example, the expression arccos1,8 looks quite respectable. Let's start decoding: arccos1.8 is an angle whose cosine is equal to 1.8... Jump-jump!? 1.8!? Cosine cannot be greater than one!!!
Right. The expression arccos1,8 does not make sense. And writing such an expression in some answer will greatly amuse the inspector.)
Elementary, as you can see.) Each angle has its own personal sine and cosine. And almost everyone has their own tangent and cotangent. Therefore, knowing the trigonometric function, we can write down the angle itself. This is what arcsines, arccosines, arctangents and arccotangents are intended for. From now on I will call this whole family by a diminutive name - arches. To type less.)
Attention! Elementary verbal and conscious deciphering arches allows you to calmly and confidently solve a variety of tasks. And in unusual Only she saves tasks.
Is it possible to switch from arcs to ordinary degrees or radians?- I hear a cautious question.)
Why not!? Easily. You can go there and back. Moreover, sometimes this must be done. Arches are a simple thing, but it’s somehow calmer without them, right?)
For example: what is arcsin 0.5?
Let's remember the decoding: arcsin 0.5 is the angle whose sine is 0.5. Now turn on your head (or Google)) and remember which angle has a sine of 0.5? Sine is equal to 0.5 y 30 degree angle. That's it: arcsin 0.5 is an angle of 30°. You can safely write:
arcsin 0.5 = 30°
Or, more formally, in terms of radians:
That's it, you can forget about the arcsine and continue working with the usual degrees or radians.
If you realized what is arcsine, arccosine... What is arctangent, arccotangent... You can easily deal with, for example, such a monster.)
An ignorant person will recoil in horror, yes...) But an informed person remember the decoding: arcsine is the angle whose sine... And so on. If a knowledgeable person also knows the table of sines... The table of cosines. Table of tangents and cotangents, then there are no problems at all!
It is enough to realize that:
I’ll decipher it, i.e. Let me translate the formula into words: angle whose tangent is 1 (arctg1)- this is an angle of 45°. Or, which is the same, Pi/4. Likewise:
and that's it... We replace all the arches with values in radians, everything is reduced, all that remains is to calculate how much 1+1 is. It will be 2.) Which is the correct answer.
This is how you can (and should) move from arcsines, arccosines, arctangents and arccotangents to ordinary degrees and radians. This greatly simplifies scary examples!
Often, in such examples, inside the arches there are negative meanings. Like, arctg(-1.3), or, for example, arccos(-0.8)... This is not a problem. Here are simple formulas for moving from negative to positive values:
You need, say, to determine the value of the expression:
This can be solved using the trigonometric circle, but you don't want to draw it. Well, okay. We move from negative values inside the arc cosine of k positive according to the second formula:
Inside the arc cosine on the right is already positive meaning. What
you simply must know. All that remains is to substitute radians instead of arc cosine and calculate the answer:
That's all.
Restrictions on arcsine, arccosine, arctangent, arccotangent.
Is there a problem with examples 7 - 9? Well, yes, there is some trick there.)
All these examples, from 1 to 9, are carefully analyzed in Section 555. What, how and why. With all the secret traps and tricks. Plus ways to dramatically simplify the solution. By the way, this section contains a lot of useful information and practical tips on trigonometry in general. And not only in trigonometry. Helps a lot.
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 functions sin, cos, tg and ctg are always accompanied by arcsine, arccosine, arctangent and arccotangent. One is a consequence of the other, and pairs of functions are equally important for working with trigonometric expressions.
Consider a drawing of a unit circle, which graphically displays the values of trigonometric functions.
If we calculate arcs OA, arcos OC, arctg DE and arcctg MK, then they will all be equal to the value of angle α. The formulas below reflect the relationship between the basic trigonometric functions and their corresponding arcs.
To understand more about the properties of the arcsine, it is necessary to consider its function. Schedule has the form of an asymmetric curve passing through the coordinate center.
Properties of arcsine:
If we compare the graphs sin And arcsin, two trigonometric functions can have common patterns.
arc cosine
Arccos of a number is the value of the angle α, the cosine of which is equal to a.
Curve y = arcos x mirrors the arcsin x graph, with the only difference being that it passes through the point π/2 on the OY axis.
Let's look at the arc cosine function in more detail:
- The function is defined on the interval [-1; 1].
- ODZ for arccos - .
- The graph is entirely located in the first and second quarters, and the function itself is neither even nor odd.
- Y = 0 at x = 1.
- The curve decreases along its entire length. Some properties of the arc cosine coincide with the cosine function.
Some properties of the arc cosine coincide with the cosine function.
Perhaps schoolchildren will find such a “detailed” study of “arches” unnecessary. However, otherwise, some elementary standard exam tasks can lead students into a dead end.
Exercise 1. Indicate the functions shown in the figure.
Answer: rice. 1 – 4, Fig. 2 – 1.
In this example, the emphasis is on the little things. Typically, students are very inattentive to the construction of graphs and the appearance of functions. Indeed, why remember the type of curve if it can always be plotted using calculated points. Do not forget that under test conditions, the time spent on drawing for a simple task will be required to solve more complex tasks.
Arctangent
Arctg the numbers a are the value of the angle α such that its tangent is equal to a.
If we consider the arctangent graph, we can highlight the following properties:
- The graph is infinite and defined on the interval (- ∞; + ∞).
- Arctangent is an odd function, therefore, arctan (- x) = - arctan x.
- Y = 0 at x = 0.
- The curve increases throughout the entire definition region.
Let us present a brief comparative analysis of tg x and arctg x in the form of a table.
Arccotangent
Arcctg of a number - takes a value α from the interval (0; π) such that its cotangent is equal to a.
Properties of the arc cotangent function:
- The function definition interval is infinity.
- The range of acceptable values is the interval (0; π).
- F(x) is neither even nor odd.
- Throughout its entire length, the graph of the function decreases.
It is very simple to compare ctg x and arctg x; you just need to make two drawings and describe the behavior of the curves.
Task 2. Match the graph and the notation form of the function.
If we think logically, it is clear from the graphs that both functions are increasing. Therefore, both figures display a certain arctan function. From the properties of the arctangent it is known that y=0 at x = 0,
Answer: rice. 1 – 1, fig. 2 – 4.
Trigonometric identities arcsin, arcos, arctg and arcctg
Previously, we have already identified the relationship between arches and the basic functions of trigonometry. This dependence can be expressed by a number of formulas that allow one to express, for example, the sine of an argument through its arcsine, arccosine, or vice versa. Knowledge of such identities can be useful when solving specific examples.
There are also relationships for arctg and arcctg:
Another useful pair of formulas sets the value for the sum of arcsin and arcos, as well as arcctg and arcctg of the same angle.
Examples of problem solving
Trigonometry tasks can be divided into four groups: calculate the numerical value of a specific expression, construct a graph of a given function, find its domain of definition or ODZ and perform analytical transformations to solve the example.
When solving the first type of problem, you must adhere to the following action plan:
When working with function graphs, the main thing is knowledge of their properties and the appearance of the curve. Solving trigonometric equations and inequalities requires identity tables. The more formulas a student remembers, the easier it is to find the answer to the task.
Let’s say in the Unified State Examination you need to find the answer for an equation like:
If you correctly transform the expression and bring it to the desired form, then solving it is very simple and quick. First, let's move arcsin x to the right side of the equality.
If you remember the formula arcsin (sin α) = α, then we can reduce the search for answers to solving a system of two equations:
The restriction on the model x arose, again from the properties of arcsin: ODZ for x [-1; 1]. When a ≠0, part of the system is a quadratic equation with roots x1 = 1 and x2 = - 1/a. When a = 0, x will be equal to 1.
Earlier in the program, students gained an idea of solving trigonometric equations, became familiar with the concepts of arc cosine and arc sine, and examples of solutions to the equations cos t = a and sin t = a. In this video tutorial we will look at solving the equations tg x = a and ctg x = a.
To begin studying this topic, consider the equations tg x = 3 and tg x = - 3. If we solve the equation tg x = 3 using a graph, we will see that the intersection of the graphs of the functions y = tg x and y = 3 has an infinite number of solutions, where x = x 1 + πk. The value x 1 is the x coordinate of the intersection point of the graphs of the functions y = tan x and y = 3. The author introduces the concept of arctangent: arctan 3 is a number whose tan is equal to 3, and this number belongs to the interval from -π/2 to π/2. Using the concept of arctangent, the solution to the equation tan x = 3 can be written as x = arctan 3 + πk.
By analogy, the equation tg x = - 3 is solved. From the constructed graphs of the functions y = tg x and y = - 3, it is clear that the intersection points of the graphs, and therefore the solutions to the equations, will be x = x 2 + πk. Using the arctangent, the solution can be written as x = arctan (- 3) + πk. In the next figure we see that arctg (- 3) = - arctg 3.
The general definition of arctangent is as follows: arctangent a is a number from the interval from -π/2 to π/2 whose tangent is equal to a. Then the solution to the equation tan x = a is x = arctan a + πk.
The author gives example 1. Find a solution to the expression arctan. Let us introduce the notation: the arctangent of a number is equal to x, then tg x will be equal to the given number, where x belongs to the segment from -π/2 to π/2. As in the examples in previous topics, we will use a table of values. According to this table, the tangent of this number corresponds to the value x = π/3. Let us write down the solution to the equation: the arctangent of a given number is equal to π/3, π/3 also belongs to the interval from -π/2 to π/2.
Example 2 - calculate the arctangent of a negative number. Using the equality arctg (- a) = - arctg a, we enter the value of x. Similar to example 2, we write down the value of x, which belongs to the segment from -π/2 to π/2. From the table of values we find that x = π/3, therefore, -- tg x = - π/3. The answer to the equation is - π/3.
Let's consider example 3. Solve the equation tg x = 1. Write that x = arctan 1 + πk. In the table, the value tg 1 corresponds to the value x = π/4, therefore, arctg 1 = π/4. Let's substitute this value into the original formula x and write the answer x = π/4 + πk.
Example 4: calculate tan x = - 4.1. In this case x = arctan (- 4.1) + πk. Because It is not possible to find the value of arctg in this case; the answer will look like x = arctg (- 4.1) + πk.
In example 5, the solution to the inequality tg x > 1 is considered. To solve it, we construct graphs of the functions y = tan x and y = 1. As can be seen in the figure, these graphs intersect at points x = π/4 + πk. Because in this case tg x > 1, on the graph we highlight the tangentoid region, which is located above the graph y = 1, where x belongs to the interval from π/4 to π/2. We write the answer as π/4 + πk< x < π/2 + πk.
Next, consider the equation cot x = a. The figure shows graphs of the functions y = cot x, y = a, y = - a, which have many intersection points. The solutions can be written as x = x 1 + πk, where x 1 = arcctg a and x = x 2 + πk, where x 2 = arcctg (- a). It is noted that x 2 = π - x 1 . This implies the equality arcctg (- a) = π - arcctg a. The following is the definition of arc cotangent: arc cotangent a is a number from the interval from 0 to π whose cotangent is equal to a. The solution to the equation сtg x = a is written as: x = arcctg a + πk.
At the end of the video lesson, another important conclusion is made - the expression ctg x = a can be written as tg x = 1/a, provided that a is not equal to zero.
TEXT DECODING:
Let's consider solving the equations tg x = 3 and tg x = - 3. Solving the first equation graphically, we see that the graphs of the functions y = tg x and y = 3 have infinitely many intersection points, the abscissas of which we write in the form
x = x 1 + πk, where x 1 is the abscissa of the point of intersection of the straight line y = 3 with the main branch of the tangentoid (Fig. 1), for which the designation was invented
arctan 3 (arc tangent of three).
How to understand arctg 3?
This is a number whose tangent is 3 and this number belongs to the interval (- ;). Then all roots of the equation tg x = 3 can be written by the formula x = arctan 3+πk.
Similarly, the solution to the equation tg x = - 3 can be written in the form x = x 2 + πk, where x 2 is the abscissa of the point of intersection of the straight line y = - 3 with the main branch of the tangentoid (Fig. 1), for which the designation arctg(- 3) (arc tangent minus three). Then all the roots of the equation can be written by the formula: x = arctan(-3)+ πk. The figure shows that arctg(- 3)= - arctg 3.
Let us formulate the definition of arctangent. The arctangent a is a number from the interval (-;) whose tangent is equal to a.
The equality is often used: arctg(-a) = -arctg a, which is valid for any a.
Knowing the definition of arctangent, we can make a general conclusion about the solution to the equation
tg x= a: the equation tg x = a has a solution x = arctan a + πk.
Let's look at examples.
EXAMPLE 1. Calculate arctan.
Solution. Let arctg = x, then tgх = and xϵ (- ;). Show table of values Therefore, x =, since tg = and ϵ (- ;).
So, arctan =.
EXAMPLE 2. Calculate arctan (-).
Solution. Using the equality arctg(- a) = - arctg a, we write:
arctg(-) = - arctg . Let - arctg = x, then - tgх = and xϵ (- ;). Therefore, x =, since tg = and ϵ (- ;). Show table of values
This means - arctg=- tgх= - .
EXAMPLE 3. Solve the equation tgх = 1.
1. Write down the solution formula: x = arctan 1 + πk.
2. Find the value of the arctangent
since tg = . Show table of values
So arctan1= .
3. Put the found value into the solution formula:
EXAMPLE 4. Solve the equation tgх = - 4.1 (tangent x is equal to minus four point one).
Solution. Let's write the solution formula: x = arctan (- 4.1) + πk.
We cannot calculate the value of the arctangent, so we will leave the solution to the equation in its obtained form.
EXAMPLE 5. Solve the inequality tgх 1.
Solution. We will solve it graphically.
- Let's construct a tangent
y = tgx and straight line y = 1 (Fig. 2). They intersect at points like x = + πk.
2. Let us select the interval of the x-axis in which the main branch of the tangentoid is located above the straight line y = 1, since by condition tgх 1. This is the interval (;).
3. We use the periodicity of the function.
Property 2. y=tg x is a periodic function with the main period π.
Taking into account the periodicity of the function y = tgх, we write the answer:
(;). The answer can be written as a double inequality:
Let's move on to the equation ctg x = a. Let us present a graphical illustration of the solution to the equation for positive and negative a (Fig. 3).
Graphs of functions y = ctg x and y = a and also
y=ctg x and y=-a
have infinitely many common points, the abscissas of which look like:
x = x 1 +, where x 1 is the abscissa of the point of intersection of the straight line y = a with the main branch of the tangentoid and
x 1 = arcctg a;
x = x 2 +, where x 2 is the abscissa of the point of intersection of the line
y = - a with the main branch of the tangentoid and x 2 = arcсtg (- a).
Note that x 2 = π - x 1. So, let’s write down an important equality:
arcсtg (-a) = π - arcсtg а.
Let us formulate the definition: arc cotangent a is a number from the interval (0;π) whose cotangent is equal to a.
The solution to the equation ctg x = a is written in the form: x = arcctg a + .
Please note that the equation ctg x = a can be transformed to the form
tg x = , except when a = 0.
This article is about finding the values of arcsine, arccosine, arctangent and arccotangent given number. First we will clarify what is called the meaning of arcsine, arccosine, arctangent and arccotangent. Next, we will obtain the main values of these arc functions, after which we will understand how the values of arc sine, arc cosine, arc tangent and arc cotangent are found using the tables of sines, cosines, tangents and Bradis cotangents. Finally, let's talk about finding the arcsine of a number when the arccosine, arctangent or arccotangent of this number, etc. is known.
Page navigation.
Values of arcsine, arccosine, arctangent and arccotangent
First of all, it’s worth figuring out what “this” actually is. the meaning of arcsine, arccosine, arctangent and arccotangent».
Bradis tables of sines and cosines, as well as tangents and cotangents, allow you to find the value of the arcsine, arccosine, arctangent and arccotangent of a positive number in degrees with an accuracy of one minute. Here it is worth mentioning that finding the values of the arcsine, arccosine, arctangent and arccotangent of negative numbers can be reduced to finding the values of the corresponding arcfunctions of positive numbers by turning to the formulas arcsin, arccos, arctg and arcctg of opposite numbers of the form arcsin(−a)=−arcsin a, arccos (−a)=π−arccos a , arctg(−a)=−arctg a and arcctg(−a)=π−arcctg a .
Let's figure out how to find the values of arcsine, arccosine, arctangent and arccotangent using the Bradis tables. We will do this with examples.
Let us need to find the arcsine value 0.2857. We find this value in the table of sines (cases when this value is not in the table will be discussed below). It corresponds to sine 16 degrees 36 minutes. Therefore, the desired value of the arcsine of the number 0.2857 is an angle of 16 degrees 36 minutes.
Often it is necessary to take into account corrections from the three columns on the right of the table. For example, if we need to find the arcsine of 0.2863. According to the table of sines, this value is obtained as 0.2857 plus a correction of 0.0006, that is, the value of 0.2863 corresponds to a sine of 16 degrees 38 minutes (16 degrees 36 minutes plus 2 minutes of correction).
If the number whose arcsine interests us is not in the table and cannot even be obtained taking into account corrections, then in the table we need to find the two values of the sines closest to it, between which this number is enclosed. For example, we are looking for the arcsine value of 0.2861573. This number is not in the table, and this number cannot be obtained using amendments either. Then we find the two closest values 0.2860 and 0.2863, between which the original number is enclosed; these numbers correspond to the sines of 16 degrees 37 minutes and 16 degrees 38 minutes. The desired arcsine value of 0.2861573 lies between them, that is, any of these angle values can be taken as an approximate arcsine value with an accuracy of 1 minute.
The arc cosine values, the arc tangent values and the arc cotangent values are found in absolutely the same way (in this case, of course, tables of cosines, tangents and cotangents are used, respectively).
Finding the value of arcsin using arccos, arctg, arcctg, etc.
For example, let us know that arcsin a=−π/12, and we need to find the value of arccos a. We calculate the arc cosine value we need: arccos a=π/2−arcsin a=π/2−(−π/12)=7π/12.
The situation is much more interesting when, using the known value of the arcsine or arccosine of a number a, you need to find the value of the arctangent or arccotangent of this number a or vice versa. Unfortunately, we do not know the formulas that define such connections. How to be? Let's understand this with an example.
Let us know that the arccosine of a number a is equal to π/10, and we need to calculate the arctangent of this number a. You can solve the problem as follows: using the known value of the arc cosine, find the number a, and then find the arc tangent of this number. To do this, we first need a table of cosines, and then a table of tangents.
The angle π/10 radians is an angle of 18 degrees; from the cosine table we find that the cosine of 18 degrees is approximately equal to 0.9511, then the number a in our example is 0.9511.
It remains to turn to the table of tangents, and with its help find the arctangent value we need 0.9511, it is approximately equal to 43 degrees 34 minutes.
This topic is logically continued by the material in the article. evaluating the values of expressions containing arcsin, arccos, arctg and arcctg.
Bibliography.
- 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
- 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.
- 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.
- I. V. Boykov, L. D. Romanova. Collection of problems for preparing for the Unified State Exam, part 1, Penza 2003.
- Bradis V. M. Four-digit math tables: For general education. textbook establishments. - 2nd ed. - M.: Bustard, 1999.- 96 p.: ill. ISBN 5-7107-2667-2
