Number circle is a unit circle whose points correspond to certain real numbers.

A unit circle is a circle of radius 1.

General form 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 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 with 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. Note that all opposite points have 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 less than 1 - 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 midpoints of all quarters have 4 in the denominator, we already know them 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 singular. For example, if point A on a straight line is equal to 3, then it can no longer be equal to any other number.

It's different on the number circle because it's a circle. For example, in order to come from point A of 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:

In this lesson we will review important property number circle and place the unit number circle in the coordinate plane according to certain rules. Let's recall the equation of a unit number circle and with its help we will solve several problems on finding the coordinates of a point on a unit number circle. At the end of the lesson, we will compile a table of coordinates for points that are multiples of π/6 and π/4.

Theme of the lesson, repetition

Earlier, we studied the number circle and found out its properties (Fig. 1).

Each real number corresponds to a single point on the circle.

Each point on the number circle corresponds not only to a number, but to all numbers of the form

Number circle in the coordinate plane

Let's put the circle in coordinate plane. As before, each number corresponds to a point on the circle. Now this point on the circle corresponds to two coordinates, like any point on the coordinate plane.

Our task is to find not only a point, but also its coordinates by a given number, and vice versa, find one or more corresponding numbers by coordinates.

Example 1. A point is given - the middle of the arc. The point corresponds to numbers of the form

Find the coordinates of the point (Fig. 3).

Coordinates can be found in two different ways, consider them in turn.

1. The point lies on the circle, R=1, so it satisfies the equation of the circle

By condition. We remember that the value of the central angle is numerically equal to the length of the arc in radians, which means the angle. This also means that the straight line divides the first quarter exactly in half, which means it is a straight line

A point lies on a straight line and therefore satisfies the equation of this straight line.

Let's compose a system of two equations.

Having solved the system, we obtain the desired coordinates.

2. Consider a rectangular one (Fig. 4).

So, we set a number, found a point and its coordinates. Let us also determine the coordinates of points symmetrical to it (Fig. 5).

Finding rectangular coordinates of points whose curvilinear coordinates are multiples

The next task is to determine the coordinates of points that are multiples of

A circle of radius R=1 is placed in the coordinate plane, Find a point on the circle and its coordinates (Fig. 6).

Consider - rectangular.

i.e. angle

Let's find the coordinates of the symmetrical points (Fig. 7).

We set a number, found a point on the circle, this point is the only one, and found its coordinates.

Problem solving

Example 1. Given a point Find its rectangular coordinates.

The point is the middle of the third quarter (Fig. 8).

Conclusion, conclusion

We placed the number circle in the coordinate plane, learned how to find a point on the circle and its coordinates by the number. This technique underlies the definition of sine and cosine, which will be discussed next.

Bibliography

Algebra and the Beginnings of Analysis, Grade 10 (in two parts). Tutorial for educational institutions(profile level) / ed.

A. G. Mordkovich. - M.: Mnemosyne, 2009. Algebra and the beginning of analysis, grade 10 (in two parts). Task book for educational institutions (profile level) / ed.

A. G. Mordkovich. - M.: Mnemozina, 2007. Vilenkin N. Ya., Ivashev-Musatov O. S., Shvartsburd S. I. Algebra and mathematical analysis for grade 10 ( tutorial for students of schools and classes with in-depth study of mathematics). - M.: Education, 1996. Galitsky M. L., Moshkovich M. M., Shvartsburd S. I. In-depth study of algebra and mathematical analysis. - M.: Enlightenment, 1997. Collection of problems in mathematics for applicants to technical universities (under the editorship of M. I. Skanavi). - M.: Higher school, 1992. Merzlyak A. G., Polonsky V. B., Yakir M. S. Algebraic simulator. - K .: A. S. K., 1997. Saakyan S. M., Goldman A. M., Denisov D. V. Tasks in algebra and the beginnings of analysis (a manual for students in grades 10-11 of general educational institutions). - M.: Education, 2003. Karp A.P. Collection of problems in algebra and principles of analysis: textbook. allowance for 10-11 cells. with a deep study mathematics. - M.: Education, 2006.

Mathematics. ru. Problems. ru. I'll solve the exam.

Homework

Algebra and the Beginnings of Analysis, Grade 10 (in two parts). Task book for educational institutions (profile level) / ed. A. G. Mordkovich. - M.: Mnemosyne, 2007.

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Slides captions:

Number circle in the coordinate plane

Let's repeat: unit circle- a number circle, the radius of which is 1. R=1 C=2 π + - y x

If the point M of the numerical circle corresponds to the number t, then it also corresponds to the number of the form t+2 π k , where k is any integer (k ϵ Z) . M(t) = M(t+2 π k), where k ϵ Z

Basic layouts First layout 0 π y x Second layout y x

x y 1 A(1, 0) B (0, 1) C (- 1, 0) D (0, -1) 0 x>0 y>0 x 0 x 0 y

Find the coordinates of the point M corresponding to the point. 1) 2) x y M P 45° O A

Coordinates of the main points of the first layout 0 2 x 1 0 -1 0 1 y 0 1 0 -1 0 0 x 1 0 -1 0 1 y 0 1 0 -1 0 D y x

M P x y O A Find the coordinates of the point M corresponding to the point. 1) 2) 30°

M P Find the coordinates of the point M corresponding to the point. 1) 2) 30° x y O A B

Using the symmetry property, we find the coordinates of points that are multiples of y x

Coordinates of the main points of the second layout x y x y y x

Example Find the coordinates of a point on a number circle. Solution: P y x

Example Find points with ordinate on a number circle Solution: y x ​​x y x y

Exercises: Find the coordinates of the points of the numerical circle: a) , b) . Find points with an abscissa on the number circle.

Key points coordinates 0 2 x 1 0 -1 0 1 y 0 1 0 -1 0 0 x 1 0 -1 0 1 y 0 1 0 -1 0 Key points coordinates of the first layout x y x y Key points coordinates of the second layout

On the topic: methodological developments, presentations and notes

Didactic material on algebra and the beginnings of analysis in grade 10 (profile level) "Number circle on the coordinate plane"

Option 1.1. Find a point on the number circle: A) -2∏ / 3B) 72. Which quarter of the number circle does the point belong to 16.3. Find which ...

Analytic geometry gives uniform methods for solving geometric problems. To do this, all given and desired points and lines are referred to the same coordinate system.

In a coordinate system, each point can be characterized by its coordinates, and each line by an equation with two unknowns, of which this line is a graph. Thus geometric problem is reduced to algebraic, where all methods of calculation are well worked out.

A circle is a locus of points with one specific property (each point of the circle is equidistant from one point, called the center). The circle equation must reflect this property, satisfy this condition.

The geometric interpretation of the equation of a circle is the line of a circle.

If we place a circle in a coordinate system, then all points of the circle satisfy one condition - the distance from them to the center of the circle must be the same and equal to the circle.

Circle centered at a point BUT and radius R placed in the coordinate plane.

If the coordinates of the center (a;b) , and the coordinates of any point on the circle (x; y) , then the circle equation has the form:

If the square of the radius of a circle is equal to the sum of the squared differences of the corresponding coordinates of any point on the circle and its center, then this equation is the equation of a circle in a plane coordinate system.

If the center of the circle coincides with the point of origin, then the square of the radius of the circle is equal to the sum of the squares of the coordinates of any point on the circle. In this case, the circle equation takes the form:

Therefore, any geometric figure how the locus of points is determined by the equation relating the coordinates of its points. Conversely, the equation relating the coordinates X and at , define the line as the locus of points in the plane whose coordinates satisfy this equation.

Examples of solving problems about the equation of a circle

Task. Write an equation for a given circle

Write an equation for a circle centered at point O (2;-3) and with radius 4.

Decision.
Let us turn to the formula of the circle equation:
R 2 \u003d (x-a) 2 + (y-b) 2

Substitute the values ​​into the formula.
Circle radius R = 4
Coordinates of the center of the circle (according to the condition)
a = 2
b=-3

We get:
(x - 2 ) 2 + (y - (-3 )) 2 = 4 2
or
(x - 2 ) 2 + (y + 3 ) 2 = 16 .

Task. Does a point belong to the equation of a circle

Check if point belongs A(2;3) circle equation (x - 2) 2 + (y + 3) 2 = 16 .

Decision.
If a point belongs to a circle, then its coordinates satisfy the circle equation.
To check whether a point with given coordinates belongs to the circle, we substitute the coordinates of the point into the equation of the given circle.

In the equation ( x - 2) 2 + (y + 3) 2 = 16
we substitute, according to the condition, the coordinates of the point A (2; 3), that is
x=2
y=3

Let's check the truth of the obtained equality
(x - 2) 2 + (y + 3) 2 = 16
(2 - 2) 2 + (3 + 3) 2 = 16
0 + 36 = 16 equality is wrong

Thus, given point not belong given equation circles.

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