Definition of a number circle on a coordinate plane. Number circle on the coordinate plane. Coordinates of points on the number circle
Slide 2
What we will study: Definition. Important coordinates of the number circle. How to find the coordinate of the number circle? Table of basic coordinates of the number circle. Examples of tasks.
Slide 3
Definition. Let's place the number circle in coordinate plane so that the center of the circle coincides with the origin of coordinates, and its radius is taken as a unit segment. The starting point of the number circle A is combined with the point (1;0). Each point on the number circle has its own coordinates x and y in the coordinate plane, and: x > 0, y > 0 in the first quarter; x 0 in the second quarter; x 0, y
Slide 4
It is important for us to learn how to find the coordinates of the points on the number circle presented in the figure below:
Slide 5
Let's find the coordinate of point π/4: Point M(π/4) is the middle of the first quarter. Let us drop the perpendicular MR from point M to straight line OA and consider the triangle OMP. Since the arc AM is half of the arc AB, then ∡MOP=45° This means that the triangle OMP is isosceles right triangle and OP=MP, i.e. at point M the abscissa and ordinate are equal: x = y Since the coordinates of the point M(x;y) satisfy the equation of the number circle, then to find them you need to solve the system of equations: Having solved this system we get: We found that the coordinates of the point M corresponding to the number π/4 will be The coordinates of the points presented on the previous slide are calculated in a similar way.
Slide 6
Slide 7
Coordinates of points on the number circle.
Slide 8
Example Find the coordinate of a point on a number circle: Р(45π/4) Solution: Because. the numbers t and t+2π k (k-integer) correspond to the same point on the number circle then: 45π/4 = (10 + 5/4) π = 10π +5π/4 = 5π/4 + 2π 5 So, the number 45π/4 corresponds to the same point on the number circle as the number 5π/4. Looking at the value of the point 5π/4 in the table we get:
Slide 9
Example Find the coordinate of a point on a number circle: Р(-37π/3) Solution: Because. the numbers t and t+2π k (k-integer) correspond to the same point on the number circle then: -37π/3 = -(12 + 1/3) π = -12π –π/3 = -π/3 + 2π (-6) This means that the number -37π/3 corresponds to the same point on the number circle as the number –π/3, and the number –π/3 corresponds to the same point as 5π/3. Looking at the value of the point 5π/3 in the table we get:
Slide 10
Find points on the number circle with ordinate y = 1/2 and write down which numbers t they correspond to. Example The straight line y = 1/2 intersects the number circle at points M and P. Point M corresponds to the number π/6 (from the table data), which means, and any number of the form π/6+2π k. Point P corresponds to the number 5π/6, and therefore to any number of the form 5π/6+2 π k. We obtained, as is often said in such cases, two series of values: π/6+2 π k and 5π/6+2 π k Answer: t= π/6+2 π k and t= 5π/6+2 π k Number circle on the coordinate plane.
Slide 11
Example Find points on the number circle with abscissa x≥ and write down which numbers t they correspond to. The straight line x= 1/2 intersects the number circle at points M and P. The inequality x ≥ corresponds to the points of the arc PM. Point M corresponds to the number 3π/4 (from the table data), which means, and any number of the form -3π/4+2π k. Point P corresponds to the number -3π/4, and therefore to any number of the form – -3π/4+2 π k Then we get -3π/4+2 π k≤t≤3π/4+2 π k Answer: -3π/ 4+2 π k≤t≤3π/4+2 π k Number circle on the coordinate plane.
Slide 12
Number circle on the coordinate plane.
Problems for independent solution. 1) Find the coordinate of a point on the number circle: P(61π/6)? 2) Find the coordinate of a point on the number circle: P(-52π/3) 3) Find points on the number circle with ordinate y = -1/2 and write down which numbers t they correspond to. 4) Find points on the number circle with ordinate y ≥-1/2 and write down which numbers t they correspond to. 5) Find the points on the number circle with the abscissa x≥ and write down which numbers t they correspond to.
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Municipal educational institution average comprehensive school № 1
KHMAO-Yugra
Lesson development
in 10th grade
on algebra and principles of analysis
Nadezhda Mikhailovna
mathematic teacher
Sovetsky
Topic: TRIGONOMETRY
Trigonometric functions
Trigonometric equations
Trigonometric transformations
Number circle on
coordinate plane
The subject is taught according to block-modular technologies.
This lesson is one of the lessons for learning new material. Therefore, the main time of the lesson is devoted to learning new material, and students do most of this work independently.
Types of student activities in the lesson: frontal, independent and individual work.
Since a lot of work needs to be done in a lesson and the results of student activities must be monitored, an interactive whiteboard is used at the stages of updating knowledge and learning new material. For a more visual representation of the imposition of the number circle on the coordinate plane and for reflection of the content educational material at the end training session Power Point presentations are also used.
educational
Learn to independently acquire knowledge
nurturing
Cultivate composure, responsibility, diligence
developing
Learn to analyze, compare, build analogies
Lesson plan:
1) Organizing time, topic, purpose of lesson 2 min.
2) Updating knowledge 4 min.
3) Learning new material 30 min.
4) Reflection 3 min.
5) Summary of Lesson 1 min.
Organizing time
Number circle
coordinate plane
consider the number circle on the coordinate plane; together find the coordinates of two points; then independently compile tables of coordinate values of other main points of the circle;
test your ability to find the coordinates of points on a number circle.
Updating knowledge
In the 9th grade geometry course we studied the following
material:
On a unit semicircle (R = 1), we considered a point M with coordinates X And at
Excerpts from a geometry textbook
Having learned to find the coordinates of a point on the unit circle,
Let’s easily move on to their other names: sines and cosines, i.e.
to the main topic - TRIGONOMETRY
The first task was given to interactive whiteboard, where students need to place dots and their corresponding numbers in place on a number circle by dragging them with their finger across the board.
Exercise 1
We got the result:
The second task is given on the interactive board. The answers are closed with a “curtain” and are revealed as they are solved.
Task 2
Result of the task:
Learning new material
Let's take a coordinate system and put a number circle on it so that their centers coincide, and the horizontal radius of the circle coincides with the positive direction of the OX axis (Power Point presentation)
As a result, we have points that belong to both the number circle and the coordinate plane. Let's consider one of these points, for example, point M (Power Point presentation)
M(t)
Let's plot the coordinates of this point
Let's find the coordinates of the points of interest to us on the unit circle, which we considered earlier with denominators 4, 3, 6 and numerator π.
Find the coordinates of the point on the unit circle corresponding to the number and, accordingly, the angle
Task 3
(Power Point presentation)
Let's depict the radius and coordinates of the point
By the Pythagorean theorem we have X 2+ x 2 = 12
But the angles of the triangle are π/4 = 45° , This means that the triangle is isosceles and x = y
Find the coordinates of a point on the unit circle corresponding to the numbers (angles)
Task 4
(Power Point presentation)
Means at= 1/2
According to the Pythagorean theorem
Triangles are equal in hypotenuse
And sharp corner, which means their legs are equal
On previous lesson Students received sheets with blanks of number circles and various tables.
Fill out the first table.
Task 5
(interactive board)
First, enter the points of the circle that are multiples of 2 and 4 into the table.
Checking the result:
(interactive board)
Fill in the ordinates and abscissas of these points yourself in the table, taking into account the coordinate signs, depending on which quarter the point is located in, using the lengths of the segments obtained above for the coordinates of the points.
Task 6
One of the students names the results obtained, the rest check their answers, then to successfully correct the results (since these tables will be used later in the work to develop skills and deepen knowledge on the topic), a correctly completed table is shown on the interactive board.
Checking the result:
(interactive board)
Fill out the second table.
Task 7
(interactive board)
First, enter into the table the points of the circle that are multiples of 3 and 6
Checking the result:
(interactive board)
Fill in the ordinates and abscissas of these points yourself in the table
Task 8
Checking the result:
(interactive board)
(Power Point presentation)
Let's do a little mathematical dictation followed by self-control.
1) Find the coordinates of the points of the unit circle:
Option 2
1 option
2) Find the abscissa of the points of the unit circle:
1) Find the coordinates of the points on the unit circle
Option 2
1 option
2) Find the abscissa of the points on the unit circle
check yourself
3) Find the ordinates of the points of the unit circle:
For yourself, you can mark “5” for 4 completed examples,
“4” for 3 examples and mark “3” for 2 examples
Summing up the lesson
1) In the future, to find the values of sine, cosine, tangent and cotangent of points and angles, it is necessary to learn from the completed tables the values of the coordinates of points belonging to the first quarter because further we will learn to express the coordinate values of all other points through the values of the points of the first quarter;
2) Prepare theoretical questions for testing.
Lesson Summary
The grade is given to the students who worked most actively in the lesson. The work of all students is not graded, since errors are corrected immediately during the lesson. The dictation was conducted for self-control; there is insufficient volume for assessment.
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Date: Lesson1
topic: Number circle on a coordinate line
Goals: introduce the concept of a number circle model in Cartesian and curvilinear coordinate systems; develop the ability to find Cartesian coordinates points of the number circle and perform the opposite action: knowing the Cartesian coordinates of the point, determine its numerical value on the number circle.
During the classes
I. Organizational moment.
II. Explanation of new material.
1. Having placed the number circle in the Cartesian coordinate system, we analyze in detail the properties of points on the number circle located in different coordinate quarters.
For a point M number circle uses the notation M(t), if we are talking about the curvilinear coordinate of a point M, or record M (X;at), if we are talking about Cartesian coordinates of a point.
2. Finding the Cartesian coordinates of “good” points on the number circle. It's about moving on from the record M(t) To M (X;at).
3. Finding the signs of the coordinates of “bad” points on the number circle. If, for example, M(2) = M (X;at), That X 0; at 0. (schoolchildren learn to identify signs trigonometric functions along the quarters of the number circle.)
1. No. 5.1 (a; b), No. 5.2 (a; b), No. 5.3 (a; b).
This group of tasks is aimed at developing the ability to find the Cartesian coordinates of “good” points on the number circle.
Solution:
№ 5.1 (A).
2. No. 5.4 (a; b), No. 5.5 (a; b).
This group of tasks is aimed at developing the skills to find the curvilinear coordinates of a point using its Cartesian coordinates.
Solution:
№ 5.5 (b).
3. No. 5.10 (a; b).
This exercise is aimed at developing the ability to find the Cartesian coordinates of “bad” points.
V. Lesson summary.
Questions for students:
– What is a model – a number circle on a coordinate plane?
– How, knowing the curvilinear coordinates of a point on the number circle, find its Cartesian coordinates and vice versa?
Homework: No. 5.1 (c; d) – 5.5 (c; d), No. 5.10 (c; d).
Date: Lesson2
TOPIC: Solving problems using the “number circle on the coordinate plane” model
Goals: continue to develop the ability to move from curvilinear coordinates of a point on a number circle to Cartesian coordinates; develop the ability to find points on the number circle whose coordinates satisfy given equation or inequality.
During the classes
I. Organizational moment.
II. Oral work.
1. Name the curvilinear and Cartesian coordinates of points on the number circle.
2. Compare the arc on the circle and its analytical notation.
III. Explanation of new material.
2. Finding points on the number circle whose coordinates satisfy the given equation.
Let's look at examples 2 and 3 with p. 41–42 textbooks.
The importance of this “game” is obvious: students are preparing to solve simple problems trigonometric equations type To understand the essence of the matter, you should first of all teach schoolchildren to solve these equations using the number circle, without moving on to ready-made formulas.
When considering an example of finding a point with an abscissa, we draw students’ attention to the possibility of combining two series of answers into one formula:
3. Finding points on the number circle whose coordinates satisfy a given inequality.
Let's look at examples 4–7 from p. 43–44 textbooks. By solving such problems, we prepare students to solve trigonometric inequalities of the form
After considering the examples, students can independently formulate algorithm solutions to inequalities of the indicated type:
1) from the analytical model we move to the geometric model - arc MR number circle;
2) make up the core of the analytical record MR; for the arc we get
3) make a general record:
IV. Formation of skills and abilities.
1st group. Finding a point on the number circle with a coordinate that satisfies a given equation.
No. 5.6 (a; b) – No. 5.9 (a; b).
In the process of working on these exercises, we practice the step-by-step execution: recording the core of a point, analytical recording.
2nd group. Finding points on the number circle with a coordinate that satisfies a given inequality.
No. 5.11 (a; b) – 5.14 (a; b).
The main skill that schoolchildren must acquire when performing these exercises is drawing up the core of an analytical notation of the arc.
V. Independent work.
Option 1
1. Mark the point on the number circle that corresponds to given number, and find its Cartesian coordinates:
2. Find points on the number circle with a given abscissa and write down which numbers t they match.
3. Mark on the number circle points with an ordinate that satisfies the inequality and write down, using the double inequality, which numbers t they match.
Option 2
1. Mark a point on the number circle that corresponds to a given number and find its Cartesian coordinates:
2. Find points on the number circle with a given ordinate at= 0.5 and write down which numbers t they match.
3. Mark on the number circle the points with the abscissa that satisfy the inequality and write down, using the double inequality, which numbers t they match.
VI. Lesson summary.
Questions for students:
– How to find a point on a circle whose abscissa satisfies a given equation?
– How to find a point on a circle whose ordinate satisfies a given equation?
– Name the algorithm for solving inequalities using the number circle.
Homework: No. 5.6 (c; d) – No. 5.9 (c; d),
No. 5.11 (c; d) – No. 5.14 (c; d).
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Slide captions:
Number circle in the coordinate plane
Let's repeat: Unit circle– a number circle whose radius is 1. R=1 C=2 π + - y x
If point M of the number circle corresponds to the number t, then it also corresponds to a 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
Let's find the coordinates of 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 Let's 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 property of symmetry, 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 the number circle Solution: y x x y x y
Exercises: Find the coordinates of the points on the number circle: a) , b) . Find the points with the abscissa on the number circle.
Coordinates of the main points 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 Coordinates of the main points of the first layout x y x y Coordinates of the main points 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 the point on the number circle: A) -2∏/3B) 72. Which quarter of the number circle does point 16.3. Find the...