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>> Number circle
While studying the algebra course for grades 7-9, we have so far dealt with algebraic functions, i.e. functions specified analytically by expressions, in the recording of which we used algebraic operations over numbers and variables (addition, subtraction, multiplication, division, exponentiation, extraction square root). But mathematical models of real situations are often associated with functions of a different type, not algebraic. We will get acquainted with the first representatives of the class of non-algebraic functions - trigonometric functions - in this chapter. You will study trigonometric functions and other types of non-algebraic functions (exponential and logarithmic) in more detail in high school.
For introduction trigonometric functions we will need a new one mathematical model- a number circle that you have not yet encountered, but you are very familiar with the number line. Recall that the number line is a straight line on which the starting point O, the scale (unit segment) and the positive direction are given. We can compare any real number with a point on a line and vice versa.
How to find the corresponding point M on a line using the number x? The number 0 corresponds to the starting point O. If x > 0, then, moving along a straight line from point 0 in the positive direction, you need to go n^th of length x; the end of this path will be the desired point M(x). If x< 0, то, двигаясь по прямой из точки О в отрицательном направлении, нужно пройти путь 1*1; конец этого пути и будет искомой точкой М(х). Число х - координата точки М.
And how did we solve the inverse problem, i.e. How did you find the x coordinate of a given point M on the number line? We found the length of the segment OM and took it with the sign “+” or * - “depending on which side of the point O the point M is located on the straight line.
But in real life You have to move not only in a straight line. Quite often, movement along circle. Here's a concrete example. Let us consider the stadium running track to be a circle (in fact, it is, of course, not a circle, but remember, as sports commentators usually say: “the runner has run a circle”, “there is half a circle left to run before the finish”, etc.), its length is 400 m. The start is marked - point A (Fig. 97). The runner from point A moves around the circle counterclockwise. Where will he be in 200 m? in 400 m? in 800 m? in 1500 m? Where should he draw the finish line if he is running a marathon distance of 42 km 195 m?
After 200 m, he will be at point C, diametrically opposite to point A (200 m is the length of half the treadmill, i.e. the length of half a circle). After running 400 m (i.e., “one lap,” as the athletes say), he will return to point A. After running 800 m (i.e., “two laps”), he will again be at point A. What is 1500 m ? This is “three circles” (1200 m) plus another 300 m, i.e. 3
Treadmill - the finish of this distance will be at point 2) (Fig. 97).
We just have to deal with the marathon. After running 105 laps, the athlete will cover a distance of 105-400 = 42,000 m, i.e. 42 km. There are 195 m left to the finish line, which is 5 m less than half the circumference. This means that the finish of the marathon distance will be at point M, located near point C (Fig. 97).
Comment. Of course you understand the convention last example. No one runs a marathon distance around the stadium, the maximum is 10,000 m, i.e. 25 laps.
You can run or walk any length along the stadium treadmill. This means that any positive number corresponds to some point - the “finish of the distance”. Moreover, anyone can negative number match a point on the circle: you just need to make the athlete run in the opposite direction, i.e. start from point A not in a counter-clockwise direction, but in a clockwise direction. Then the stadium running track can be considered as a number circle.
In principle, any circle can be considered as a numerical circle, but in mathematics it was agreed to use a unit circle for this purpose - a circle with a radius of 1. This will be our “treadmill”. The length b of a circle with radius K is calculated by the formula The length of a half circle is n, and the length of a quarter circle is AB, BC, SB, DA in Fig. 98 - equal Let us agree to call arc AB the first quarter of the unit circle, arc BC the second quarter, arc CB the third quarter, arc DA the fourth quarter (Fig. 98). In this case, we are usually talking about an Open arc, i.e. about an arc without its ends (something like an interval on a number line).
Definition. A unit circle is given, and the starting point A is marked on it - the right end of the horizontal diameter (Fig. 98). Let's match each one real number I point of the circle according to the following rule:
1) if x > 0, then, moving from point A in a counterclockwise direction (the positive direction of going around the circle), we will describe a path along the circle with length and the end point M of this path will be the desired point: M = M(x);
2) if x< 0, то, двигаясь из точки А в направлении по часовой стрелке (отрицательное направление обхода окружности), опишем по окружности путь длиной и |; конечная точка М этого пути и будет искомой точкой: М = М(1);
Let us associate point A with 0: A = A(0).
A unit circle with an established correspondence (between real numbers and points on the circle) will be called a number circle.
Example 1. Find on number circle 
Since the first six of the given seven numbers are positive, then to find the corresponding points on the circle, you need to walk along the circle given length, moving from point A in a positive direction. Let us take into account that
The number 2 corresponds to point A, since, having passed along the circle a path of length 2, i.e. exactly one circle, we will again get to the starting point A So, A = A(2).
What's happened 
This means that moving from point A in a positive direction, you need to go through a whole circle.
Comment. When we are in 7th and 8th grades worked with the number line, then we agreed, for the sake of brevity, not to say “the point on the line corresponding to the number x,” but to say “point x.” We will adhere to exactly the same agreement when working with the number circle: “point f” - this means that we are talking about a point on the circle that corresponds to the number
Example 2.
Dividing the first quarter AB into three equal parts by points K and P, we get: 
Example 3. Find points on the number circle that correspond to numbers 
We will make constructions using Fig. 99. Depositing arc AM (its length is -) from point A five times in the negative direction, we obtain point!, - the middle of arc BC. So,
Comment. Please note some of the liberties we take in using mathematical language. It is clear that the arc AK and the length of the arc AK are different things (the first concept is geometric figure, and the second concept is number). But both are designated the same way: AK. Moreover, if points A and K are connected by a segment, then both the resulting segment and its length are denoted in the same way: AK. It is usually clear from the context what meaning is intended in the designation (arc, arc length, segment or segment length).
Therefore, two number circle layouts will be very useful to us.
FIRST LAYOUT
Each of the four quarters of the number circle is divided into two equal parts, and near each of the available eight points their “names” are written (Fig. 100).
SECOND LAYOUT Each of the four quarters of the number circle is divided into three equal parts, and near each of the available twelve points their “names” are written (Fig. 101).
Please note that on both layouts we could given points assign other “names”.
Have you noticed that in all the examples of arc lengths
expressed by some fractions of the number n? This is not surprising: after all, the length of a unit circle is 2n, and if we divide a circle or its quarter into equal parts, we get arcs whose lengths are expressed in fractions of the number and. Do you think it is possible to find a point E on the unit circle such that the length of the arc AE is equal to 1? Let's figure it out:
Reasoning in a similar way, we conclude that on the unit circle one can find point Eg, for which AE = 1, and point E2, for which AEr = 2, and point E3, for which AE3 = 3, and point E4, for which AE4 = 4, and point Eb, for which AEb = 5, and point E6, for which AE6 = 6. In Fig. 102 the corresponding points are marked (approximately) (for orientation, each of the quarters of the unit circle is divided by dashes into three equal parts).
Example 4. Find the point on the number circle corresponding to the number -7.
We need, starting from point A(0) and moving in a negative direction (clockwise direction), to go along a circle of length 7. If we go through one circle, we get (approximately) 6.28, which means we still need to go through ( in the same direction) a path of length 0.72. What kind of arc is this? A little less than half a quarter circle, i.e. its length is less than the number -. 
So, on a number circle, like on a number line, each real number corresponds to one point (only, of course, it is easier to find it on a line than on a circle). But for a straight line the opposite is also true: each point corresponds to a single number. For a number circle, such a statement is not true; we have repeatedly seen this above. The following statement is true for the number circle.
If point M of the number circle corresponds to the number I, then it also corresponds to a number of the form I + 2k, where k is any integer (k e 2).
In fact, 2n is the length of the numerical (unit) circle, and the integer |th| can be considered as the number of complete rounds of the circle in one direction or another. If, for example, k = 3, then this means that we make three rounds of the circle in the positive direction; if k = -7, then this means that we are doing seven (| k | = | -71 = 7) rounds of the circle in the negative direction. But if we are at point M(1), then, having also completed | to | full circles around the circle, we will again find ourselves at point M.
A.G. Mordkovich Algebra 10th grade
Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for a year guidelines discussion programs Integrated LessonsIn 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 number sets.
First we introduced a set of natural numbers
Then a set of integers 
A bunch 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. Many 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 straight 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). Tutorial for educational 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 10th grade ( tutorial 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 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 to prepare for exams ().
Number circle is a unit circle whose points correspond to certain real numbers.
A unit circle is a circle of radius 1.
General view of the number circle.
1) Its radius is taken as a unit of measurement.
2) The horizontal and vertical diameters divide the number circle into four quarters (see figure). They are respectively called the first, second, third and fourth quarter.
3) The horizontal diameter is denoted by AC, with A being the far right point.
The vertical diameter is designated BD, with B being the highest point.
Respectively:
the first quarter is the arc AB
second quarter – arc BC
third quarter – arc CD
fourth quarter – arc DA
4) The starting point of the number circle is point A.
Counting along the number circle can be done either clockwise or counterclockwise.
Counting from point A counterclockwise is called positive direction.
Counting from point A clockwise is called negative direction.
Number circle on the coordinate plane.
The center of the radius of the number circle corresponds to the origin (number 0).
The horizontal diameter corresponds to the axis x, vertical – axes y.
The starting point A of the number circle is on the axis x and has coordinates (1; 0).
Valuesx Andy in quarters of a number circle:
Basic quantities of the number circle:
Names and locations of the main points on the number circle:
How to remember number circle names.
There are several simple patterns that will help you easily remember the basic names of the number circle.
Before we begin, let us remind you: the counting is carried out in the positive direction, that is, from point A (2π) counterclockwise.
1) Let's start with the extreme points on the coordinate axes.
The starting point is 2π (the rightmost point on the axis X, equal to 1).
As you know, 2π is the circumference of a circle. This means that half a circle is 1π or π. Axis X divides the circle exactly in half. Accordingly, the leftmost point on the axis X equal to -1 is called π.
The highest point on the axis at, equal to 1, divides the upper semicircle in half. This means that if a semicircle is π, then half a semicircle is π/2.
At the same time, π/2 is also a quarter of a circle. Let's count three such quarters from the first to the third - and we will come to the lowest point on the axis at, equal to -1. But if it includes three quarters, then its name is 3π/2.
2) Now let's move on to the remaining points. Please note: all opposite points have the same numerator - and these are opposite points relative to the axis at, both relative to the center of the axes, and relative to the axis X. This will help us know their point values without cramming.
You only need to remember the meaning of the points of the first quarter: π/6, π/4 and π/3. And then we will “see” some patterns:
- Relative to the y axis at points of the second quarter opposite to the points of the first quarter, the numbers in the numerators are by 1 less than the value denominators. For example, take the point π/6. The point opposite to it relative to the axis at also has 6 in the denominator and 5 in the numerator (1 less). That is, the name of this point is: 5π/6. The point opposite π/4 also has 4 in the denominator and 3 in the numerator (1 less than 4) - that is, it is a 3π/4 point.
The point opposite π/3 also has 3 in the denominator, and 1 less in the numerator: 2π/3.
- Relative to the center of the coordinate axes everything is the other way around: the numbers in the numerators of opposite points (in the third quarter) are 1 greater than the value of the denominators. Let's take the point π/6 again. The point opposite to it relative to the center also has 6 in the denominator, and in the numerator the number is 1 larger - that is, it is 7π/6.
The point opposite the point π/4 also has 4 in the denominator, and in the numerator the number is 1 more: 5π/4.
The point opposite the point π/3 also has 3 in the denominator, and in the numerator the number is 1 more: 4π/3.
- Relative to axis X(fourth quarter) the matter is more complicated. Here you need to add to the value of the denominator a number that is 1 less - this sum will be equal to the numerical part of the numerator of the opposite point. Let's start again with π/6. Let's add to the denominator value equal to 6 a number that is 1 less than this number - that is, 5. We get: 6 + 5 = 11. This means that it is opposite to the axis X the point will have 6 in the denominator and 11 in the numerator - that is, 11π/6.
Point π/4. We add to the value of the denominator a number 1 less: 4 + 3 = 7. This means that it is opposite to the axis X the point has 4 in the denominator and 7 in the numerator - that is, 7π/4.
Point π/3. The denominator is 3. We add to 3 a smaller number by one - that is, 2. We get 5. This means that the point opposite to it has 5 in the numerator - and this is the point 5π/3.
3) Another pattern for the points of the midpoints of the quarters. It is clear that their denominator is 4. Let's pay attention to the numerators. The numerator of the middle of the first quarter is 1π (but it is not customary to write 1). The numerator of the middle of the second quarter is 3π. The numerator of the middle of the third quarter is 5π. The numerator of the middle of the fourth quarter is 7π. It turns out that the numerators of the middle quarters contain the first four odd numbers in ascending order:
(1)π, 3π, 5π, 7π.
This is also very simple. Since the midpoints of all quarters have 4 in the denominator, we already know their full names: π/4, 3π/4, 5π/4, 7π/4.
Features of the number circle. Comparison with the number line.
As you know, on the number line, each point corresponds to a single number. For example, if point A on a line is equal to 3, then it can no longer be equal to any other number.
It's different on the number circle because it's a circle. For example, in order to come from point A of a circle to point M, you can do it as if on a straight line (only passing an arc), or you can go around a whole circle, and then come to point M. Conclusion:
Let point M be equal to some number t. As we know, the circumference of a circle is 2π. This means that we can write a point t on a circle in two ways: t or t + 2π. These are equivalent values.
That is, t = t + 2π. The only difference is that in the first case you came to point M immediately without making a circle, and in the second case you made a circle, but ended up at the same point M. You can make two, three, or two hundred such circles . If we denote the number of circles by the letter k, then we get a new expression:
t = t + 2π k.
Hence the formula:
Number circle equation
(the second equation is in the section “Sine, cosine, tangent, cotangent”):
x 2 + y 2 = 1 |
Coordinates x points lying on the circle are equal to cos(θ), and the coordinates y correspond to sin(θ), where θ is the magnitude of the angle.
- If you find it difficult to remember this rule, just remember that in the pair (cos; sin) “the sine comes last.”
- This rule can be derived by considering right triangles and definition of trigonometric functions data (sine of angle equal to the ratio the length of the opposite side, and the cosine is the length of the adjacent side to the hypotenuse).
Write down the coordinates of four points on the circle. A “unit circle” is a circle whose radius is equal to one. Use this to determine the coordinates x And y at four points of intersection of the coordinate axes with the circle. Above, for clarity, we designated these points as “east”, “north”, “west” and “south”, although they do not have established names.
- "East" corresponds to the point with coordinates (1; 0) .
- "North" corresponds to the point with coordinates (0; 1) .
- "West" corresponds to the point with coordinates (-1; 0) .
- "South" corresponds to the point with coordinates (0; -1) .
- This is similar to a regular graph, so there is no need to memorize these values, just remember the basic principle.
Remember the coordinates of the points in the first quadrant. The first quadrant is located in the upper right part of the circle, where the coordinates x And y accept positive values. These are the only coordinates you need to remember:
- the point π / 6 has coordinates () ;
- the point π/4 has coordinates () ;
- the point π/3 has coordinates () ;
- Note that the numerator only takes three values. If you move in a positive direction (from left to right along the axis x and from bottom to top along the axis y), the numerator takes the values 1 → √2 → √3.
Draw straight lines and determine the coordinates of the points of their intersection with the circle. If you draw straight horizontal and vertical lines from the points of one quadrant, the second points of intersection of these lines with the circle will have the coordinates x And y with the same absolute values, but different signs. In other words, you can draw horizontal and vertical lines from the points of the first quadrant and label the points of intersection with the circle with the same coordinates, but at the same time leave space on the left for the correct sign ("+" or "-").
- For example, one can draw a horizontal line between points π/3 and 2π/3. Since the first point has coordinates ( 1 2 , 3 2 (\displaystyle (\frac (1)(2)),(\frac (\sqrt (3))(2)))), the coordinates of the second point will be (? 12 , ? 3 2 (\displaystyle (\frac (1)(2)),?(\frac (\sqrt (3))(2)))), where instead of the "+" or "-" sign there is a question mark.
- Use the simplest method: pay attention to the denominators of the coordinates of the point in radians. All points with a denominator of 3 have the same absolute coordinate values. The same applies to points with denominators 4 and 6.
To determine the sign of the coordinates, use the rules of symmetry. There are several ways to determine where to place the "-" sign:
- Remember the basic rules for regular charts. Axis x negative on the left and positive on the right. Axis y negative from below and positive from above;
- start with the first quadrant and draw lines to other points. If the line crosses the axis y, coordinate x will change its sign. If the line crosses the axis x, the sign of the coordinate will change y;
- remember that in the first quadrant all functions are positive, in the second quadrant only the sine is positive, in the third quadrant only the tangent is positive, and in the fourth quadrant only the cosine is positive;
- Whichever method you use, you should get (+,+) in the first quadrant, (-,+) in the second, (-,-) in the third, and (+,-) in the fourth.
Check if you made a mistake. Below is full list coordinates of "special" points (except for four points on coordinate axes), if you move along the unit circle counterclockwise. Remember that to determine all these values, it is enough to remember the coordinates of the points only in the first quadrant:
- first quadrant: ( 3 2 , 1 2 (\displaystyle (\frac (\sqrt (3))(2)),(\frac (1)(2)))); (2 2 , 2 2 (\displaystyle (\frac (\sqrt (2))(2)),(\frac (\sqrt (2))(2)))); (1 2 , 3 2 (\displaystyle (\frac (1)(2)),(\frac (\sqrt (3))(2))));
- second quadrant: ( − 1 2 , 3 2 (\displaystyle -(\frac (1)(2)),(\frac (\sqrt (3))(2)))); (− 2 2 , 2 2 (\displaystyle -(\frac (\sqrt (2))(2)),(\frac (\sqrt (2))(2)))); (− 3 2 , 1 2 (\displaystyle -(\frac (\sqrt (3))(2)),(\frac (1)(2))));
- third quadrant: ( − 3 2 , − 1 2 (\displaystyle -(\frac (\sqrt (3))(2)),-(\frac (1)(2)))); (− 2 2 , − 2 2 (\displaystyle -(\frac (\sqrt (2))(2)),-(\frac (\sqrt (2))(2)))); (− 1 2 , − 3 2 (\displaystyle -(\frac (1)(2)),-(\frac (\sqrt (3))(2))));
- fourth quadrant: ( 1 2 , − 3 2 (\displaystyle (\frac (1)(2)),-(\frac (\sqrt (3))(2)))); (2 2 , − 2 2 (\displaystyle (\frac (\sqrt (2))(2)),-(\frac (\sqrt (2))(2)))); (3 2 , − 1 2 (\displaystyle (\frac (\sqrt (3))(2)),-(\frac (1)(2)))).
