Negative numbers on a number circle. Number circle lesson

If you are already familiar with trigonometric circle , and you just want to refresh individual elements in your memory, 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 are associated with an impassable thicket. Suddenly, so many values ​​​​of trigonometric functions pile up, so many formulas ... But it’s like, it didn’t work out at first, and ... off and on ... sheer misunderstanding ...

It is very important not to wave your hand at values ​​of trigonometric functions, - they say, you can always look at the spur with a table of values.

If you constantly look at the table with the values ​​​​of trigonometric formulas, let's get rid of this habit!

Will save us! You will work with it several times, and then it will pop up in your head on its own. Why 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, looking at standard table of values ​​of trigonometric formulas , which is the sine of, 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 solving trigonometric equations and inequalities without a trigonometric circle - nowhere at all.

Introduction to the trigonometric circle

Let's go in order.

First, write down the following series of numbers:

And now this:

And finally this one:

Of course, it is clear that, in fact, in the first place is, in the second place is, and in the last -. That is, we will be more interested in the chain .

But how beautiful it turned out! In which case, we will restore this “wonderful ladder”.

And why do we need it?

This chain is the main values ​​​​of sine and cosine in the first quarter.

Let's draw a circle of unit radius in a rectangular coordinate system (that is, we take any radius along the length, and declare its length to be unit).

From the “0-Start” beam, we set aside in the direction of the arrow (see Fig.) corners.

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 that, you ask?

Let's not take everything apart. Consider principle, which will allow you to cope with other, similar situations.

Triangle AOB is a right triangle with . And we know that opposite the angle at lies a leg twice as small as the hypotenuse (our hypotenuse = the radius of the circle, that is, 1).

Hence, AB= (and hence OM=). And by the Pythagorean theorem

I hope something is clear now.

So point B will correspond to the value, and point M will correspond to the value

Similarly with the rest of the values ​​of the first quarter.

As you understand, the axis familiar to us (ox) will be cosine axis, and the axis (oy) - sinus axis . later.

To the left of zero on the cosine axis (below zero on the sine axis) there will, of course, be negative values.

So, here it is, the ALL-POWERFUL, without which nowhere in trigonometry.

But how to use the trigonometric circle, we'll talk in.

Chapter 2
3) number

let's match the dot.

The unit circle with the established correspondence will be called

number circle.

This is the second geometric model for the set of real

numbers. The first model - the number line - students already know. There is

analogy: for the number line, the correspondence rule (from number to point)

almost verbatim the same. But there is also a fundamental difference - the source

main difficulties in working with a number circle: on a straight line, each

dot corresponds the only number, on a circle it is not. If a

circle corresponds to a number, then it corresponds to all

numbers of the form

Where is the length of the unit circle, and is an integer

Rice. one

a number indicating the number of complete rounds of the circle in one direction or another

side.

This moment is difficult for students. They should be offered

understanding the essence of the real task:

The stadium running track is 400m long, the runner is 100m away

from the starting point. What path did he take? If he just started running, then

ran 100 m; if you managed to run one lap, then - (

Two circles - () ; if you can run

circles, then the path will be (

) . Now you can compare

the result obtained with the expression

Example 1 What numbers does the dot correspond to

number circle

Decision. Since the length of the whole circle

That is the length of her quarter

Therefore, for all numbers of the form

Similarly, it is established which numbers correspond to the points

called respectively the first, second, third,

fourth quarters of the number circle.

All school trigonometry is based on a numerical model

circles. Experience shows that the shortcomings with this model are too

hasty introduction of trigonometric functions do not allow to create

a solid foundation for the successful assimilation of the material. Therefore, not

you need to hurry, and take some time to consider the following

five different types of problems with a number circle.

The first type of tasks. Finding points on the numerical circle,

corresponding to given numbers, expressed in fractions of a number

Example 2

numbers

Decision. Let's split the arc

in half with a point into three equal parts -

dots

(Fig. 2). Then

So the number

Corresponding point

number
Example

3.
on the

numerical

circles

points,

corresponding numbers:

Decision. We will build

a) Postponing the arc

(its length

) Five times

from the point

in the negative direction

get a point

b) Postponing the arc

(its length

) seven times from

in the positive direction, we get a point separating

third part of the arc

It will correspond to the number

c) Postponing the arc

(its length

) five times from the point

positive

direction, we get a point

Separating the third part of the arc. She and

will match the number

(experience shows that it is better to postpone not

five times over

And 10 times

After this example, it is appropriate to give two main layouts of the numeric

circles: on the first of them (Fig. 3) all quarters are divided in half, on

the second (Fig. 4) - into three equal parts. These layouts are useful to have in the office

mathematics.

Rice. 2

Rice. 3 Rice. 4

Be sure to discuss with students the question: what will happen if

each of the layouts move not in positive, but in negative

direction? On the first layout, the selected points will have to be assigned

other "names": respectively

etc.; on the second layout:

The second type of tasks. Finding points on the numerical circle,

corresponding to given numbers, not expressed in fractions of a number

Example 4 Find points on the number circle corresponding to

numbers 1; 2; 3; -5.

Decision.

Here we have to rely on the fact that

Therefore point 1

located on the arc

closer to the point

Points 2 and 3 are on the arc, the first one is

The second is closer to (Fig. 5).

Let's take a closer look

on finding the point corresponding to the number - 5.

Move from a point

in the negative direction, i.e. clockwise

Rice. 5

arrow. If we go in this direction to the point

Get

This means that the point corresponding to the number - 5 is located

slightly to the right of the dot

(see fig.5).

The third type of tasks. Preparation of analytical records (double

inequalities) for arcs of a numerical circle.

In fact, we are acting on

the same plan that was used in 5-8

classes for studying the number line:

first find a point by number, then by

dot - number, then use double

inequalities for writing gaps on

number line.

Consider, for example, an open

Where is the middle of the first

quarters of a number circle, and

- its middle

second quarter (Fig. 6).

The inequalities characterizing the arc, i.e. representing

An analytical model of the arc is proposed to be compiled in two stages. On the first

stage constitute the core analytical record(this is the main thing to follow

teach students) for a given arc

On the second

stage make up a general record:

If we are talking about arc

Then, when writing the kernel, you need to take into account that

() lies inside the arc, and therefore you have to move to the beginning of the arc

in the negative direction. Hence, the kernel of the analytical notation of the arc

has the form

Rice. 6

The terms "kernel of the analytical

arc records", "analytical record

arcs" are not generally accepted,

considerations.

Fourth

tasks.

Finding

Cartesian

coordinates

number circle points, center

which is combined with the beginning of the system

coordinates.

Let us first consider one rather subtle point, until now

practically not mentioned in current school textbooks.

Starting to study the model "numerical circle on a coordinate

plane", teachers should be clearly aware of what difficulties await

students here. These difficulties are related to the fact that in the study of this

models from schoolchildren are required to have a sufficiently high level

mathematical culture, because they have to work simultaneously in

two coordinate systems - in the "curvilinear", when information about

the position of the point is taken along the circle (number

corresponds to

circle point

(); is the “curvilinear coordinate” of the point), and in

Cartesian rectangular coordinate system (at the point

Like every point

coordinate plane, there is an abscissa and an ordinate). The task of the teacher is to help

schoolchildren in overcoming these natural difficulties. Unfortunately,

usually in school textbooks they do not pay attention to this and from the very

first lessons use notes

Not considering that the letter in

in the mind of a schoolchild is clearly associated with the abscissa in the Cartesian

rectangular coordinate system, and not with the length traveled along the numerical

path circles. Therefore, when working with a number circle, one should not

use symbols

Rice. 7

Let's return to the fourth type of tasks. It's about moving from writing

records

(), i.e. from curvilinear to cartesian coordinates.

Let's combine the number circle with the Cartesian rectangular system

coordinates as shown in Fig. 7. Then dots

will have

the following coordinates:

() () () (). Very important

teach students to determine the coordinates of all those points that

marked on two main layouts (see Fig.3,4). For point

It all comes down to

considering an isosceles right triangle with a hypotenuse

His legs are equal

So the coordinates

). The same is true for points.

But the only difference is that you need to take into account

abscissa and ordinate signs. Specifically:

What should students remember? Only that the modules of the abscissa and

the ordinates at the midpoints of all quarters are equal

And they must know the signs

determine for each point directly from the drawing.

For point

It all comes down to considering a rectangular

triangle with hypotenuse 1 and angle

(Fig. 9). Then the cathet

opposite corner

Will be equal

adjacent

√

Means,

point coordinates

The same is true for the point

only the legs "change places", and therefore

Rice. eight

Rice. nine

we get

). It is the meanings

(up to signs) and will be

“serve” all points of the second layout (see Fig. 4), except for points

as abscissa and ordinate. Suggested way of remembering: "where is shorter,

; where it is longer

Example 5 Find coordinates of a point

(see Fig.4).

Decision. Dot

Closer to the vertical axis than to

horizontal, i.e. the modulus of its abscissa is less than the modulus of its ordinate.

So the modulus of the abscissa is

The module of the ordinate is

signs in both

cases are negative (third quarter). Conclusion: dot

Has coordinates

In the fourth type of problems, the Cartesian coordinates of all

points presented on the first and second layouts mentioned

In fact, in the course of this type of tasks, we prepare students for

calculation of values ​​of trigonometric functions. If everything is here

worked out quite reliably, then the transition to a new level of abstraction

(ordinate - sine, abscissa - cosine) will be less painful than

The fourth type includes tasks of this type: for a point

find signs of cartesian coordinates

The decision should not cause difficulties for students: the number

match point

Fourth quarter means .

Fifth type of tasks. Finding points on the numerical circle by

given coordinates.

Example 6 Find points with ordinate on a number circle

write down what numbers they correspond to.

Decision. Straight

Crosses the number circle at points
(Fig. 11). With the help of the second layout (see Fig. 4) we set that the point

corresponds to the number

So she

matches all numbers of the form
corresponds to the number

And that means

all numbers of the form

Answer:

Example 7 Find on numeric

circle point with abscissa

write down what numbers they correspond to.

Decision. Straight
√

intersects the number circle at points

- in the middle of the second and third quarters (Fig. 10). With the help of the first

layout set that point

corresponds to the number

And that means everyone

numbers of the form

corresponds to the number

And that means everyone

numbers of the form

Answer:

You must show the second option.

record the answer for example 7. After all, the point

corresponds to the number

Those. all numbers of the form

we get:

Rice. ten

Fig.11

Emphasize the undeniable importance

the fifth type of tasks. In fact, we teach

schoolchildren

decision

protozoa

trigonometric equations: in example 6

it's about the equation

And in the example

- about the equation

understanding of the essence of the matter is important to teach

schoolchildren solve equations of the types

along the number circle

don't rush into formulas

Experience shows that if the first stage (work on

numerical circle) is not worked out reliably enough, then the second stage

(work on formulas) is perceived by schoolchildren formally, that,

Naturally, it must be overcome.

Similar to examples 6 and 7 should be found on the number circle

points with all "major" ordinates and abscissas

As special subjects, it is appropriate to single out the following:

Remark 1. In propaedeutic terms, preparatory

work on the topic "Length of a circle" in the course of geometry of the 9th grade. Important

advice: the system of exercises should include tasks of the type proposed

below. The unit circle is divided into four equal parts by points

the arc is bisected by a point and the arc is bisected by points

into three equal parts (Fig. 12). What are the lengths of the arcs

(it is assumed that the circumnavigation of the circle is carried out in a positive

direction)?

Rice. 12

The fifth type of tasks includes working with conditions like

means
to

decision

protozoa

trigonometric inequalities, we also “fit” gradually.

five lessons and only in the sixth lesson should the definitions of sine and

cosine as the coordinates of a point on a numerical circle. Wherein

it is advisable to solve all types of problems with schoolchildren again, but with

using the introduced notation, offering to perform such

for example, tasks: calculate

solve the equation

inequality

etc. We emphasize that in the first lessons

trigonometry simple trigonometric equations and inequalities

are not purpose training, but used as facilities for

mastering the main thing - the definitions of sine and cosine as coordinates of points

number circle.

Let the number

match point

number circle. Then its abscissa

called cosine of a number

and denoted

And its ordinate is called the sine of a number

and is marked. (Fig. 13).

From this definition one can immediately

set the signs of sine and cosine according to

quarters: for sine

For cosine

Dedicate a whole lesson to this (as it is

accepted) is hardly appropriate. Do not do it

force schoolchildren to memorize these signs: any mechanical

memorization, memorization is a violent technique to which students,

>> Number circle

While studying the algebra course of grades 7-9, we have so far dealt with algebraic functions, i.e. functions given analytically by expressions, in the notation of which algebraic operations on numbers and a variable were used (addition, subtraction, multiplication, division, exponentiation, square root extraction). But mathematical models of real situations are often associated with functions of a different type, not algebraic. With the first representatives of the class of non-algebraic functions - trigonometric functions - we will get acquainted in this chapter. You will study trigonometric functions and other types of non-algebraic functions (exponential and logarithmic) in more detail in high school.
To introduce trigonometric functions, we need a new mathematical model- a number circle, which you have not yet met, but are well acquainted with the number line. Recall that a number line is a line on which the starting point O, the scale (single segment) and the positive direction are given. We can associate any real number with a point on a straight line and vice versa.

How to find the corresponding point M on the line given the number x? The number 0 corresponds to the starting point O. If x > 0, then, moving in a straight line from the point 0 in the positive direction, you need to go n^th 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 is considered circles. Here is a specific example. We will consider the stadium treadmill as a circle (in fact, it is, of course, not a circle, but remember how sports commentators usually say: “the runner ran a circle”, “there is half a circle left to run to the finish line”, etc.), its length is 400 m. The start is marked - point A (Fig. 97). The runner from point A moves in a counterclockwise circle. Where will he be in 200 meters? after 400 m? after 800 m? after 1500 m? And where to draw the finish line if he runs a marathon distance of 42 km 195 m?

After 200 m, he will be at point C, diametrically opposite point A (200 m is the length of half the treadmill, i.e. the length of half the 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. And 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 have to deal with the marathon. After running 105 laps, the athlete will overcome the distance 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 of the last example. Nobody runs the marathon distance around the stadium, the maximum is 10,000 m, i.e. 25 circles.

You can run or walk a path of any length along the stadium's running track. This means that any positive number corresponds to some point - the “finish of the distance”. Moreover, any negative number can be associated with a circle point: you just need to make the athlete run in the opposite direction, i.e. start from point A not in the opposite direction, but in the clockwise direction. Then the stadium running track can be considered as a numerical circle.

In principle, any circle can be considered as a numerical one, 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 We agree to call the arc AB the first quarter of a unit circle, the arc BC - the second quarter, the arc CB - the third quarter, the 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, the starting point A is marked on it - the right end of the horizontal diameter (Fig. 98). Associate each real number I with a circle point 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 describe a path along the circle with a 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);

0 we assign point A: A = A(0).

A unit circle with an established correspondence (between real numbers and points of the circle) will be called a number circle.
Example 1 Find on the number circle
Since the first six of the given seven numbers are positive, then in order to find the corresponding points on the circle, you need to go along the circle a path of a given length, moving from point A in a positive direction. At the same time, we take into account that

Point A corresponds to the number 2, since, having passed a path of length 2 along the circle, i.e. exactly one circle, we again get to the starting point A So, A \u003d A (2).
What So, moving from point A in a positive direction, you need to go through a whole circle.

Comment. When we're in 7th or 8th grade worked with the number line, we agreed, for the sake of brevity, not to say "point of the line corresponding to the number x", but to say "point x". We will adhere to exactly the same agreement when working with a numerical circle: "point f" - this means that we are talking about a circle point 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. Postponing the arc AM (its length is equal to -) from the point A five times in the negative direction, we get the point!, - the middle of the arc BC. So,

Comment. Notice some 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 a geometric figure, and the second concept is a number). But both are denoted 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 attached to the designation (arc, arc length, segment or segment length).

Therefore, two layouts of the number circle will be very useful to us.

FIRST LAYOUT
Each of the four quarters of the numerical circle is divided into two equal parts, and their “names” are written near each of the eight available points (Fig. 100).

SECOND LAYOUT Each of the four quarters of the numerical circle is divided into three equal parts, and their “names” are written near each of the twelve points available (Fig. 101).

Please note that on both layouts, we could assign other “names” to the given points.
Have you noticed that in all the examples analyzed, the lengths of the arcs
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 the circle or its quarter into equal parts, then we get arcs whose lengths are expressed as fractions of the number and. And what do you think, is it possible to find such a point E on the unit circle that the length of the arc AE will be equal to 1? Let's guess:

Arguing in a similar way, we conclude that on the unit circle one can find both the point Eg, for which AE, = 1, and the point E2, for which AEg = 2, and the point E3, for which AE3 = 3, and the point E4, for which AE4 = 4, and point Eb, for which AEb = 5, and point E6, for which AE6 = 6. In fig. 102 (approximately) the corresponding points are marked (moreover, for orientation, each of the quarters of the unit circle is divided by dashes into three equal parts).

Example 4 Find on the number circle the point corresponding to the number -7.

We need, starting from the point A (0) and moving in a negative direction (in a clockwise direction), go around the circle path of length 7. If we go through one circle, we get (approximately) 6.28, which means we still need to go ( in the same direction) a path of length 0.72. What is this arc? Slightly less than half a quarter of a circle, i.e. its length is less than number -.

So, a numerical circle, like a numerical straight line, each real number corresponds to one point (only, of course, it is easier to find it on a straight line than on a circle). But for a straight line, the opposite is also true: each point corresponds to a single number. For a numerical circle, such a statement is not true, we have repeatedly convinced ourselves of this above. For a number circle, the following statement is true.
If the point M of the numerical circle corresponds to the number I, then it also corresponds to a number of the form I + 2n, where k is any integer (k e 2).

Indeed, 2n is the length of the numerical (unit) circle, and the integer |d| 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 \u003d -7, then this means that we make seven (| k | \u003d | -71 \u003d 7) rounds of the circle in the negative direction. But if we are at the point M(1), then by doing more | to | full circles, we will again find ourselves at the point M.

A.G. Mordkovich Algebra Grade 10

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When studying trigonometry at school, each student is faced with a very interesting concept of "numerical circle". It depends on the ability of the school teacher to explain what it is and why it is needed, how well the student will go about trigonometry later. Unfortunately, not every teacher can explain this material in an accessible way. As a result, many students get confused even with how to celebrate points on the number circle. If you read this article to the end, you will learn how to do it without problems.

So let's get started. Let's draw a circle, the radius of which is equal to 1. The most "right" point of this circle will be denoted by the letter O:

Congratulations, you have just drawn a unit circle. Since the radius of this circle is 1, then its length is .

Each real number can be associated with the length of the trajectory along the number circle from the point O. The direction of movement is counterclockwise as the positive direction. For negative - clockwise:

Arrangement of points on a number circle

As we have already noted, the length of the numerical circle (unit circle) is equal to. Where then will the number be located on this circle? Obviously from the point O counterclockwise, you need to go half the length of the circle, and we will find ourselves at the desired point. Let's denote it with a letter B:

Note that the same point could be reached by passing the semicircle in the negative direction. Then we would put the number on the unit circle. That is, the numbers and correspond to the same point.

Moreover, the same point also corresponds to the numbers , , , and, in general, an infinite set of numbers that can be written in the form , where , that is, belongs to the set of integers. All this is because from the point B you can make a "round the world" trip in any direction (add or subtract the circumference) and get to the same point. We get an important conclusion that needs to be understood and remembered.

Each number corresponds to a single point on the number circle. But each point on the number circle corresponds to infinitely many numbers.

Let us now divide the upper semicircle of the numerical circle into arcs of equal length with a point C. It is easy to see that the arc length OC is equal to . Let's put aside now from the point C an arc of the same length in a counterclockwise direction. As a result, we get to the point B. The result is quite expected, since . Let's postpone this arc in the same direction again, but now from the point B. As a result, we get to the point D, which will already match the number :

Note again that this point corresponds not only to the number , but also, for example, to the number , because this point can be reached by setting aside from the point O quarter circle in the clockwise direction (in the negative direction).

And, in general, we note again that this point corresponds to an infinite number of numbers that can be written in the form . But they can also be written as . Or, if you like, in the form of . All these records are absolutely equivalent, and they can be obtained from one another.

Let us now break the arc into OC halved dot M. Think now what is the length of the arc OM? That's right, half the arc OC. I.e . What numbers does the dot correspond to M on a number circle? I am sure that now you will realize that these numbers can be written in the form.

But it is possible otherwise. Let's take in the presented formula. Then we get that . That is, these numbers can be written as . The same result could be obtained using a number circle. As I said, both entries are equivalent, and they can be obtained from one another.

Now you can easily give an example of numbers that correspond to points N, P and K on the number circle. For example, numbers , and :

Often it is precisely the minimal positive numbers that are taken to designate the corresponding points on the number circle. Although this is not at all necessary, and the point N, as you already know, corresponds to an infinite number of other numbers. Including, for example, the number .

If you break the arc OC into three equal arcs with dots S and L, so the point S will lie between the points O and L, then the arc length OS will be equal to , and the length of the arc OL will be equal to . Using the knowledge that you received in the previous part of the lesson, you can easily figure out how the rest of the points on the number circle turned out:

Numbers that are not multiples of π on the number circle

Let us now ask ourselves the question, where on the number line to mark the point corresponding to the number 1? To do this, it is necessary from the most "right" point of the unit circle O set aside an arc whose length would be equal to 1. We can only approximately indicate the location of the desired point. Let's proceed as follows.

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 numerical circle into four quarters. They are respectively called the first, second, third and fourth quarter.

3) The horizontal diameter is designated AC, with A being the extreme right dot.
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 numerical circle is point A.

The number circle can be counted either clockwise or counterclockwise.

Counting from point A against clockwise is called positive direction.

Counting from point A on clockwise is called negative direction.

Number circle on the coordinate plane.

The center of the radius of the numerical circle corresponds to the origin (number 0).

Horizontal diameter corresponds to the axis x, vertical - axes y.

Starting point A number circleti is on the axisxand has coordinates (1; 0).

Names and locations of the main points of the number circle:

How to remember the names of the number circle.

There are a few simple patterns that will help you easily remember the basic names of the number circle.

Before we start, we recall: the countdown is in the positive direction, that is, from point A (2π) counterclockwise.

1) Let's start from 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. So half the circle is 1π or π. Axis X divides the circle in half. Accordingly, the leftmost point on the axis X equal to -1 is called π.

Highest point on the axis at, equal to 1, bisects the upper semicircle. So if the semicircle is π, then half of the semicircle is π/2.

At the same time, π/2 is also a quarter of a circle. We 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 rest of the points. Please note: all opposite points have the same denominator - moreover, these are opposite points and relative to the axis at, and relative to the center of the axes, and relative to the axis X. This will help us to know their point values ​​without cramming.

It is necessary to remember only the value of the points of the first quarter: π / 6, π / 4 and π / 3. And then we will “see” some patterns:

- Axis Relative at at the points of the second quarter, opposite to the points of the first quarter, the numbers in the numerators are 1 less than the denominators. For example, take the point π/6. The opposite point about the axis at also has 6 in the denominator, and 5 in the numerator (1 less). That is, the name of this point: 5π/6. The point opposite to π/4 also has 4 in the denominator, and 3 in the numerator (1 less than 4) - that is, this is the point 3π/4.
The point opposite to π/3 also has 3 in the denominator, and 1 less in the numerator: 2π/3.

- Relative to the center of the coordinate axes the opposite is true: the numbers in the numerators of the opposite points (in the third quarter) are 1 more than the values ​​of the denominators. 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 more - that is, it is 7π / 6.
The point opposite to the point π/4 also has 4 in the denominator, and the number in the numerator is 1 more: 5π/4.
The point opposite to the point π/3 also has 3 in the denominator, and the number in the numerator is 1 more: 4π/3.

- Axis Relative X(fourth quarter) the matter is more difficult. Here it is necessary 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 value of the denominator, equal to 6, a number that is 1 less than this number - that is, 5. We get: 6 + 5 = 11. Hence, opposite to it with respect 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. Hence, opposite to it with respect to the axis X the point has 4 in the denominator and 7 in the numerator, i.e. 7π/4.
Point π/3. The denominator is 3. We add to 3 one less number - that is, 2. We get 5. Hence, the opposite point has 5 in the numerator - and this is the point 5π / 3.

3) Another regularity for 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 1 is not customary to write). 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 in the numerators of the midpoints of the quarters there are the first four odd numbers in ascending order:
(1)π, 3π, 5π, 7π.
It's also very simple. Since the middles 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 a number line.

As you know, on the number line, each point corresponds to a single number. For example, if point A on a straight line is equal to 3, then it cannot equal 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 the circle to point M, you can do it as on a straight line (only after passing the arc), or you can go around the whole circle, and then come to point M. Conclusion:

Let the point M be equal to some number t. As we know, the circumference of a circle is 2π. Hence, we can write the point of the circle t 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, and two hundred such circles. . If we denote the number of circles by the letter n, we get a new expression:
t = t + 2π n.

Hence the formula: