In this article we will analyze in great detail the definition of the number circle, find out its main property and arrange the numbers 1,2,3, etc. About how to mark other numbers on the circle (for example, \(\frac(π)(2), \frac(π)(3), \frac(7π)(4), 10π, -\frac(29π)( 6)\)) understands .
Number circle called a circle of unit radius whose points correspond , arranged according to the following rules:
1) The origin is at the extreme right point of the circle;
2) Counterclockwise - positive direction; clockwise – negative;
3) If we plot the distance \(t\) on the circle in the positive direction, then we will get to a point with the value \(t\);
4) If we plot the distance \(t\) on the circle in the negative direction, then we will get to a point with the value \(–t\).
Why is the circle called a number circle?
Because it has numbers on it. In this way, the circle is similar to the number axis - on the circle, like on the axis, there is a specific point for each number.
Why know what a number circle is?
Using the number circle, the values of sines, cosines, tangents and cotangents are determined. Therefore, to know trigonometry and pass the Unified State Exam with 60+ points, you must understand what a number circle is and how to place dots on it.
What do the words “...of unit radius...” mean in the definition?
This means that the radius of this circle is equal to \(1\). And if we construct such a circle with the center at the origin, then it will intersect with the axes at points \(1\) and \(-1\).
It doesn’t have to be drawn small; you can change the “size” of the divisions along the axes, then the picture will be larger (see below).
Why is the radius exactly one? This is more convenient, because in this case, when calculating the circumference using the formula \(l=2πR\), we get:
The length of the number circle is \(2π\) or approximately \(6.28\).
What does “...the points of which correspond to real numbers” mean?
As we said above, on the number circle for any real number there will definitely be its “place” - a point that corresponds to this number.
Why determine the origin and direction on the number circle?
The main purpose of the number circle is to uniquely determine its point for each number. But how can you determine where to put the point if you don’t know where to count from and where to move?
Here it is important not to confuse the origin on the coordinate line and on the number circle - these are two different reference systems! And also do not confuse \(1\) on the \(x\) axis and \(0\) on the circle - these are points on different objects.
Which points correspond to the numbers \(1\), \(2\), etc.?
Remember, we assumed that the number circle has a radius of \(1\)? This will be our unit segment (by analogy with the number axis), which we will plot on the circle.
To mark a point on the number circle corresponding to the number 1, you need to go from 0 to a distance equal to the radius in the positive direction.
To mark a point on the circle corresponding to the number \(2\), you need to travel a distance equal to two radii from the origin, so that \(3\) is a distance equal to three radii, etc.
When looking at this picture, you may have 2 questions:
1. What happens when the circle “ends” (i.e. we make a full revolution)?
Answer: let's go for the second round! And when the second one is over, we’ll go to the third one and so on. Therefore, an infinite number of numbers can be plotted on a circle.
2. Where will the negative numbers be?
Answer: right there! They can also be arranged, counting from zero the required number of radii, but now in a negative direction.
Unfortunately, it is difficult to denote integers on the number circle. This is due to the fact that the length of the number circle will not be equal to an integer: \(2π\). And at the most convenient places (at the points of intersection with the axes) there will also be fractions, not integers
Video lessons are among the most effective teaching tools, especially in school subjects such as mathematics. Therefore, the author of this material has collected only useful, important and competent information into a single whole.
This lesson is 11:52 minutes long. It takes almost the same amount of time for a teacher to explain new material on a given topic in class. Although the main advantage of the video lesson will be the fact that students will listen carefully to what the author is talking about, without being distracted by extraneous topics and conversations. After all, if students do not listen carefully, they will miss an important point of the lesson. And if the teacher explains the material himself, then his students can easily distract from the main thing with their conversations on abstract topics. And, of course, it becomes clear which method will be more rational.
The author devotes the beginning of the lesson to repeating those functions that students were familiar with earlier in the algebra course. And the first to start studying are trigonometric functions. To consider and study them, a new mathematical model is required. And this model becomes the number circle, which is precisely what is stated in the topic of the lesson. To do this, the concept of a unit circle is introduced and its definition is given. Further in the figure, the author shows all the components of such a circle, and what will be useful to students for further learning. Arcs indicate quarters.
Then the author suggests considering the number circle. Here he makes the remark that it is more convenient to use a unit circle. This circle shows how point M is obtained if t>0, t<0 или t=0. После этого вводится понятие самой числовой окружности.
Next, the author reminds students how to find the circumference of a circle. And then it outputs the length of the unit circle. It is proposed to apply these theoretical data in practice. To do this, consider an example where you need to find a point on a circle that corresponds to certain number values. The solution to the example is accompanied by an illustration in the form of a picture, as well as the necessary mathematical notations.
According to the condition of the second example, it is necessary to find points on the number circle. Here, too, the entire solution is accompanied by comments, illustrations and mathematical notation. This contributes to the development and improvement of students’ mathematical literacy. The third example is constructed similarly.
Next, the author notes those numbers on the circle that occur more often than others. Here he suggests making two models of a number circle. When both layouts are ready, the next, fourth example is considered, where you need to find a point on the number circle corresponding to the number 1. After this example, a statement is formulated according to which you can find the point M corresponding to the number t.
Next, a remark is introduced according to which students learn that the number “pi” corresponds to all numbers that fall on a given point when it passes the entire circle. This information is supported by the fifth example. His solution contains logically correct reasoning and drawings illustrating the situation.
TEXT DECODING:
NUMERIC CIRCLE
Previously, we studied functions defined by analytical expressions. And these functions were called algebraic. But in the school mathematics course, functions of other classes, not algebraic ones, are studied. Let's start learning trigonometric functions.
In order to introduce trigonometric functions, we need a new mathematical model - the number circle. Let's consider the unit circle. A circle whose radius is equal to the scale segment, without indicating specific units of measurement, will be called unit. The radius of such a circle is considered equal to 1.
We will use a unit circle in which the horizontal and vertical diameters CA and DB (ce a and de be) are drawn (see Figure 1).
We will call arc AB the first quarter, arc BC the second quarter, arc CD the third quarter, and arc DA the fourth quarter.
Consider the number circle. In general, any circle can be considered as a numerical circle, but it is more convenient to use the unit circle for this purpose.
DEFINITION A unit circle is given, and the starting point A is marked on it - the right end of the horizontal diameter. Let us associate each real number t (te) with a point on the circle according to the following rule:
1) If t>0 (te is greater than zero), then, moving from point A in a counterclockwise direction (positive direction of the circle), we describe a path AM (a em) of length t along the circle. Point M will be the desired point M(t) (em from te).
2) If t<0(тэ меньше нуля), то, двигаясь из точки А в направлении по часовой стрелке (отрицательное направление обхода окружности), опишем по окружности путь АМ (а эм) длины |t| (модуль тэ). Точка М и будет искомой точкой М(t) (эм от тэ).
3) Let us assign point A to the number t = 0.
A unit circle with an established correspondence (between real numbers and points on the circle) will be called a number circle.
It is known that the circumference L (el) is calculated by the formula L = 2πR (el equals two pi er), where π≈3.14, R is the radius of the circle. For a unit circle R=1cm, that means L=2π≈6.28 cm (el is equal to two pi approximately 6.28).
Let's look at examples.
EXAMPLE 1. Find a point on the number circle that corresponds to the given number: ,.(pi by two, pi, three pi by two, two pi, eleven pi by two, seven pi, minus five pi by two)
Solution. The first six numbers are positive, therefore, to find the corresponding points on the circle, you need to walk a path of a given length along the circle, moving from point A in the positive direction. The length of each quarter of a unit circle is equal. This means AB =, that is, point B corresponds to the number (see Fig. 1). AC = , that is, point C corresponds to the number. AD = , that is, point D corresponds to the number. And point A again corresponds to the number, because after walking a path along the circle we ended up at the starting point A.
Let's consider where the point will be located. Since we already know what the length of the circle is, we will reduce it to the form (four pi plus three pi by two). That is, moving from point A in the positive direction, you need to describe a whole circle twice (a path of length 4π) and additionally a path of length that ends at point D.
What's happened? This is 3∙2π + π (three times two pi plus pi). This means that moving from point A in the positive direction, you need to describe a whole circle three times and additionally a path of length π, which will end at point C.
To find a point on the number circle that corresponds to a negative number, you need to walk from point A along the circle in the negative direction (clockwise) a path of length, and this corresponds to 2π +. This path will end at point D.
EXAMPLE 2. Find points on the number circle (pi by six, pi by four, pi by three).
Solution. Dividing arc AB in half, we get point E, which corresponds. And dividing the arc AB into three equal parts by points F and O, we obtain that point F corresponds, and point T corresponds
(see figure 2).
EXAMPLE 3. Find points on the number circle (minus thirteen pi by four, nineteen pi by six).
Solution. Depositing the arc AE (a em) of length (pi by four) from point A thirteen times in the negative direction, we obtain point H (ash) - the middle of the arc BC.
Depositing an arc AF of length (pi by six) from point A nineteen times in the positive direction, we get to point N (en), which belongs to the third quarter (arc CD) and CN is equal to the third part of the arc CD (se de).
(see figure example 2).
Most often you have to look for points on the number circle that correspond to the numbers (pi by six, pi by four, pi by three, pi by two), as well as those that are multiples of them, that is, (seven pi by six, five pi by four, four pi by three, eleven pi by two). Therefore, in order to quickly navigate, it is advisable to make two layouts of the number circle.
On the first layout, each of the quarters of the number circle will be divided into two equal parts and near each of the resulting points we will write their “names”:
On the second layout, each of the quarters is divided into three equal parts and near each of the resulting twelve points we write down their “names”:
If we move clockwise, we will get the same “names” for the points on the drawings, only with a minus value. For the first layout:
Similarly, if you move along the second layout clockwise from point O.
EXAMPLE 4. Find points on the number circle that correspond to the numbers 1 (one).
Solution. Knowing that π≈3.14 (pi is approximately equal to three point fourteen hundredths), ≈ 1.05 (pi times three is approximately equal to one point five hundredths), ≈ 0.79 (pi times four is approximately equal to zero point seventy nine hundredths) . Means,< 1 < (один больше, чем пи на четыре, но меньше, чем пи на три), то есть число 1 находится в первой четверти.
The following statement is true: if a point M on the number circle corresponds to a number t, then it corresponds to any number of the form t + 2πk(te plus two pi ka), where ka is any integer and kϵ Z(ka belongs to Zet).
Using this statement, we can conclude that the point corresponds to all points of the form t =+ 2πk (te is equal to pi times three plus two peaks), where kϵZ ( ka belongs to zet), and to the point (five pi by four) - points of the form t = + 2πk (te is equal to five pi by four plus two pi ka), where kϵZ ( ka belongs to zet) and so on.
EXAMPLE 5. Find the point on the number circle: a) ; b) .
Solution. a) We have: = =(6 +) ∙ π = 6π + = + 3∙ 2π.(twenty pi times three equals twenty times three pi equals six plus two thirds, multiplied by pi equals six pi plus two pi times three equals two pi times three plus three times two pi).
This means that the number corresponds to the same point on the number circle as the number (this is the second quarter) (see the second layout in Fig. 4).
b) We have: = - (8 +) ∙ π = + 2π ∙ (- 4). (minus thirty-five pi times four equals minus eight plus three fourths times pi equals minus three pi times four plus two pi times minus four). That is, the number corresponds to the same point on the number circle as the number
In this lesson we will recall the definition of a number line and give a new definition of a number circle. We will also consider in detail an important property of the number circle and important points on the circle. Let us define the direct and inverse problems for the number circle and solve several examples of such problems.
Topic: Trigonometric functions
Lesson: Number Circle
For any function, the independent argument is deferred either by number line, or on a circle. Let us characterize both the number line and number circle.
The straight line becomes a number (coordinate) line if the origin of coordinates is marked and the direction and scale are selected (Fig. 1).
The number line establishes a one-to-one correspondence between all points on the line and all real numbers.
For example, we take a number and put it on the coordinate axis, we get a point. We take a number and put it on the axis, we get a point (Fig. 2).
And vice versa, if we take any point on the coordinate line, then there is a unique real number corresponding to it (Fig. 2).
People did not come to such a correspondence right away. To understand this, let's remember the basic numerical sets.
First we introduced a set of natural numbers
Then a set of integers 
Set of rational numbers
It was assumed that these sets would be sufficient, and that there would be a one-to-one correspondence between all rational numbers and points on a line. But it turned out that there are countless points on the number line that cannot be described by numbers of the form
An example is the hypotenuse of a right triangle with legs 1 and 1. It is equal (Fig. 3).
Among the set of rational numbers, is there a number exactly equal to No, there is not. Let's prove this fact.
Let's prove it by contradiction. Let us assume that there is a fraction equal to i.e.
Then we square both sides. Obviously, the right side of the equality is divisible by 2, . This means and Then But then and A means Then it turns out that the fraction is reducible. This contradicts the condition, which means
The number is irrational. The set of rational and irrational numbers form the set of real numbers 
If we take any point on a line, some real number will correspond to it. And if we take any real number, there will be a single point corresponding to it on the coordinate line.
Let us clarify what a number circle is and what are the relationships between the set of points on the circle and the set of real numbers.
Origin - point A. Counting direction - counterclockwise - positive, clockwise - negative. Scale - circumference (Fig. 4).
Introducing these three provisions, we have number circle. We will indicate how to assign a point on a circle to each number and vice versa.
By setting the number we get a point on the circle
Each real number corresponds to a point on the circle. What about the other way around?
The dot corresponds to the number. And if we take numbers, all these numbers have only one point in their image on the circle
For example, corresponds to the point B(Fig. 4).
Let's take all the numbers. They all correspond to the point. B. There is no one-to-one correspondence between all real numbers and points on a circle.
If there is a fixed number, then only one point on the circle corresponds to it
If there is a point on a circle, then there is a set of numbers corresponding to it
Unlike a straight line, a coordinate circle does not have a one-to-one correspondence between points and numbers. Each number corresponds to only one point, but each point corresponds to an infinite number of numbers, and we can write them down.
Let's look at the main points on the circle.
Given a number, find which point on the circle it corresponds to.
Dividing the arc in half, we get a point (Fig. 5).
Inverse problem: given a point in the middle of an arc, find all real numbers that correspond to it.
Let us mark all multiple arcs on the number circle (Fig. 6).
Arcs that are multiples of
A number is given. You need to find the corresponding point.
Inverse problem - given a point, you need to find which numbers it corresponds to.
We looked at two standard tasks at two critical points.
a) Find a point on the number circle with coordinate
Delay from the point A this is two whole turns and another half, and we get a point M- this is the middle of the third quarter (Fig. 8).
Answer. Dot M- mid-third quarter.
b) Find a point on the number circle with coordinate
Delay from the point A a full turn and we still get a point N(Fig. 9).
Answer: Point N is in the first quarter.
We looked at the number line and the number circle and remembered their features. A special feature of the number line is the one-to-one correspondence between the points of this line and the set of real numbers. There is no such one-to-one correspondence on the circle. Each real number on the circle corresponds to a single point, but each point on the number circle corresponds to an infinite number of real numbers.
In the next lesson we will look at the number circle in the coordinate plane.
List of references on the topic "Number circle", "Point on a circle"
1. Algebra and beginning of analysis, grade 10 (in two parts). Textbook for general education institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
3. Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I. Algebra and mathematical analysis for grade 10 (textbook for students of schools and classes with in-depth study of mathematics). - M.: Prosveshchenie, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. In-depth study of algebra and mathematical analysis.-M.: Education, 1997.
5. Collection of problems in mathematics for applicants to higher educational institutions (edited by M.I. Skanavi). - M.: Higher School, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic simulator.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Problems on algebra and principles of analysis (a manual for students in grades 10-11 of general education institutions). - M.: Prosveshchenie, 2003.
8. Karp A.P. Collection of problems on algebra and principles of analysis: textbook. allowance for 10-11 grades. with depth studied Mathematics.-M.: Education, 2006.
Homework
Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
№№ 11.6 - 11.12, 11.15 - 11.17.
Additional web resources
3. Educational portal for exam preparation ().
Item name Algebra and beginning of mathematical analysis
Class 10
UMK Algebra and beginnings of mathematical analysis, grades 10-11. AT 2 . Part 1. Textbook for general education institutions (basic level) / A.G. Mordkovich. – 10th edition, ster. - M.: Mnemosyne, 2012. Part 2. Problem book for educational institutions (basic level) /[ A.G. Mordkovich et al.]; edited by A.G. Mordkovich. – 10th edition, ster. - M.: Mnemosyne, 2012.
Level of study. Base
Lesson topic Number circle (2 o'clock)
Lesson #1
Target: introduce the concept of a number circle as a model of a curvilinear coordinate system.
Tasks : to develop the ability to use the number circle when solving problems.
Planned results:
During the classes
Organizing time.
2. Checking homework that caused difficulties for students
II. Oral work.
1. Match each interval on the number line with an inequality and an analytical notation for the interval. Enter the data in the table.
A (– ; –5] D (–5; 5)
B [–5; 5] E (– ; –5)
IN [–5; + ) AND [–5; 5)
G (–5; 5] Z (–5; + )
1 –5 < X < 5 5 –5 X 5
2 X –5 6 X –5
3 –5 < X 5 7 5 X < 5
4 X < –5 8 X > –5
A1. Unlike the studied number line, the number circle is a more complex model. The concept of an arc, which underlies it, is not reliably worked out in geometry.
2 . Working with the textbook . Let's look at a practical example with. 23–24 textbooks (stadium running track). You can ask students to give similar examples (the movement of a satellite in orbit, the rotation of a gear, etc.).
3. We justify the convenience of using the unit circle as a numerical one.
4. Working with the textbook. Let's look at examples from p. 25–31 textbooks. The authors emphasize that for successful mastery of the number circle model, both the textbook and the problem book provide a system of special “didactic games.” There are six of them, in this lesson we will use the first four.
(Mordkovich A. G. M79 Algebra and the beginnings of mathematical analysis. Grades 10-11 (basic level): methodological manual for teachers / A. G. Mordkovich, P. V. Semenov. - M.: Mnemosyna, 2010. - 202 p. : ill.)
1st "game" – calculation of the arc length of a unit circle. Students should get used to the fact that the length of the entire circle is 2 , half a circle – , quarter circle – etc.
2nd "game" – finding points on the number circle corresponding to given numbers, expressed in fractions of a number
for example, points 
etc. (“good” numbers and points).
3rd "game" – finding points on the number circle that correspond to given numbers, not expressed in fractions of a number for example, points M (1), M (–5), etc. (“bad” numbers and points).
4th "game" – recording of numbers corresponding to a given “good” point on the number circle, for example, the middle of the first quarter is “good”, the numbers corresponding to it have the form 
Dynamic pause
The exercises solved in this lesson correspond to the four designated didactic games. Students use a number circle layout with diametersAC (horizontal) andBD(vertical).
1. № 4.1, № 4.3.
Solution:
№ 4.3.
2. № 4.5 (a; b) – 4.11 (a; b).
3. № 4.12.
4. № 4.13 (a; b), № 4.14.
Solution:
№ 4.13.
V. Test work.
Option 1
Option 2
1. Mark the point on the number circle that corresponds to this number:
2. Find all the numbers that correspond to the points marked on the number circle.
VI. Lesson summary.
Questions for students:
– Give the definition of a number circle.
– What is the length of a unit circle? Length of half a unit circle? Her quarters?
– How can you find a point on the number circle that corresponds to a number? Number 5?
Homework:, page 23. No. 4.2, No. 4.4, No. 4.5 (c; d) – No. 4.11 (c; d), No. 4.13 (c; d), No. 4.15.
Lesson #2
Goals : consolidate the concept of the number circle as a model of a curvilinear coordinate system.
Tasks : continue to develop the ability to find points on the number circle that correspond to given “good” and “bad” numbers; write down the number corresponding to a point on the number circle; develop the ability to compose an analytical notation of the arc of a number circle in the form of a double inequality.
To develop computational skills, correct mathematical speech, and logical thinking of students.
Instill independence, attention and accuracy. Foster a responsible attitude towards learning.
Planned results:
Know, understand: - number circle.
Be able to: - find points on a circle according to given coordinates; - find the coordinates of a point located on a number circle.
Be able to apply the studied theoretical material when performing written work.
Lesson technical support Computer, screen, projector, textbook, problem book.
Additional methodological and didactic support for the lesson: Mordkovich A. G. M79 Algebra and the beginnings of mathematical analysis. Grades 10-11 (basic level): methodological manual for teachers / A. G. Mordkovich, P. V. Semenov. - M.: Mnemosyna, 2010. - 202 p. : silt
During the classes
Organizing time.
Psychological mood of students.
Checking homeworkNo. 4.2, No. 4.4, No. 4.5 (c; d) – No. 4.11 (c; d), No. 4.13 (c; d),
№ 4.15. Analyze the solution to tasks that caused difficulty.
Oral work.
(on slide)
1. Match the points on the number circle and the given numbers:
b)
V)
G)
d)
e)
and)
h)
2. Find the points on the number circle.
–2; 4; –8;
13.
III. Explanation of new material.
As already noted, students master a system of six didactic “games” that provide the ability to solve problems of four main types associated with the number circle (from number to point; from point to number; from arc to double inequality; from double inequality to arc).
(Mordkovich A. G. M79 Algebra and the beginnings of mathematical analysis. Grades 10-11 (basic level): methodological manual for teachers / A. G. Mordkovich, P. V. Semenov. - M.: Mnemosyne, 2010. - 202 p. : ill.)
In this lesson we will use the last two games:
5th "game" – compilation of analytical records (double inequalities) for arcs of the number circle. For example, if an arc is given connecting the middle of the first quarter (the beginning of the arc) and the lowest point of the two that divide the second quarter into three equal parts (the end of the arc), then the corresponding analytical notation has the form:
If the beginning and end of the same arc are swapped, then the corresponding analytical record of the arc will look like:
The authors of the textbook note that the terms “core of analytical notation of an arc”, “analytical notation of an arc” are not generally recognized, they were introduced for purely methodological reasons, and whether to use them or not is up to the teacher.
6th "game" – from this analytical notation of the arc (double inequality) move to its geometric image.
The explanation should be carried out using the technique of analogy. You can use a movable number line model that can be “collapsed” into a number circle.
Working with the textbook .
Let's look at example 8 from p. 33 textbooks.
Dynamic pause
IV. Formation of skills and abilities.
When completing assignments, students must ensure that when writing an arc analytically, the left side of the double inequality is less than the right side. To do this, when recording, you need to move in a positive direction, that is, counterclockwise.
1st group . Exercises to find “bad” points on the number circle.
№ 4.16, No. 4.17 (a; b).
2nd group . Exercises on the analytical recording of an arc and the construction of an arc based on its analytical recording.
№ 4.18 (a; b), No. 4.19 (a; b), No. 4.20 (a; b).
V. Independent work.
Option 1
3. According to the analytical model 
write down the designation of the number arc and build its geometric model.
Option 2
1. Based on the geometric model of the arc of the number circle, write the analytical model in the form of a double inequality.
2. According to the given designation of the arc of the number circle 
indicate its geometric and analytical models.
3. According to the analytical model 
write down the designation of the arc of the number circle and build its geometric model.
VI. Lesson summary.
Questions for students:
– In what ways can you write analytically the arc of the number circle?
– What is called the core of the analytical recording of an arc?
– What conditions must the numbers on the left and right of a double inequality meet?
Homework:
1. , page 23. No. 4.17 (c; d), No. 4.18 (c; d), No. 4.19 (c; d), No. 4.20 (c; d).
2. Based on the geometric model of the arc of the number circle, write down its analytical model in the form of a double inequality.
3. According to the given designation of the arc of the number circle 
indicate its geometric and analytical models.
