Lesson on the topic of solving trigonometric inequalities. Development of a lesson "trigonometric inequalities"
LESSONS No. 27-28
Methods for solving trigonometric inequalities
Goals and objectives of the lesson:
Educational:
Explore ways to solve trigonometric inequalities.
Organize the work of students at a level corresponding to the level of acquired knowledge and skills.
Developmental:
To develop in students the ability to construct mathematical models, in this case a graphical model for solving inequalities.
Educational:
To promote the development of students' cognitive interest in the subject, influencing the interest of high school students in self-knowledge.
Lesson type: combined lesson.
Lesson methods: verbal, practical, control and generalization of knowledge.
Forms of organizing student activities in the classroom: frontal, work in groups, controlled independent work.
Method of acquiring knowledge : heuristic, exploratory.
Presentation for the lesson.
During the classes
1. Self-determination for activity (3 min)
Psychological mood of students. Announcing the topic of the lesson, commenting on the objectives of the lesson.
2. Checking homework (5 minutes)
Comment on homework; if necessary, students who have completed it show the solution at the board
3. Updating students' theoretical knowledge ( 12 min)
Frontal survey of students:
Range of values of trigonometric functions
Domain of trigonometric functions
The values of trigonometric functions of angles are 0 0, 30 0, 45 0, 60 0, 90 0, 120 0, 135 0, 150 0, 180 0.
List the types of simple trigonometric equations.
Methods for solving trigonometric equations.
Methods for solving systems of trigonometric equations.
Working with the trigonometric circle. Using the values of trigonometric functions, determine the angle, find the values of the inverse trigonometric functions.
4. Explanation of new material ( 20 minutes).
Types of the simplest trigonometric inequalities and their interpretation on the trigonometric circle:
1) cost > A
Answer: (- arccos A +2π k ; arccos a+ 2π k ), k ЄZ
2)sint< A
Answer: (-( π +arcsin A )+2π k ; arcsin A +2π k ), k ЄZ
3) tgt > - A
Answer: (- arctg A +π k ; π/2 +π k ), k ЄZ
4) ctgt> A
Answer: (0+ π k ; arcctg A+π k ), k ЄZ .
Let's look at examples of solutions (on the slides):
Students independently comment on the proposed solution
Algorithm for solving simple trigonometric inequalities
Using the simplest algebraic transformations and trigonometric transformations, reduce the given trigonometric inequality to the simplest.
Mark on the axis corresponding to the trigonometric function located on the left side of the inequality the value from the right side of the inequality.
Draw a line through this point perpendicular to this axis.
Mark the points of intersection of the straight line with the trigonometric circle (prick them out in the case of a strict inequality and paint them in otherwise).
Select the corresponding arc with boundaries at these points according to the inequality sign.
Indicate the counting direction (counterclockwise).
We find the beginning of the arc and the angle it corresponding.
Finding the angle corresponding to the end of the arc.
We write the answer in the form of an interval, taking into account the periodicity of the function.
5. Practical part. Reinforcing the material learned (30 min)
№136(a,c), No.137(a,c), No.138(a,c),No.140(a,c), No.142(a,c), No.144(a,c), No.142, No. 145 (textbook Algebra and beginnings of analysis 10, A.E. Abylkasymova)
Students solve at the board in twos (or different examples if the class level is above average, and the same example otherwise - in order to create a competitive effect).
6. Independent work (12 min)
Option -1 Option -2
1) sin
x
< 
/2 1)sin
x
< 1/2
2) cos
x
< -1/2 2) cos
x
≥ -
/2
3) tg 2 x -1 3) tg 3 x ≤ 1
4)
sin
(2
x
–
π
/6)
-
/2 4)
cos (3
x
–
π
/4) ≤ -
/2
5) 2cos (4x– π/6) > 1 5)2 sin (x/2 + π/4) ≥ -1
Independent work tests students' ability to reduce inequalities to the simplest and solve simple trigonometric inequalities. The following situations are provided: strict – non-strict inequality; an arc highlighted on a circle above - below, to the right - to the left of a given number.
7. Homework (2 minutes)
§11 (p.80) – explore how to solve trigonometric inequalities using graphs of trigonometric functions
Perform in any way No. 136(b,d), No.137(b,d), No.138(b,d),No.140(b,d), No.142(b,d), No.144(b,d) (textbook Algebra and beginnings of analysis 10, A.E. Abylkasymova)
8. Lesson summary (3 min)
Briefly describe the work of the class during the lesson. Draw students' attention to ways to solve trigonometric inequalities discussed in the lesson. Give a comment on the ratings.
9. Reflection(3 min)
Fill out the table:
Availability of explanation
Level of understanding of the topic
What grade did you work for today?
Who, in your opinion, worked actively in the lesson (indicate grades)
What type of inequality is problematic?
Are you interested in the topic you studied?
Are you satisfied with the pace of the lesson? Is there a need to reduce or increase it?
Lesson topic :
Lesson Objectives :
Lesson type : combined.
During the classes
1.Organizational part
2.Knowledge test:
3.Repetition.
4.New theme .
Solving the simplest trigonometric inequalities sinx < 0, sin x > 0
sin x≤ 0, sin x ≥ 0
Students are invited to use card No. 1 (format A-4) with the following content.
Card No. 1.
Algorithm for solving trigonometric inequalities.
On the ordinate axis of the unit circle we mark the point corresponding to the valueA(approximately).
Through the resulting point we draw a straight line parallel to the other axis of the coordinate system until it intersects with the circle (Intersection points can be connected to the center of the circle).
On the unit circle at the intersection points we write down the numbers corresponding to these points.
Mentally move our straight line parallel to the coordinate axis depending on the valueA.
We highlight by hatching that part of the arc of the unit circle that the moving straight line intersects. If the inequality is strict, then the points at the ends of the arc are not shaded (punctured points).
We write down the answer.
Solving the inequality sinx>
Further, according to the algorithm, the teacher on the board, and the students on the card, carry out sequential operations on unit circles (Fig. 1, a, b, c), considering the solution to the inequality sinx >
Rice. 1
The answer is recorded: 
Solving the inequality cosx>
The solution to the inequality is carried out by one of the students on the board. With maximum independence, using a drawing, students write down the solution to this inequality on a card (Rice. 2, a ). If necessary, the teacher provides assistance to the student at the blackboard and to the students in the class. The algorithm for solving the inequality is fixed.
Rice. 2
Answer: 
5. Consolidation.
Students are asked to solve the inequality themselves (Rice. 6, b )
Answer:
6. Homework clause 8.1, card material.
7. Monitoring and evaluation of work. Lesson summary.
Repeat the algorithm for solving trigonometric inequalities using any example from the textbook § 8 p. 8.1 (A.N. Shynybekov. Algebra and the beginnings of mathematical analysis. Textbook for grade 10 of secondary school. Almaty “Atamura” 2012).
Mathematics teacher Lorenz Olga Vasilievna _________________________
Lesson topic : Solving simple trigonometric inequalities.
Lesson Objectives : a) organize work on studying ways to solve trigonometric inequalities;
contribute to the formation of skills and abilities to solve simple trigonometric inequalities;
b) create conditions for the development of memory, attention, counting techniques, intuition, speech, curiosity, independence of logical thinking;
c) to promote tactfulness, respect for classmates, willpower, responsible attitude to learning, self-discipline and perseverance.
Lesson type : combined.
During the classes
1.Organizational part : dividing class students into groups, distributing roles in groups.
2.Knowledge test:
D/Z orally: frontal check, explanation of solutions to tasks that caused difficulties.
3.Repetition.
For which function is there an inverse function? Give an example of a function for which there is an inverse function over the entire domain of definition; there is no inverse function over the entire domain of definition.
What is the relationship between the domain of definition and the range of values of the direct and inverse functions?
How are the graphs of direct and inverse functions located in a rectangular coordinate system?
Is it possible to say that trigonometric functions have inverse functions throughout their entire domain of definition? Justify your answer.
4.New topic.
Students - group leaders prepare presentations at home on the topic: “Solving the simplest trigonometric inequalities.” During the explanation, these students explain the new topic using their presentations.
5. Fastening. Independent work in groups.
Cos X<-
( + 2 k; + 2 k), k
Sin X ≥
[ + 2 k, + 2 k], k
Sin X< -
(- ;- + 2 k) , k
Sin X< -
(- ;- + 2 k) , k
Sin X ≥
X + 2 n, + 2 k], n
LESSON TOPIC: Solving simple trigonometric inequalities
The purpose of the lesson: show an algorithm for solving trigonometric inequalities using the unit circle.
Lesson Objectives:
Educational – ensure repetition and systematization of the topic material; create conditions for monitoring the acquisition of knowledge and skills;
Developmental - to promote the formation of skills to apply techniques: comparison, generalization, identification of the main thing, transfer of knowledge to a new situation, development of mathematical horizons, thinking and speech, attention and memory;
Educational – to promote interest in mathematics and its applications, activity, mobility, communication skills, and general culture.
Students' knowledge and skills:
- know the algorithm for solving trigonometric inequalities;
Be able to solve simple trigonometric inequalities.
Equipment: interactive whiteboard, lesson presentation, cards with independent work tasks.
DURING THE CLASSES:
1. Organizational moment(1 min)
I propose the words of Sukhomlinsky as the motto of the lesson: “Today we are learning together: me, your teacher and you are my students. But in the future the student must surpass the teacher, otherwise there will be no progress in science.”
2. Warm up. Dictation “True - False”
3. Repetition
For each option - task on the slide, continue each entry. Running time 3 min.
Let's cross-check this work of ours using the answer table on the board.
Evaluation criterion:“5” - all 9 “+”, “4” - 8 “+”, “3” - 6-7 “+”
4. Updating students’ knowledge(8 min)
Today in class we must learn the concept of trigonometric inequalities and master the skills of solving such inequalities.
– Let’s first remember what a unit circle is, a radian measure of an angle, and how the angle of rotation of a point on a unit circle is related to the radian measure of an angle. (working with presentation)
Unit circle is a circle with radius 1 and center at the origin.
The angle formed by the positive direction of the axis OX and the ray OA is called the rotation angle. It's important to remember where the 0 corners are; 90; 180; 270; 360.
If A is moved counterclockwise, positive angles are obtained.
If A is moved clockwise, negative angles are obtained.
сos t is the abscissa of a point on the unit circle, sin t is the ordinate of a point on the unit circle, t is the angle of rotation with coordinates (1;0).
5 . Explanation of new material (17 min.)
Today we will get acquainted with the simplest trigonometric inequalities.
Definition.
The simplest trigonometric inequalities are inequalities of the form:
The guys will tell us how to solve such inequalities (presentation of projects by students with examples). Students write down definitions and examples in their notebooks.
During the presentation, students explain the solution to the inequality, and the teacher completes the drawings on the board.
An algorithm for solving simple trigonometric inequalities is given after the students' presentation. Students see all stages of solving an inequality on the screen. This promotes visual memorization of the algorithm for solving a given problem.
Algorithm for solving trigonometric inequalities using the unit circle:
1. On the axis corresponding to a given trigonometric function, mark the given numerical value of this function.
2. Draw a line through the marked point intersecting the unit circle.
3. Select the points of intersection of the line and the circle, taking into account the strict or non-strict inequality sign.
4. Select the arc of the circle on which the solutions to the inequality are located.
5. Determine the values of the angles at the starting and ending points of the circular arc.
6. Write down the solution to the inequality taking into account the periodicity of the given trigonometric function.
To solve inequalities with tangent and cotangent, the concept of a line of tangents and cotangents is useful. These are the lines x = 1 and y = 1, respectively, tangent to the trigonometric circle.
6. Practical part(12 min)
To practice and consolidate theoretical knowledge, we will complete small tasks. Each student receives task cards. Having solved the inequalities, you need to choose an answer and write down its number.
7. Reflection on activities in the lesson
-What was our goal?
- Name the topic of the lesson
- We managed to use a well-known algorithm
- Analyze your work in class.
8. Homework(2 minutes)
Solve the inequality:
9. Lesson summary(2 minutes)
I propose to end the lesson with the words of Y.A. Komensky: “Consider unhappy that day or that hour in which you have not learned anything new and have not added anything to your education.”
Lesson topic: Solving trigonometric inequalities
The lesson was held in the 11th grade of school No. 4 named after. Gorky, Bryansk (2007).
The class works according to the textbook
https://pandia.ru/text/80/202/images/image002_105.jpg" width="142 height=189" height="189">
Teacher: teacher of the highest category, honored teacher of the Russian Federation Nina Vladimirovna Kusacheva.
Goals lesson:
1) Identify techniques for reducing trigonometric inequalities to the simplest: considering a complex argument as simple; use of equivalent transformations; application of trigonometric formulas.
2) Identify ways to solve trigonometric inequalities: reduction to the simplest; introduction of a new variable.
3) Learn to recognize ways to solve trigonometric inequalities.
4) Learn to write the answer if tabular values of trigonometric functions are not used.
5) Improve the ability to solve trigonometric inequalities.
6) Test your ability to solve simple trigonometric inequalities.
Lesson type: a lesson in improving skills.
Lesson Plan:
1. Identification of techniques and methods for solving trigonometric inequalities, difficulties in completing homework through analysis of solutions to the most complex inequalities.
2. Improving the ability to solve trigonometric inequalities:
a) recognition of solution methods and repetition of the algorithm for solving simple trigonometric inequalities;
b) working with the simplest inequality, where tabular values are not used to record the answer;
c) improving the ability to solve inequalities that can be reduced to the simplest trigonometric ones using equivalent transformations through comparison of inequalities;
d) improving the ability to solve inequalities that can be reduced to simple trigonometric ones using reduction formulas;
e) improving the ability to solve trigonometric inequalities through the use of several solution methods.
3. Independent work on solving trigonometric inequalities.
4. Setting homework.
During the classes:
1. Identification of techniques and methods for solving trigonometric inequalities, difficulties in completing homework through analysis of solutions to the most complex inequalities.
Teacher:(The solutions to inequalities No. 7, 8, 10 from the home card are written on the board).
Look at the solution to inequality No. 7. What questions do you have about any of the steps in the solution?
№7 sin x ≤ - cos x;
sin x + cos x ≤0;
https://pandia.ru/text/80/202/images/image004_95.gif" width="24" height="41 src="> sin x + cos x) ≤ 0;
https://pandia.ru/text/80/202/images/image005_84.gif" width="17" height="41">) ≤ 0;
sin(x + ) ≤ 0;
x+ О [ - π +2π n, 2π n], nО Z
xО [ -5π/4 + 2π n,- π/4+ 2π n], nО Z
Answer: xО [ -5π/4 +2π n,- π/4+ 2π n], nО Z
Teacher: Then I have a few questions. How was the 3rd line obtained?
Students: We multiplied and divided each term by .
Teacher: Is it possible to perform such an inequality transformation?
Students: Yes, this conversion is equivalent.
Teacher: For what purpose did we do this?
Students: So that you can apply the trigonometric addition formula - the sine of the sum of two angles.
Teacher: What is another name for this technique?
Students: Technique for introducing an auxiliary angle.
Teacher: How did you guess that you need to multiply and divide each term exactly by?
Students: is the square root of the sum of the squares of the coefficients in the transformed inequality.
Teacher: Name the inequality that can be considered the simplest and give reasons for your answer.
Students: Inequality sin(x+ ) ≤ 0 can be considered the simplest if we consider the complex argument ( x+ ) as simple, for example, t.
Teacher: So, the main idea of solving inequality No. 7 is to reduce it to the simplest trigonometric inequality. Let's repeat what techniques were used?
Students: 1) equivalent transformations (transfer of terms; multiplication and division of each term by the same number; introduction of an auxiliary angle);
(The teacher helps students by pointing to one or another line of the solution.)
Teacher: Look at the solution to inequality #8.
№ 8 sin 2x+ https://pandia.ru/text/80/202/images/image007_69.gif" width="21" height="22">/2 cos 2x) ≥ 1;
2 sin (2x+ π/3) ≥ 1;
sin (2x+ π/3) ≥ 1/2;
2x+ π/3 О [π/6 + 2π n, 5π/6 + 2π n], nО Z;
xО [-π/12 + π n, π/4 + π n], n О Z;
Answer: xО [-π/12 + π n, π/4 + π n], nО Z.
What questions do you have about any of the solution steps? (pause) What techniques were used to solve this inequality?
Students: 1) equivalent transformations (transfer of terms; multiplication and division of each term by the same number; introduction of an auxiliary angle, division of both sides of the inequality by a positive number);
2) application of the trigonometric formula,
3) treated a complex argument as simple.
Teacher: Consider the solution to inequality #10:
№10 cos 2 x – 2cosx >0;
Let cos x= t;
t 2 – 2t >0;
https://pandia.ru/text/80/202/images/image003_118.gif" width="22" height="21">;
2. cos(3π/2 + x) < -/2;
3. cos(π + 2 x) – 1 ≥ 0;
4. sin x > 2/3;
5. 5cos(x– π/6) – 1 ≥ 0;
6. 4sin 2 3x < 3.
Teacher: Highlight the inequalities that require the use of equivalent transformations when reducing a trigonometric inequality to its simplest form?
Students: 1, 3, 5.
Teacher: What are the inequalities in which you need to consider a complex argument as a simple one?
Students: 1, 2, 3, 5, 6.
Teacher: What are the inequalities where trigonometric formulas can be applied?
Students: 2, 3, 6.
Teacher: Name the inequalities where the method of introducing a new variable can be applied?
Students: 6.
Teacher: Now we will start solving inequalities from the simplest and learn how to write the answer if tabular values are not used. But first, answer whether it is true that the simplest trigonometric inequalities can be solved using the algorithm written on the board:
Algorithm for solving simple trigonometric inequalities
1. Orally replace the inequality with an equation. Draw a unit circle and mark the points on it that correspond to the equation.
2. Mark the points of the circle corresponding to the inequality, i.e. select the corresponding arc.
3. Indicate the counting direction.
4. Find the beginning of the arc and the angle corresponding to it.
5. Find the angle corresponding to the end of the arc.
6. We write the answer in the form of an interval, taking into account the periodicity of the function.
Teacher: Is this the order in which you solved the simplest inequalities?
Students: Yes.
A comment. The task of analyzing a list of inequalities from the standpoint of methods for solving them allows you to practice their recognition. When developing skills, it is important to identify the stages of its implementation and formulate them in a general form, which is presented in the algorithm for solving the simplest trigonometric inequalities.
b) Working with the simplest inequality, where tabular values are not used to record the answer.
Teacher: Let's start solving with inequality No. 4.
Organization of further work:
https://pandia.ru/text/80/202/images/image010_58.gif" width="204" height="130">One student solves the inequality at the board, saying each step of the algorithm out loud
5cos(x– π/6) – 1 ≥ 0;
cos(x– π/6) ≥ 1/5;
x– π/6 О [- arccos 1/5 + 2π n, arccos 1/5 + 2π n], nО Z;
xО [π/6 – arccos 1/5 + 2π n, π/6 + arccos 1/5 + 2π n], nО Z.
Upon completion of the solution, the teacher asks the student who solved the inequality at the board the following questions:
Teacher: How would the answer change if a strict inequality were given?
Student: Then the square brackets would be replaced with round brackets.
Teacher: How would you write down the answer if an inequality was given? cos (x– π/6) ≤ 1/5?
Student: xО [π/6 + arccos 1/5 + 2π n, 13π/6 – arccos 1/5 + 2π n], nО Z.
Teacher: What methods of reduction to the simplest trigonometric inequality were used?
Student: Equivalent transformations were used (transferring terms from one part of the equation to another, dividing both sides of the inequality by a positive number); treated a complex argument as simple.
Teacher:(addressing the class); Do you have any questions or comments for the respondent? (the student answers the students’ questions and agrees or disagrees with the comments, then sits down).
Teacher: What inequality is inequality No. 1 similar to and in what ways?
Students: To inequality No. 5 by reducing it to the simplest; to inequality No. 4 by the location of the arc.
Teacher: Solve orally inequality No. 1: 2 sin (x– π/4) ≥ .
Students: Answer: xО [ π/2 + 2π n, π + 2π n], nО Z.
A comment. Improving the ability to solve trigonometric inequalities is facilitated by the following questions: “How will we solve a group of inequalities?”; “How does one inequality differ from another?”; “How is one inequality similar to another?”; How would the answer change if strict inequality were given?"; How would the answer change if instead of the ">" sign there was a "<»?»; «Какие способы сведения к простейшему тригонометрическому неравенству использовались при решении данного неравенства?»; «Есть ли вопросы или замечания к отвечающему?». Оправдана такая организация работы, когда один ученик у доски решает неравенство, проговаривая каждый шаг алгоритма вслух, поскольку предложенное неравенство № 5 содержит косинус, а не синус, как это было на предыдущем этапе. Совершенствованию умения решать тригонометрические неравенства способствует и устное решение с предварительным обсуждением некоторых опор: «На какое неравенство похоже данное и чем?».
d) Improving the ability to solve inequalities that can be reduced to the simplest trigonometric ones using reduction formulas.
Teacher: Consider inequality No. 2 cos(3π/2 + x)< -https://pandia.ru/text/80/202/images/image011_55.gif" width="217" height="126 src=">A willing student solves the inequality at the board without saying the solution:
cos(3π/2 + x)< -https://pandia.ru/text/80/202/images/image007_69.gif" width="21" height="22 src=">/2;
Answer: xО (- 2π/3 + 2π n,-π/3 + 2π n), nО Z.
Upon completion of the solution, students check the formatting and make comments if necessary. After which the teacher asks the respondent the following questions:
Teacher: How does this inequality differ from those solved previously?
Student: This inequality has been reduced to its simplest form using the reduction formula.
Teacher: Are there other inequalities that can be solved this way?
Student: № 3.
Teacher: We will solve the inequality orally, commenting on the progress of the solution.
Students:(they comment on the progress of the solution in order, the teacher makes changes to the inequality)
№ 3 cos(π + 2 x) – 1 ≥ 0;
cos(π + 2 x) ≥ 1;
- cos 2x ≥ 1;
cos 2x ≤ -1
2x= -π + 2π n , nО Z;
x= -π/2 + π n , nО Z.
Teacher: So, what is the peculiarity of solving this inequality?
Students: His solution came down to solving an equation.
Teacher: So, what do you do next when you see that the argument of a trigonometric function is complex?
Students: We'll see if we can use reduction formulas to simplify the argument.
Academic discipline: Mathematics. Subject: “Solving simple trigonometric inequalities” Lesson type: a lesson in mastering new material with elements of primary consolidation. Lesson objectives: 1) educational:
- show an algorithm for solving trigonometric inequalities using the unit circle. learn to solve simple trigonometric inequalities.
- development of the ability to generalize acquired knowledge; development of logical thinking;
- development of attention; development of students' competent oral and written mathematical speech.
- learn to express your ideas and opinions; develop the ability to help and support friends; develop the ability to determine how the views of comrades differ from their own.
- visually - illustrative;
- to connect new information with already studied material; to develop the ability to analyze and select the necessary information; to develop the ability to share your ideas and opinions. for the development of logic and reflection skills.
- textbook by A. N. Kolmogorov “Algebra and the beginnings of analysis”, grades 10-11; projector, board; MS PowerPoint presentation.
- Organizing time (1 min); Checking homework (7 min); Learning new material (31 min); Homework (3 min); Summarizing (3 min)
Lesson topic: Solving simple trigonometric inequalities.
Completed by: mathematics teacher of KGBOU NPO “PU No. 44” Moser O. S.
Stages of activity
Teacher: - In the last lesson we solved the simplest trigonometric equations, today we will learn how to solve the simplest trigonometric inequality using the unit circle. The solution of inequalities containing trigonometric functions is reduced, as a rule, to the solution of the simplest trigonometric inequalities of the formsin x ≤ a, cos x > a, tg x ≥ a, ctg x aAnd etc. Let us consider the solution of trigonometric inequalities using specific examples using the unit circle: Algorithm for solving this inequality: Using the same algorithm, the teacher and students solve the following examples:
