Item name Algebra and beginning of mathematical analysis
Class 10
WMC Algebra and the beginning of mathematical analysis, grades 10-11. IN 2 . Part 1. Textbook for educational institutions (basic level) / A.G. Mordkovich. - 10th ed., ster. - M.: Mnemozina, 2012. Part 2. Task book for educational institutions (basic level) /[ A.G. Mordkovich and others.]; ed. A.G. Mordkovich. - 10th ed., ster. - M.: Mnemozina, 2012.
Level of study. Base
Lesson topic Number circle (2 hours)
Lesson #1
Target: introduce the concept of a numerical circle as a model of a curvilinear coordinate system.
Tasks : to form 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. Associate each interval on the number line with an inequality and an analytical record of the interval. Enter the data on the table.
BUT (– ; –5] D (–5; 5)
B [–5; 5] E (– ; –5)
AT [–5; + ) AND [–5; 5)
G (–5; 5] W (–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. In contrast to 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 . Consider a practical example from p. 23–24 textbook (stadium running track). You can ask students to give similar examples (satellite orbiting, gear turning, etc.).
3. We substantiate the convenience of using the unit circle as a numerical one.
4. Work with the textbook. Consider examples from p. 25–31 of the textbook. The authors emphasize that in order to successfully master the model of a numerical circle, both in the textbook and in the problem book, a system of special "didactic games" is provided. There are six of them, in this lesson we will use the first four.
(Mordkovich A. G. M79 Algebra and the beginning of mathematical analysis. Grades 10-11 (basic level): a teaching aid for a teacher / A. G. Mordkovich, P. V. Semenov. - M.: Mnemozina, 2010. - 202 p. : ill.)
1st "game" – calculation of the length of the arc of a unit circle. Students should get used to the fact that the length of the entire circle is 2 , half circle - , quarters of a circle - etc.
2nd "game" - finding points on the numerical circle corresponding to given numbers, expressed in fractions of a number
e.g. dots 
and so on ("good" numbers and points).
3rd "game" - finding points on the numerical circle that correspond to given numbers, expressed not in fractions of a number e.g. dots M (1), M (-5), etc. ("bad" numbers and points).
4th "game" - a record of numbers corresponding to a given "good" point of the numerical 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. Verification work.
Option 1
Option 2
1. Mark on the number circle the point that corresponds to the given number:
2. Find all the numbers that correspond to the points marked on the number circle.
VI. Lesson results.
Questions for students:
– Define a number circle.
– What is the length of the unit circle? The length of one half of the unit circle? Her quarters?
– How can you find a point on a 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 : to fix the concept of a numerical circle as a model of a curvilinear coordinate system.
Tasks : to continue the formation of the ability to find points on the numerical circle corresponding to the given "good" and "bad" numbers; write down the number corresponding to a point on the number circle; to form the ability to make an analytical record of the arc of a numerical circle in the form of a double inequality.
To develop computational skills, correct mathematical speech, logical thinking of students.
To instill independence, attention and accuracy. Cultivate a responsible attitude to learning.
Planned results:
To know, to understand: - a numerical circle.
Be able to: - find points on a circle according to given coordinates; - find the coordinates of a point located on a number circle.
To be able to apply the studied theoretical material in the performance of written work.
Technical support of the lesson 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): a teaching aid for a teacher / A. G. Mordkovich, P. V. Semenov. - M.: Mnemozina, 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 problem solving problems.
oral work.
(on slide)
1. Match the points on the number circle and the given numbers:
b)
in)
G)
e)
e)
and)
h)
2. Find points on the number circle.
–2; 4; –8;
13 .
III. Explanation of new material.
As already noted, students master the system of six didactic "games" that provide the ability to solve problems of four main types related to 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 beginning of mathematical analysis. Grades 10-11 (basic level): a teaching aid for a teacher / A. G. Mordkovich, P. V. Semenov. - M. : Mnemozina, 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 a numerical circle. For example, if an arc is given that connects the middle of the first quarter (the beginning of the arc) and the lower point of those 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 interchanged, then the corresponding analytical record of the arc will look like:
The authors of the textbook note that the terms "kernel of the analytical notation of the arc", "analytical notation of the arc" are not generally recognized, they are introduced for purely methodological reasons, and it is up to the teacher to use them or not.
6th "game" – from the given analytical record of the arc (double inequality) go to its geometric representation.
The explanation should be carried out with the help of analogy. You can use a moving model of the number line, which can be "folded" into a number circle.
Working with the textbook .
Consider example 8 from p. 33 textbooks.
Dynamic pause
IV. Formation of skills and abilities.
When completing assignments, students must ensure that when the arc is analytically recorded, the left side of the double inequality is less than the right side. To do this, you need to move in the positive direction, that is, counterclockwise, when writing.
1st group . Exercises to find "bad" points on the numerical circle.
№ 4.16, No. 4.17 (a; b).
2nd group . Exercises on the analytical notation of an arc and the construction of an arc from its analytical notation.
№ 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 numerical arc and build its geometric model.
Option 2
1. According to the geometric model of the arc of a numerical circle, write down the analytical model in the form of a double inequality.
2. According to the given designation of the arc of the numerical circle 
indicate its geometric and analytical models.
3. According to the analytical model 
write down the designation of the arc of a numerical circle and build its geometric model.
VI. Lesson results.
Questions for students:
– In what ways can an arc of a number circle be written analytically?
– What is called the core of the analytical record of the arc?
– What conditions must be met by the numbers on the left and right in the notation of a double inequality?
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. According to the geometric model of the arc of a numerical circle, write down its analytical model in the form of a double inequality.
3. According to the given designation of the arc of the numerical circle 
indicate its geometric and analytical models.
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 with a radius of 1. Let's denote the "right" point of this circle 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 be 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. That is . 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 the minimum positive numbers that are taken to denote 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.
In this article, we will analyze in great detail the definition of a numerical 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 call a circle of unit radius, the points of which correspond to 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 the point with the value \(t\);
4) If we plot the distance \(t\) on the circle in the negative direction, then we will get to the point with the value \(–t\).
Why is a circle called a number?
Because it has numbers on it. In this, the circle is similar to a number axis - on the circle, as well as on the axis, for each number there is a certain point.
Why know what a number circle is?
With the help of a numerical circle, the value of sines, cosines, tangents and cotangents is determined. Therefore, in order to know trigonometry and pass the exam with 60+ points, it is imperative to 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 \(1\). And if we construct such a circle centered at the origin, then it will intersect with the axes at the points \(1\) and \(-1\).
It is not necessary to draw it small, you can change the “size” of divisions along the axes, then the picture will be larger (see below).
Why is the radius exactly one? It’s 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 \(6,28\).
And what does "... the points of which correspond to real numbers" mean?
As mentioned 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 an end 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! Also, don't confuse \(1\) on the \(x\) axis and \(0\) on the circle - these are points on different objects.
What points correspond to the numbers \(1\), \(2\), etc?
Remember, we assumed that the radius of a number circle is \(1\)? This will be our single segment (by analogy with the number axis), which we will put on the circle.
To mark a point on the number circle corresponding to the number 1, you need to travel from 0 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.
Looking at this picture, you may have 2 questions:
1. What will happen when the circle "ends" (i.e. we make a full turn)?
Answer: let's go to the second round! And when the second is over, we will go to the third and so on. Therefore, an infinite number of numbers can be applied to 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 designate integers on the number circle. This is due to the fact that the length of the numerical circle will not be an integer: \ (2π \). And at the most convenient places (at the points of intersection with the axes) there will also be not integers, but fractions
Chapter 23) 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 circle, 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
Solution. 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
Solution. 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:
Solution. 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. four
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.
Solution.
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. 9
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).
Solution. 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.
Solution. 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.
Solution.
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 funds 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. It does not follow
force schoolchildren to memorize these signs: any mechanical
memorization, memorization is a violent technique to which students,
We present to your attention a video lesson on the topic "Numeric Circle". A definition is given of what sine, cosine, tangent, cotangent and functions are y= sin x, y= cos x, y= tg x, y= ctg x for any numeric argument. We consider standard tasks for the correspondence between numbers and points in a unit number circle to find a single point for each number, and, conversely, to find for each point a set of numbers that correspond to it.
Topic: Elements of the theory of trigonometric functions
Lesson: Number Circle
Our immediate goal is to define trigonometric functions: sinus, cosine, tangent, cotangent-
A numerical argument can be plotted on a coordinate line or on a circle.
Such a circle is called a numerical or unit circle, because. for convenience, take a circle with
For example, given a point, mark it on the coordinate line
and on number circle.
When working with a number circle, it was agreed that counterclockwise movement is a positive direction, clockwise movement is negative.
Typical tasks - you need to determine the coordinates of a given point or, conversely, find a point by its coordinates.
The coordinate line establishes a one-to-one correspondence between points and numbers. For example, a number corresponds to point A with coordinate
Each point B with a coordinate is characterized by only one number - the distance from 0 to taken with a plus or minus sign.
On the number circle, one-to-one correspondence only works in one direction.
For example, there is a point B on the coordinate circle (Fig. 2), the length of the arc is 1, i.e. this point corresponds to 1.
Given a circle, the circumference of a circle. If then is the length of the unit circle.
If we add , we get the same point B, more - we also get to point B, subtract - also point B.
Consider point B: arc length =1, then the numbers characterize point B on the number circle.
Thus, the number 1 corresponds to the only point of the numerical circle - point B, and the point B corresponds to an uncountable set of points of the form 
.
The following is true for a number circle:
If T. M number circle corresponds to a number then it also corresponds to a number of the form
You can make as many full turns around the number circle in a positive or negative direction as you like - the point is the same. Therefore, trigonometric equations have an infinite number of solutions.
For example, given point D. What are the numbers it corresponds to?
We measure the arc.
the set of all numbers corresponding to the point D.
Consider the main points on the number circle.
The length of the whole circle.
Those. the record of the set of coordinates can be different .
Consider typical tasks on the number circle.
1. Given: . Find: a point on a number circle.
We select the whole part:
It is necessary to find m. on the number circle. 
, then 
.
This set also includes the point .
2. Given: . Find: a point on a number circle.
Need to find t.
m. also belongs to this set.
Solving standard problems on the correspondence between numbers and points on a number circle, we found out that it is possible to find a single point for each number, and it is possible to find for each point a set of numbers that are characterized by a given point.
Let's divide the arc into three equal parts and mark the points M and N.
Let's find all the coordinates of these points.
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So, our goal is to define trigonometric functions. To do this, we need to learn how to set a function argument. We considered the points of the unit circle and solved two typical problems - to find a point on the number circle and write down all the coordinates of the point of the unit circle.
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