Lesson properties of trigonometric functions and their graphs. Methodical development of the lesson trigonometric functions, their properties and graphs
Class: 10
The purpose of the lesson:
- Educational:
- practice building skills function graphs, using the periodicity of trigonometric functions;
- consolidate the studied material about even and odd functions
- Developing:
- develop skills, analyze, apply the existing knowledge of students in a changed situation.
- Educational:
- to educate students in accuracy, curiosity, careful attitude to the world around, moral qualities;
- create conditions for the development of cognitive activity of students, the implementation of the personal functions of each student, his free development, taking into account individual characteristics and potential opportunities.
Equipment:
- multimedia projector;
- worksheets for students;
- evaluation sheets;
- board;
- chalk, drawing tools;
- notebooks;
- coordinate system blanks
DURING THE CLASSES
Students at the entrance to the class for the lesson choose tokens in which the trigonometric functions sine, cosine, tangent are written. Then they are seated at round tables in groups with tokens of the same function.
The objectives of the lesson are stated. Throughout the lesson, students independently assess their preparation for the lesson. To do this, each group is given evaluation sheets, the criteria for evaluating their activities at each stage of the lesson are reflected on the slides ( Annex 1
).
Evaluation sheets are filled out by students and are handed in at the end of the lesson with written work For checking.
Evaluation paper
| № | F.I | Theoretical warm-up, "math lotto" | group Work | Test | Grade per lesson |
| 1 | |||||
| 2 | |||||
| 3 | |||||
| 4 | |||||
| 5 |
II. Frontal survey "Theoretical warm-up"
In order to complete the practical tasks of the lesson, it is necessary to remember the theoretical material. To do this, we will "Theoretical Workout" on slide ( Annex 1 ) a table with question numbers is given, in turn, each group chooses a question number, reads out the question and immediately gives an answer to it.
At this stage, the students' knowledge is updated, which is necessary for further work in the lesson.
- What is called a function?
- What is the scope of a function?
- What is the scope of a function?
- What is an even function?
- Which function is called odd?
- What properties does a graph have? even function?
- What property does the graph of an odd function have?
- Define the basic trigonometric functions.
- What can be said about the parity of trigonometric functions?
- What is a periodic function?
- What is the smallest positive period for the sine and cosine function?
- What is the smallest positive period for the tangent (cotangent) function?
- What is the domain of the sine function?
- What is the domain of the cosine function?
- What is the domain of the tangent function?
- What is the domain of the cotangent function?
- What is the range of the sine function?
- What is the range of the cone function?
- What is the range of the tangent function?
- What is the range of the cotangent function?
- Which of the functions takes highest value y = sin 2x or y = 2 sin x&
- We repeated the theoretical material with you. And now I invite you to show your knowledge in determining an even or odd function, while performing the "mathematical loto". Each group receives a sheet - a task with a "mathematical lotto". ( Annex 2 ).
Exercise: in the resulting table, shade those cells in which the even (odd) function is located.
"Math Lotto"
Option 1.
Exercise: Shade in the table those cells in which the even function is located
Option 2.
Exercise: Shade in the table those cells in which the odd function is located
Evaluation criteria for a frontal survey, participation in the joint work of the class:
- 2 points, did not actively participate;
- 3 points, answered questions, made suggestions when completing the task "mathematical loto"
- 4 points, actively answered questions, offered correct answers when solving the "mathematical loto"
III. Group work on plotting trigonometric functions
Working together in a group on a task, the student correlates his “I” with himself and others, comparing different or identical visions of the task and the process of solving it, evaluating his capabilities and claims. Pupils have to act in different roles and in the role of "student" and in the role of "teacher". Here the ability to work in a group, the ability to defend one's point of view and accept the point of view of comrades is formed.
Each group is invited to independently draw graphs of trigonometric functions in notebooks, having previously determined its domain of definition, range of value, period. Each group also receives blanks of the coordinate system on an A4 or A3 sheet on which they need to depict the completed task (you can use felt-tip pens of different colors when plotting charts)
After completing their assignment, each group presents their work in front of the class. The work of everyone in the group is evaluated by the whole group, the assessment is recorded in the evaluation sheet.
Criteria for evaluating work in a group:
- 3 points, did not actively participate in the work;
- 4 points, made suggestions in solving the problem;
- 5 points, actively participated in the work of the group, suggested the right ways to solve the problem.
IV. test work
Before students can take the test, they must choose a level of difficulty appropriate to their ability.
At this stage of work, a situation is created for students in which they need to assess their real knowledge and capabilities.
1) If the student believes that he has mastered the material at "3", then it is enough for him to complete 1 - 5 test tasks.
2) If you have mastered the material on "4", then you need to complete 6 - 7 test tasks.
3) If the material is mastered at "5", then you must complete all the tasks of the test.
Test key:
| job number | I option | II option |
| A1 | V | V |
| A2 | B | G |
| A3 | V | B |
| A4 | G | G |
| A5 | A | G |
| A6 | A | V |
| A7 | B | A |
| IN 1 | – 7 | – 6 |
| IN 2 | 5 | – 4 |
Notebooks and assessment sheets are handed over to the teacher.
V. Summary of the lesson
Grades in the journal are set after checking the work by the teacher, comparing it with the results of the assessment sheets of knowledge accounting.
VI. Homework
Group I: p.93 No. 18
II group: p.93 No. 19
III group: p.93 No. 20
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How to capture the attention of a student in the digital age, when the flow of information completely absorbs a person? Probably, many are familiar with the situation when there is a lesson, and the students look at the phone screen and do not want to do anything else. How to capture the attention of students and make the lesson interesting? Let's take a look at a few ways to create interest in the lesson.
Municipal state-owned evening (shift) educational institution "Evening (shift) comprehensive school No. 4 at the penal colony "
Outline and presentation for the lesson
algebra and the beginning of analysis in grade 10 on the topic
« Trigonometric functions and their properties"
Lesson topic: "Trigonometric functions and their properties"
Educational: to generalize and systematize students' knowledge on the topic under study, to control the level of assimilation of the material;
Developing: development of mathematical thinking, intellectual and cognitive abilities, development of the ability to justify one's decision, control and evaluate the results of one's actions;
Educational: education of a culture of communication, cognitive activity, a sense of responsibility for the work done, discipline, accuracy, independence.
Equipment and materials for the lesson: multi-projector, presentation to accompany the lesson, self-control sheets, cards with the text of independent work.
Lesson type: knowledge review lesson
During the classes.
I. Organizing time.
II. Presentation of the topic and objectives of the lesson.
I would like to start today's lesson with the words of the great physiologist I.P. Pavlov:
“Learn the basics of science before climbing to its heights. Never take on the next without mastering the previous. slide 2
We live in real world and to know it, we need knowledge. But before climbing to the next step, we need to make sure that we are firmly on our feet, that we have good, solid knowledge on the topic being studied.
Can you tell me what topic we are studying?
And any knowledge should turn into skill and habit. Today in the lesson we will summarize and systematize the existing knowledge on this topic. We will test our knowledge, skills and abilities, find out the gaps and try to eliminate them.
Updating of basic knowledge.
one . front poll.
What are the trigonometric functions that you know?
And now we repeat the properties of the trigonometric functions known to us.
(Students name the properties of trigonometric functions, each correct answer is highlighted on the slide. As a result of the discussion, a table appears.) Slide 3-6
2. Oral work on solving the simplest problems on the transformation of graphs of trigonometric functions. Slide 7-9
Work with self-control sheets . (Annex 1)
In the lesson, you will various tasks, and gradually fill in the student's self-control sheet. Sign the self-control sheet and familiarize yourself with its contents. Assess how ready you are to complete tasks and set a predictive score. And put the sheet aside for now.
Graphic dictation.
The result of the dictation on the sheets of self-control of students will be such a record.
where the signs indicate: + yes, no. After the end of the dictation, the students exchange the dictation with a desk mate for verification. Each correct answer is worth 1 point, for an incorrect answer and no answer, 0 points are given. Slide 10
Independent work by options . (Annex 2)
I option.
no intersection points
II option.
Specify the set of function values:
4) no intersection points
Find the smallest positive period of a function
Self-test. slide 11
Each correct answer is worth 1 point, for an incorrect answer and no answer, 0 points are given.
Group work. slide 12
Performing tasks of increased complexity.
I remind you the order of work in groups: 10 minutes you solve the task yourself, 5 minutes discuss the solution of tasks collectively. Do not forget to put a self-assessment and determine your level of knowledge. For the error-free execution of the task, 2 points are set, the solution with shortcomings is estimated at 1 point.
I group
Plot the Function
II group
Plot the Function
2) Find the smallest positive period of the function:
Who wants to explain their decision? Slide 13-15
Summary of the lesson.
Let's summarize our work. Calculate the points and, according to the criteria, put the final grade. If you are satisfied with your results, then put a signature under your assessment. Analyze your level of knowledge. If not everything worked out, think about what else you need to work on.
The homework task is to once again analyze what worked, what didn’t work, and what needs to be worked on. For tasks in which you made mistakes, select similar tasks and solve them. The results of your work in the lesson will show me your self-control sheets. Thank you for the lesson!
Annex 1
Student self-control sheet ______________________________________________
(last name, first name)
To the lesson of algebra and the beginning of analysis on the topic "Trigonometric functions and their properties"
Predictive score ________
No. 1. Graphic dictation.
No. 2. Independent work.
No. 3. Group work. Tasks of increased complexity.
If you scored 21-23 points - score "5"
16-20 points - score "4" I scored _________ points
10-15 points - score "3" My mark "____"___________________
(student's signature)
Answer the questions and rate it out of 5 point system
How, in your opinion, did the lesson go, did you understand everything? _______________
Did you feel confident in class? ___________________
Was the previous knowledge enough for you?? ____________
Annex 2
Independent work.
I option.
1. Specify the set of function values: y= 4x.
1) The set of real numbers;
2) The set of real numbers, except for numbers of the form ;
3) The set of real numbers, except for numbers of the form
Determine the sign of the number sin 1 cos 9 tg (-2)
3) impossible to determine
;
no intersection points.
II option.
The set of real numbers;
2) The set of real numbers, except for numbers of the form
3) The set of real numbers, except for numbers of the form
3) impossible to determine
4) no intersection points
Find the smallest positive period of a function
View presentation content
"trigonometric functions"
"Trigonometric Functions and Their Properties"
Pugacheva A.V., teacher of mathematics, Moscow State Educational Institution "V (C) School No. 4 at IC"
Mariinsk, Kemerovo region
“Learn the basics of science before climbing to its heights. Never take on the next one without mastering the previous one. .
I.P. Pavlov
y x
Function Graph
at
X
O
Function Properties
Function Properties
Domain
Domain
Points of intersection of the graph with the axes
coordinates
coordinates
Even / Odd
Even / Odd
gaps
gaps
increasing
increasing
monotony
odd
monotony
odd
descending
descending
Extremes
Extremes
Periodicity
Periodicity
Constancy intervals
Constancy intervals
Many values
Many values
y x
Function Graph
at
X
O
Function Properties
Function Properties
Domain
Domain
Points of intersection of the graph with the axes
Points of intersection of the graph with the axes
coordinates
coordinates
Even / Odd
Even / Odd
Intervals of monotonicity
Intervals of monotonicity
increasing
increasing
descending
descending
Extremes
Extremes
Periodicity
Periodicity
Constancy intervals
Constancy intervals
Many values
Many values
Function Graph
ytg x
Function Properties
Function Properties
Domain
Domain
Points of intersection of the graph with the axes
Points of intersection of the graph with the axes
coordinates
coordinates
Even / Odd
Even / Odd
gaps
gaps
increasing
odd
monotony
odd
increasing
monotony
descending
descending
Extremes
Extremes
Periodicity
Periodicity
Constancy intervals
Constancy intervals
Many values
Many values
at
O
X
Function Properties
Function Properties
Domain
Domain
Points of intersection of the graph with the axes
Points of intersection of the graph with the axes
coordinates
coordinates
Even / Odd
Even / Odd
gaps
gaps
monotony
increasing
monotony
increasing
odd
odd
Extremes
descending
descending
Extremes
Periodicity
Periodicity
Constancy intervals
Constancy intervals
Many values
Many values
Function Graph
ytg x
at
X
O
1. The graph of what function is shown in the figure?
at
1
X
O
-1
1) ycos x
2) y2 cos x
3) y2cos x
4) y2 sin x
2. The graph of which function is shown in the figure?
at
O
X
1) y x
2) y 2x
4) ycos x
3) y 2x
3. The graph of which function is shown in the figure?
at
X
O
1) y2cos x
2) ycos(x+
3) ycosx+ 1
4) ycos(x+
Checking the graphic dictation:
Independent work.
Let's check:
I option.
II option.
I group
1) Plot the function graph:
a) y=
b) y= 3
function period:
y(x)=
II group
1) Plot the function
a) y=
b) y \u003d 2
2) Find the smallest positive
function period:
y(x)=cos5x
Let's check:
I group
at
y=
X
O
y=3
at
X
Let's check:
II group
at
y=
X
O
at
y=
X
I group
We use the formula for the sine of the difference of two angles
and get
y(x)==
T=2
II group
We use the formula for the cosine of the difference of two angles
and get
y(x)==
The smallest positive period of a function is
T=2
Lesson topic: trigonometric functions, their properties and graphs.
Lesson type: learning and primary consolidation of new knowledge.
Form of study: classroom class.
Form of activity: frontal and individual.
The purpose of the lesson: familiarity with trigonometric functions; the formation of knowledge and skills in the construction of graphs of trigonometric functions.
Lesson objectives:
1. Educational:
Give definitions of trigonometric functions;
Consider the basic properties of trigonometric functions;
Show graphs of trigonometric functions.
2. Developing:
To promote the development of skills to analyze, establish connections, cause and effect;
Anticipate possible errors and ways to eliminate them;
Contribute to increased concentration, development of memory and speech.
3. Educational:
To promote the development of interest in the subject "Mathematics";
To promote the development of independent thinking;
In order to solve the problems of aesthetic education, to contribute to the neat and competent construction of graphs of functions during the lesson.
Teaching methods: verbal methods (story, explanation); visual methods (demonstration, TCO); practical methods.
Equipment: computer, projector, handout.
Download:
Preview:
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In this lesson, we'll look at basic trigonometric functions, their properties and graphs, and also list main types of trigonometric equations and systems. In addition, we indicate general solutions of the simplest trigonometric equations and their special cases.
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Theory
Lesson summary
Trigonometric functions and their properties
We have already repeatedly used the term "trigonometric function". Back in the first lesson of this topic, we defined them with right triangle and the unit trigonometric circle. Using such methods of specifying trigonometric functions, we can already conclude that for them one value of the argument (or angle) corresponds to exactly one value of the function, i.e. we have the right to call sine, cosine, tangent and cotangent exactly functions.
In this lesson, it's time to try to abstract from the previously discussed methods for calculating the values of trigonometric functions. Today we will move on to the usual algebraic approach to working with functions, we will consider their properties and draw graphs.
As for the properties of trigonometric functions, then Special attention should be referred to:
The domain of definition and the domain of values, because for the sine and cosine there are restrictions on the domain of values, and for the tangent and cotangent there are restrictions on the domain of definition;
The periodicity of all trigonometric functions, since we have already noted the presence of the smallest non-zero argument, the addition of which does not change the value of the function. Such an argument is called the period of the function and is denoted by the letter . For sine/cosine and tangent/cotangent, these periods are different.
The sine function and its graph
Consider a function:
1) Domain of definition;
2) Range of values 
;
3) The function is odd 
;
Let's plot the function. In this case, it is convenient to start the construction from the image of the area, which limits the graph from above by the number 1 and from below by the number , which is related to the range of the function. In addition, for construction it is useful to remember the values of the sines of several basic tabular angles, for example, that. This will allow you to build the first full “wave” of the chart and then redraw it to the right and left, taking advantage of the fact that the picture will be repeated with a period shift, i.e. by .
Cosine function and its graph
Now let's look at the function:
The main properties of this function:
1) Domain of definition;
2) Range of values ;
3) The function is even 
This implies the symmetry of the graph of the function with respect to the y-axis;
4) The function is not monotone throughout its domain of definition;
5) The function is periodic with period .
Let's plot the function. As well as when constructing a sine, it is convenient to start with the image of the area that limits the graph from above by the number 1 and from below by the number , which is related to the range of the function. We will also plot the coordinates of several points on the graph, for which it is necessary to remember the cosine values \u200b\u200bof several basic tabular angles, for example, that . With the help of these points, we can build the first complete "wave" of the chart and then redraw it to the right and left, taking advantage of the fact that the picture will be repeated with an offset by a period, i.e. by .
Tangent function and its graph
Let's move on to the function:
The main properties of this function:
1) Domain of definition except for , where . We have already indicated in previous lessons that does not exist. This statement can be generalized by taking into account the period of the tangent;
2) Range of values, i.e., the values of the tangent are not limited;
3) The function is odd 
;
4) The function monotonically increases within its so-called tangent branches, which we will now see in the figure;
5) The function is periodic with period .
Let's plot the function. In this case, it is convenient to start the construction with the image of the vertical asymptotes of the graph at points that are not included in the domain of definition, i.e., etc. Next, we depict the branches of the tangent inside each of the strips formed by the asymptotes, pressing them to the left asymptote and to the right one. At the same time, do not forget that each branch is monotonically increasing. All branches are depicted in the same way, since the function has a period equal to . This can be seen from the fact that each branch is obtained by shifting the neighboring one along the x-axis.
Cotangent function and its graph
And we conclude with a look at the function:
The main properties of this function:
1) Domain of definition except for , where . According to the table of values of trigonometric functions, we already know that it does not exist. This statement can be generalized by taking into account the period of the cotangent;
2) Range of values, i.e., the values of the cotangent are not limited;
3) The function is odd 
;
4) The function monotonically decreases within its branches, which are similar to the tangent branches;
5) The function is periodic with period .
Let's plot the function. In this case, as for the tangent, it is convenient to start the construction from the image of the vertical asymptotes of the graph at points that are not included in the domain of definition, i.e., etc. Next, we depict the branches of the cotangent inside each of the strips formed by the asymptotes, pressing them to the left asymptote and to the right. In this case, we take into account that each branch is monotonically decreasing. All branches, similarly to the tangent, are depicted in the same way, since the function has a period equal to.
Calculation of periods of trigonometric functions with a complex argument
Separately, it should be noted that trigonometric functions with a complex argument may have a non-standard period. It's about about view functions:
They have the same period. And about functions:
They have the same period.
As you can see, to calculate a new period, the standard period is simply divided by the factor in the argument. It does not depend on other modifications of the function.
You can understand and understand in more detail where these formulas come from in the lesson about constructing and converting function graphs.
Trigonometric equations and methods for their solution
We have come to one of the most important parts of the topic "Trigonometry", which we will devote to solving trigonometric equations. The ability to solve such equations is important, for example, when describing oscillatory processes in physics. Let's imagine that you have driven a few laps on a kart in a sports car, solving a trigonometric equation will help determine how long you have already been participating in the race, depending on the position of the car on the track.
Let's write the simplest trigonometric equation:
The solution of such an equation is the arguments, the sine of which is equal to. But we already know that because of the periodicity of the sine, there are an infinite number of such arguments. Thus, the solution to this equation will be, etc. The same applies to the solution of any other simple trigonometric equation, there will be an infinite number of them.
Trigonometric equations are divided into several basic types. Separately, one should dwell on the simplest, since all the rest are reduced to them. There are four such equations (according to the number of basic trigonometric functions). For them, common solutions are known, they must be remembered.
Protozoa trigonometric equations and their general solutions look like this:
Please note that the sine and cosine values must take into account the limitations known to us. If, for example, , then the equation has no solutions and this formula should not be applied.
In addition, these root formulas contain a parameter in the form of an arbitrary integer . V school curriculum this is the only case when the solution of an equation without a parameter contains a parameter. This arbitrary integer shows that it is possible to write down an infinite number of roots of any of the indicated equations simply by substituting all the integers in turn.
You can get acquainted with the detailed receipt of these formulas by repeating the chapter “Trigonometric Equations” in the 10th grade algebra program.
Separately, it is necessary to pay attention to the solution of particular cases of the simplest equations with sine and cosine. These equations look like:
Formulas for finding general solutions should not be applied to them. Such equations are most conveniently solved using a trigonometric circle, which gives a simpler result than general solution formulas.
For example, the solution to the equation is 
. Try to get this answer yourself and solve the rest of the indicated equations.
In addition to the most common type of trigonometric equations indicated, there are several more standard ones. We list them, taking into account those that we have already indicated:
1) Protozoa, For example, ;
2) Particular cases of the simplest equations, For example, ;
3) Complex Argument Equations, For example, 
;
4) Equations reduced to their simplest form by taking out a common factor, For example, ;
5) Equations reduced to their simplest form by transforming trigonometric functions, For example, ;
6) Equations Reducible to the Simplest by Substitution, For example, ;
7) Homogeneous equations , For example, ;
8) Equations that are solved using the properties of functions, For example, 
. Don't be intimidated by the fact that this equation has two variables, it is solved at the same time;
As well as equations that are solved using various methods.
Systems of trigonometric equations and methods for their solution
In addition to solving trigonometric equations, it is necessary to be able to solve their systems.
The most common types of systems are:
1) In which one of the equations is a power law, For example, 
;
2) Systems of simple trigonometric equations, For example, 
.
In today's lesson, we looked at the basic trigonometric functions, their properties and graphs. And also got acquainted with the general formulas for solving the simplest trigonometric equations, indicated the main types of such equations and their systems.
In the practical part of the lesson, we will analyze the methods for solving trigonometric equations and their systems.
Box 1.Solution of special cases of the simplest trigonometric equations.
As we said in the main part of the lesson, special cases of trigonometric equations with sine and cosine of the form:
have more simple solutions than give formulas for general solutions.
For this, a trigonometric circle is used. Let us analyze the method for solving them using the equation as an example.
Draw a point on a trigonometric circle at which the cosine value is zero, which is also the coordinate along the x-axis. As you can see, there are two such points. Our task is to point out is equal to the angle, which corresponds to these points on the circle.
 |
We start counting from the positive direction of the abscissa axis (cosine axis) and when postponing the angle we get to the first point shown, i.e. one of the solutions will be this angle value. But we are still satisfied with the angle that corresponds to the second point. How to get into it?
To do this, you need to add a developed corner to the already set aside corner. The second angle, which is the solution to the equation, is . But we must not forget that this is not all, because we can build an angle larger than a full circle, and it will again fall into the first point and will also be a solution to our equation. To do this, add to the second calculated angle again, and get the value. You can continue these actions an infinite number of times.
If we write out the first three roots of the equation we obtained, then we can see a pattern:
, , , ... and write out the formula for all roots:
As you can see, this formula really looks simpler than the general solution of the equation with cosine, if only because it does not contain "". However, this does not mean that general formula will give the wrong solution.
Similarly, solutions can be obtained for all other indicated particular cases of trigonometric equations.
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