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Manuals and simulators in the online store "Integral" for grade 10 from 1C
Algebraic problems with parameters, grades 9–11
Software environment "1C: Mathematical constructor 6.1"
1. Number circle in life.
2. Definition of a numerical circle.
3. General view and length of the numerical circle.
4. Location of the main points of the circle.
Number circle and life
In real life, circular motion is common. For example, cycling competitions that complete a certain lap against the clock, or racing car competitions that need to complete the most laps in the allotted time.
Consider a specific example…
A runner runs in a circle 400 meters long. The athlete starts at point A (fig. 1) and moves counterclockwise. Where will he be in 200 m, 800 m, 1500 m? And where to draw the finish line if the runner needs to run 4195 m? 
Solution:
After 200 m, the runner will be at point C. Since he will run exactly half the distance.
After running 800 m, the runner will make exactly two laps and end up at point A.
1500m is 3 laps of 400m (1200m) and another 300m, i.e. $\frac(3)(4)$ from the track, finishing this distance at point D.
Where will our runner be after running 4195 m? 10 laps is 4000m, 195m remains to be run, which is 5m less than half the distance. So the finish line will be at point K, located near point C.
Definition of a number circle
Remember!is a unit circle whose points correspond to certain real numbers. unit circle called a circle of radius 1.
General view of the number circle
1) Radius circle is taken as the unit of measurement.
2) Horizontal the diameter is denoted AC, with A being the rightmost point. Vertical the diameter is designated BD, with B being the highest point.
Diameters AC and BD divide the circle into four quarters:
first quarter is the arc AB.
second quarter- arc BC.
third quarter– arc CD.
fourth quarter– arc DA.
3) starting point number circle - point A.
Counting from point A counterclockwise is called the positive direction. Counting from point A clockwise is called the negative direction.
Number circle length
The length of the numerical circle is calculated by the formula:$L = 2 π * R = 2 π * 1 = 2 π$.
Since this is the unit circle, then $R = 1$.
If we take $π ≈ 3.14$, then the circumference L can be expressed as a number:
$2 π ≈ 2 * 3.14 = $6.28.
The length of each quarter is: $\frac(1)(4)*2π=\frac(π)(2)$.
Location of the main points of the circle
The main points on the circle and their names are shown in the figure:
Each of the four quarters of the numerical circle is divided into three equal parts. Near each of the twelve points obtained, a number is written to which it corresponds.
The following statement is true for a number circle:If a point $M$ of a number circle corresponds to a number $t$ , then it also corresponds to a number of the form $t+2π *k$, where $k$ is an integer. $M(t) = M(t+2π*k)$.
Consider an example.
In the unit circle, the arc AB is divided by the point M into two equal parts, and by the points K and P into three equal parts. What is the length of the arc: AM, MB, AK, KR, RB, AR, KM?
Arc length $AB =\frac(π)(2)$. Dividing it into two equal parts by the point M, we get two arcs, each of length $\frac(π)(4)$. Hence $AM =MV=\frac(π)(4)$.
The arc AB is divided into three equal parts by points K and P. The length of each resulting part is equal to $\frac(1)(3)* \frac(π)(2)$, i.e. $\frac(π)(6) $. Hence, $AK = CR = RV =\frac(π)(6)$.
The arc АР consists of two arcs AK and КР of length - $\frac(π)(6)$. Hence $AP = 2 *\frac(π)(6) =\frac(π)(3)$.
It remains to calculate the length of the KM arc. This arc is obtained from the arc AM by eliminating the arc AK. Thus, $KM = AM – AK =\frac(π)(4) - \frac(π)(6) = \frac(π)(12)$.
A task:
Find a point on the number circle that corresponds to a given number:
$2π$, $\frac(7π)(2)$, $\frac(π)(4)$, $-\frac(3π)(2)$.
Solution:
The point A corresponds to the number $2π$, because passing along the circle a path of length $2π$, i.e. exactly one circle, we again get to point A.
The number $\frac(7π)(2)$ corresponds to the point D, because $\frac(7π)(2)=2π+\frac(3π)(2)$, i.e. moving in the positive direction, you need to go through a whole circle and additionally a path of length $\frac(3π)(2)$, which will end at point D.
The point M corresponds to the number $\frac(π)(4)$, because moving in the positive direction, you need to go through a path of half the arc AB of length $\frac(π)(2)$, which will end at the point M.
The number $-\frac(3π)(2)$ corresponds to point B, because moving in a negative direction from point A, you need to go through a path of length $\frac(3π)(2)$, which will end at point B.
Example.
Find points on the number circle:
a) $21\frac(π)(4)$;
b) $-37\frac(π)(6)$.
Solution:
Let's use the formula: $M(t) = M(t+2π*k)$ (8 slide) we get:
a) $\frac(21π)(4) = (4+\frac(5)(4))*π = 4π +\frac(5π)(4) = 2*2π +\frac(5π)(4) $, then the number $\frac(21π)(4)$ corresponds to the same number as the number $\frac(5)(4π)$ - the middle of the third quarter.
b) $-\frac(37π)(6)=-(6+\frac(1)(6))*π =-(6π +\frac(π)(6)) = -3*2π - \frac (π )(6)$. Hence, the number $-\frac(37π)(6)$ corresponds to the same number as the number $-\frac(1)(6π)$. Same as $\frac(11π)(6)$.
Example.
Find all numbers t that correspond to points on the number circle that belong to a given arc:
a) VA;
b) MK.
Solution:
a) Arc BA is an arc with a beginning at point B and an end at point A, while moving along a circle counterclockwise. Point B is respectively equal to $\frac(π)(2)$, and point A is equal to $2π$. Hence, for points t we have: $\frac(π)(2) ≤ t ≤ 2π$. But according to the formula on slide 8, the numbers $\frac(π)(2)$ and $2π$ correspond to numbers of the form $\frac(π)(2)+2π*k$ and $2π+2π*k$, respectively.
$\frac(π)(2) +2π*k ≤ t ≤ 2π +2π*k$, where $k$ is an integer.
b) The arc MK is an arc with the beginning at the point M and the end at the point K. The point M, respectively, is equal to $-\frac(3π)(4)$, and the point K is equal to $\frac(π)(4)$.
So for points t we have:
$\frac(-3π)(4) ≤ t ≤\frac(π)(4)$.
According to the formula on slide 8, the numbers $-\frac(3π)(4)$ and $\frac(π)(4)$ correspond to numbers of the form: $-\frac(3π)(4)+2π*k$ and $\ frac(π)(4)+2π*k$ respectively.
Then our number t takes the values:
$-\frac(3π)(4)+2π*k ≤ t ≤ \frac(π)(4) +2π*k$, where $k$ is an integer.
Tasks for independent solution
1) On the unit circle, the arc BC is divided by the point T into two equal parts, and by the points K and P into three equal parts. What is the length of the arc: BT, TS, VC, CR, RS, BP, CT?
2) Find a point on the number circle that corresponds to a given number:
$π$, $\frac(11π)(2)$, $\frac(21π)(4)$, $-\frac(7π)(2)$, $\frac(17π)(6)$.
3) Find all the numbers t, which on the number circle correspond to the points belonging to the given arc:
a) AB;
b) AC;
c) PM, where P is the midpoint of the arc AB and point M is the midpoint of DA.
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
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 goal 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,
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 the 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 circle)?
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
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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.
 |
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.
1. Mordkovich A.G. and others. Algebra 9th grade: Proc. For general education Institutions. - 4th ed. - M.: Mnemosyne, 2002.-192 p.: ill.
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