Identity transformations of trigonometric expressions. Lesson "simplification of trigonometric expressions" Trigonometric expressions and their transformations

Sections: Maths

Class: 11

Lesson 1

Topic: Grade 11 (preparation for the exam)

Simplification of trigonometric expressions.

Solution of the simplest trigonometric equations. (2 hours)

Goals:

  • Systematize, generalize, expand the knowledge and skills of students related to the use of trigonometry formulas and the solution of the simplest trigonometric equations.

Equipment for the lesson:

Lesson structure:

  1. Orgmoment
  2. Testing on laptops. The discussion of the results.
  3. Simplifying trigonometric expressions
  4. Solution of the simplest trigonometric equations
  5. Independent work.
  6. Summary of the lesson. Explanation of homework.

1. Organizational moment. (2 minutes.)

The teacher greets the audience, announces the topic of the lesson, recalls that the task was previously given to repeat the trigonometry formulas and sets the students up for testing.

2. Testing. (15min + 3min discussion)

The goal is to test the knowledge of trigonometric formulas and the ability to apply them. Each student has a laptop on his desk in which there is a test option.

There can be any number of options, I will give an example of one of them:

I option.

Simplify expressions:

a) basic trigonometric identities

1. sin 2 3y + cos 2 3y + 1;

b) addition formulas

3. sin5x - sin3x;

c) converting a product to a sum

6. 2sin8y cos3y;

d) double angle formulas

7.2sin5x cos5x;

e) half angle formulas

f) triple angle formulas

g) universal substitution

h) lowering the degree

16. cos 2 (3x/7);

Students on a laptop in front of each formula see their answers.

The work is instantly checked by the computer. The results are displayed on a large screen for everyone to see.

Also, after the end of the work, the correct answers are shown on the students' laptops. Each student sees where the mistake was made and what formulas he needs to repeat.

3. Simplification of trigonometric expressions. (25 min.)

The goal is to repeat, work out and consolidate the application of the basic formulas of trigonometry. Solving problems B7 from the exam.

At this stage, it is advisable to divide the class into groups of strong (work independently with subsequent verification) and weak students who work with the teacher.

Assignment for strong students (prepared in advance on a printed basis). The main emphasis is on the reduction and double angle formulas, according to the USE 2011.

Simplify expressions (for strong learners):

In parallel, the teacher works with weak students, discussing and solving tasks on the screen under the dictation of the students.

Calculate:

5) sin(270º - α) + cos(270º + α)

6)

Simplify:

It was the turn to discuss the results of the work of the strong group.

Answers appear on the screen, and also, with the help of a video camera, the work of 5 different students is displayed (one task for each).

The weak group sees the condition and the solution method. There is discussion and analysis. With the use of technical means, this happens quickly.

4. Solution of the simplest trigonometric equations. (30 minutes.)

The goal is to repeat, systematize and generalize the solution of the simplest trigonometric equations, recording their roots. Solution of problem B3.

Any trigonometric equation, no matter how we solve it, leads to the simplest.

When completing the assignment, students should pay attention to writing the roots of equations of particular cases and general form and to the selection of roots in the last equation.

Solve Equations:

Write down the smallest positive root of the answer.

5. Independent work (10 min.)

The goal is to test the acquired skills, identify problems, errors and ways to eliminate them.

A variety of work is offered at the student's choice.

Option for "3"

1) Find the value of the expression

2) Simplify the expression 1 - sin 2 3α - cos 2 3α

3) Solve the equation

Option for "4"

1) Find the value of the expression

2) Solve the equation Write down the smallest positive root of your answer.

Option for "5"

1) Find tgα if

2) Find the root of the equation Write down the smallest positive root of your answer.

6. Summary of the lesson (5 min.)

The teacher sums up the fact that the lesson repeated and consolidated the trigonometric formulas, the solution of the simplest trigonometric equations.

Homework is assigned (prepared on a printed basis in advance) with a spot check in the next lesson.

Solve Equations:

9)

10) Give your answer as the smallest positive root.

Lesson 2

Topic: Grade 11 (preparation for the exam)

Methods for solving trigonometric equations. Root selection. (2 hours)

Goals:

  • Generalize and systematize knowledge on solving trigonometric equations of various types.
  • To promote the development of mathematical thinking of students, the ability to observe, compare, generalize, classify.
  • Encourage students to overcome difficulties in the process of mental activity, to self-control, introspection of their activities.

Equipment for the lesson: KRMu, laptops for each student.

Lesson structure:

  1. Orgmoment
  2. Discussion d / s and samot. the work of the last lesson
  3. Repetition of methods for solving trigonometric equations.
  4. Solving trigonometric equations
  5. Selection of roots in trigonometric equations.
  6. Independent work.
  7. Summary of the lesson. Homework.

1. Organizing moment (2 min.)

The teacher greets the audience, announces the topic of the lesson and the work plan.

2. a) Analysis of homework (5 min.)

The goal is to check performance. One work is shown on the screen with the help of a video camera, the rest are selectively collected for the teacher to check.

b) Analysis of independent work (3 min.)

The goal is to sort out the mistakes, indicate ways to overcome them.

On the screen are the answers and solutions, the students have pre-issued their work. The analysis is going fast.

3. Repetition of methods for solving trigonometric equations (5 min.)

The goal is to recall methods for solving trigonometric equations.

Ask students what methods of solving trigonometric equations they know. Emphasize that there are so-called basic (frequently used) methods:

  • variable substitution,
  • factorization,
  • homogeneous equations,

and there are applied methods:

  • according to the formulas for converting a sum to a product and a product to a sum,
  • by the reduction formulas,
  • universal trigonometric substitution
  • introduction of an auxiliary angle,
  • multiplication by some trigonometric function.

It should also be recalled that one equation can be solved in different ways.

4. Solving trigonometric equations (30 min.)

The goal is to generalize and consolidate knowledge and skills on this topic, to prepare for solving C1 from the USE.

I consider it expedient to solve equations for each method together with students.

The student dictates the solution, the teacher writes down on the tablet, the whole process is displayed on the screen. This will allow you to quickly and efficiently restore previously covered material in your memory.

Solve Equations:

1) variable change 6cos 2 x + 5sinx - 7 = 0

2) factorization 3cos(x/3) + 4cos 2 (x/3) = 0

3) homogeneous equations sin 2 x + 3cos 2 x - 2sin2x = 0

4) converting the sum to the product cos5x + cos7x = cos(π + 6x)

5) converting the product to the sum 2sinx sin2x + cos3x = 0

6) lowering the degree of sin2x - sin 2 2x + sin 2 3x \u003d 0.5

7) universal trigonometric substitution sinx + 5cosx + 5 = 0.

When solving this equation, it should be noted that the use of this method leads to a narrowing of the domain of definition, since the sine and cosine are replaced by tg(x/2). Therefore, before writing out the answer, it is necessary to check whether the numbers from the set π + 2πn, n Z are horses of this equation.

8) introduction of an auxiliary angle √3sinx + cosx - √2 = 0

9) multiplication by some trigonometric function cosx cos2x cos4x = 1/8.

5. Selection of roots of trigonometric equations (20 min.)

Since in the conditions of fierce competition when entering universities, the solution of one first part of the exam is not enough, most students should pay attention to the tasks of the second part (C1, C2, C3).

Therefore, the purpose of this stage of the lesson is to recall the previously studied material, to prepare for solving problem C1 from the USE in 2011.

There are trigonometric equations in which you need to select the roots when writing out the answer. This is due to some restrictions, for example: the denominator of a fraction is not equal to zero, the expression under the root of an even degree is non-negative, the expression under the sign of the logarithm is positive, etc.

Such equations are considered to be equations of increased complexity and in the USE version they are in the second part, namely C1.

Solve the equation:

The fraction is zero if then using the unit circle, we will select the roots (see Figure 1)

Picture 1.

we get x = π + 2πn, n Z

Answer: π + 2πn, n Z

On the screen, the selection of roots is shown on a circle in a color image.

The product is equal to zero when at least one of the factors is equal to zero, and the arc, at the same time, does not lose its meaning. Then

Using the unit circle, select the roots (see Figure 2)

Figure 2.

5)

Let's go to the system:

In the first equation of the system, we make the change log 2 (sinx) = y, we obtain the equation then , back to the system

using the unit circle, we select the roots (see Figure 5),

Figure 5

6. Independent work (15 min.)

The goal is to consolidate and check the assimilation of the material, identify errors, and outline ways to correct them.

The work is offered in three versions, prepared in advance on a printed basis, at the choice of students.

Equations can be solved in any way.

Option for "3"

Solve Equations:

1) 2sin 2 x + sinx - 1 = 0

2) sin2x = √3cosx

Option for "4"

Solve Equations:

1) cos2x = 11sinx - 5

2) (2sinx + √3)log 8 (cosx) = 0

Option for "5"

Solve Equations:

1) 2sinx - 3cosx = 2

2)

7. Summary of the lesson, homework (5 min.)

The teacher sums up the lesson, once again draws attention to the fact that the trigonometric equation can be solved in several ways. The best way to achieve a quick result is the one that is best learned by a particular student.

When preparing for the exam, you need to systematically repeat the formulas and methods for solving equations.

Homework (prepared in advance on a printed basis) is distributed and ways of solving some equations are commented.

Solve Equations:

1) cosx + cos5x = cos3x + cos7x

2) 5sin(x/6) - cos(x/3) + 3 = 0

3) 4sin 2x + sin2x = 3

4) sin 2 x + sin 2 2x - sin 2 3x - sin 2 4x = 0

5) cos3x cos6x = cos4x cos7x

6) 4sinx - 6cosx = 1

7) 3sin2x + 4 cos2x = 5

8) cosx cos2x cos4x cos8x = (1/8) cos15x

9) (2sin 2 x - sinx)log 3 (2cos 2 x + cosx) = 0

10) (2cos 2 x - √3cosx)log 7 (-tgx) = 0

11)

The video lesson "Simplification of trigonometric expressions" is designed to form students' skills in solving trigonometric problems using basic trigonometric identities. During the video lesson, types of trigonometric identities are considered, examples of solving problems using them. Using visual aids, it is easier for the teacher to achieve the objectives of the lesson. A vivid presentation of the material contributes to the memorization of important points. The use of animation effects and voice acting allow you to completely replace the teacher at the stage of explaining the material. Thus, using this visual aid in mathematics lessons, the teacher can increase the effectiveness of teaching.

At the beginning of the video lesson, its topic is announced. Then the trigonometric identities studied earlier are recalled. The screen displays the equalities sin 2 t+cos 2 t=1, tg t=sin t/cos t, where t≠π/2+πk for kϵZ, ctg t=cos t/sin t, true for t≠πk, where kϵZ, tg t · ctg t=1, at t≠πk/2, where kϵZ, called basic trigonometric identities. It is noted that these identities are often used in solving problems where it is necessary to prove equality or simplify the expression.

Further, examples of the application of these identities in solving problems are considered. First, it is proposed to consider solving problems of simplifying expressions. In example 1, it is necessary to simplify the expression cos 2 t- cos 4 t+ sin 4 t. To solve the example, the common factor cos 2 t is first bracketed. As a result of such a transformation in brackets, the expression 1- cos 2 t is obtained, the value of which from the basic identity of trigonometry is equal to sin 2 t. After the transformation of the expression, it is obvious that one more common factor sin 2 t can be taken out of brackets, after which the expression takes the form sin 2 t (sin 2 t + cos 2 t). From the same basic identity, we deduce the value of the expression in brackets equal to 1. As a result of simplification, we obtain cos 2 t- cos 4 t+ sin 4 t= sin 2 t.

In example 2, the expression cost/(1- sint)+ cost/(1+ sint) also needs to be simplified. Since the expression cost is in the numerators of both fractions, it can be bracketed out as a common factor. Then the fractions in brackets are reduced to a common denominator by multiplying (1- sint)(1+ sint). After reduction of similar terms, 2 remains in the numerator, and 1 - sin 2 t in the denominator. On the right side of the screen, the basic trigonometric identity sin 2 t+cos 2 t=1 is recalled. Using it, we find the denominator of the fraction cos 2 t. After reducing the fraction, we get a simplified form of the expression cost/(1- sint)+ cost/(1+ sint)=2/cost.

Next, we consider examples of proving identities in which the acquired knowledge about the basic identities of trigonometry is applied. In Example 3, it is necessary to prove the identity (tg 2 t-sin 2 t)·ctg 2 t=sin 2 t. The right side of the screen displays three identities that will be needed for the proof - tg t ctg t=1, ctg t=cos t/sin t and tg t=sin t/cos t with restrictions. To prove the identity, the brackets are first opened, after which a product is formed that reflects the expression of the main trigonometric identity tg t·ctg t=1. Then, according to the identity from the definition of cotangent, ctg 2 t is transformed. As a result of transformations, the expression 1-cos 2 t is obtained. Using the basic identity, we find the value of the expression. Thus, it is proved that (tg 2 t-sin 2 t)·ctg 2 t=sin 2 t.

In example 4, you need to find the value of the expression tg 2 t+ctg 2 t if tg t+ctg t=6. To evaluate the expression, the right and left sides of the equation (tg t+ctg t) 2 =6 2 are first squared. The abbreviated multiplication formula is displayed on the right side of the screen. After opening the brackets on the left side of the expression, the sum tg 2 t+2 tg t ctg t+ctg 2 t is formed, for the transformation of which one of the trigonometric identities tg t ctg t=1 can be applied, the form of which is recalled on the right side of the screen. After the transformation, the equality tg 2 t+ctg 2 t=34 is obtained. The left side of the equality coincides with the condition of the problem, so the answer is 34. The problem is solved.

The video lesson "Simplifying trigonometric expressions" is recommended for use in a traditional school mathematics lesson. Also, the material will be useful to a teacher who provides distance learning. In order to form a skill in solving trigonometric problems.

TEXT EXPLANATION:

"Simplification of trigonometric expressions".

Equality

1)sin 2 t + cos 2 t = 1 (sine squared te plus cosine squared te equals one)

2) tgt =, at t ≠ + πk, kϵZ (the tangent of te is equal to the ratio of the sine of te to the cosine of te when te is not equal to pi by two plus pi ka, ka belongs to zet)

3) ctgt = , at t ≠ πk, kϵZ (the cotangent of te is equal to the ratio of the cosine of te to the sine of te when te is not equal to the peak of ka, which belongs to z).

4)tgt ∙ ctgt = 1 for t ≠ , kϵZ

are called basic trigonometric identities.

Often they are used in simplifying and proving trigonometric expressions.

Consider examples of using these formulas when simplifying trigonometric expressions.

EXAMPLE 1. Simplify the expression: cos 2 t - cos 4 t + sin 4 t. (expression a cosine squared te minus cosine of the fourth degree of te plus sine of the fourth degree of te).

Solution. cos 2 t - cos 4 t + sin 4 t = cos 2 t∙ (1 - cos 2 t) + sin 4 t = cos 2 t ∙ sin 2 t + sin 4 t = sin 2 t (cos 2 t + sin 2 t) = sin 2 t 1= sin 2 t

(we take out the common factor cosine square te, in parentheses we get the difference between unity and the square of cosine te, which is equal to the square of sine te by the first identity. We get the sum of the sine of the fourth degree te of the product of cosine square te and sine square te. We take out the common factor sine square te outside the brackets, in brackets we get the sum of the squares of the cosine and the sine, which, according to the basic trigonometric identity, is equal to 1. As a result, we get the square of the sine te).

EXAMPLE 2. Simplify the expression: + .

(expression be the sum of two fractions in the numerator of the first cosine te in the denominator one minus sine te, in the numerator of the second cosine te in the denominator of the second one plus sine te).

(We take the common factor cosine te out of brackets, and in brackets we bring it to a common denominator, which is the product of one minus sine te by one plus sine te.

In the numerator we get: one plus sine te plus one minus sine te, we give similar ones, the numerator is equal to two after bringing similar ones.

In the denominator, you can apply the abbreviated multiplication formula (difference of squares) and get the difference between the unit and the square of the sine te, which, according to the basic trigonometric identity

is equal to the square of the cosine te. After reducing by cosine te, we get the final answer: two divided by cosine te).

Consider examples of the use of these formulas in the proof of trigonometric expressions.

EXAMPLE 3. Prove the identity (tg 2 t - sin 2 t) ∙ ctg 2 t \u003d sin 2 t (the product of the difference between the squares of the tangent of te and the sine of te and the square of the cotangent of te is equal to the square of the sine of te).

Proof.

Let's transform the left side of the equality:

(tg 2 t - sin 2 t) ∙ ctg 2 t = tg 2 t ∙ ctg 2 t - sin 2 t ∙ ctg 2 t = 1 - sin 2 t ∙ ctg 2 t =1 - sin 2 t ∙ = 1 - cos 2 t = sin 2 t

(Let's open the brackets, from the previously obtained relation it is known that the product of the squares of the tangent of te by the cotangent of te is equal to one. Recall that the cotangent of te is equal to the ratio of the cosine of te to the sine of te, which means that the square of the cotangent is the ratio of the square of the cosine of te to the square of the sine of te.

After reduction by the sine square of te, we obtain the difference between unity and the cosine of the square of te, which is equal to the sine of the square of te). Q.E.D.

EXAMPLE 4. Find the value of the expression tg 2 t + ctg 2 t if tgt + ctgt = 6.

(the sum of the squares of the tangent of te and the cotangent of te, if the sum of the tangent and cotangent is six).

Solution. (tgt + ctgt) 2 = 6 2

tg 2 t + 2 ∙ tgt ∙ctgt + ctg 2 t = 36

tg 2 t + 2 + ctg 2 t = 36

tg 2 t + ctg 2 t = 36-2

tg 2 t + ctg 2 t = 34

Let's square both parts of the original equality:

(tgt + ctgt) 2 = 6 2 (the square of the sum of the tangent of te and the cotangent of te is six squared). Recall the abbreviated multiplication formula: The square of the sum of two quantities is equal to the square of the first plus twice the product of the first and the second plus the square of the second. (a+b) 2 =a 2 +2ab+b 2 We get tg 2 t + 2 ∙ tgt ∙ctgt + ctg 2 t = 36 .

Since the product of the tangent of te and the cotangent of te is equal to one, then tg 2 t + 2 + ctg 2 t \u003d 36 (the sum of the squares of the tangent of te and the cotangent of te and two is thirty-six),

Sections: Maths

Class: 11

Lesson 1

Topic: Grade 11 (preparation for the exam)

Simplification of trigonometric expressions.

Solution of the simplest trigonometric equations. (2 hours)

Goals:

  • Systematize, generalize, expand the knowledge and skills of students related to the use of trigonometry formulas and the solution of the simplest trigonometric equations.

Equipment for the lesson:

Lesson structure:

  1. Orgmoment
  2. Testing on laptops. The discussion of the results.
  3. Simplifying trigonometric expressions
  4. Solution of the simplest trigonometric equations
  5. Independent work.
  6. Summary of the lesson. Explanation of homework.

1. Organizational moment. (2 minutes.)

The teacher greets the audience, announces the topic of the lesson, recalls that the task was previously given to repeat the trigonometry formulas and sets the students up for testing.

2. Testing. (15min + 3min discussion)

The goal is to test the knowledge of trigonometric formulas and the ability to apply them. Each student has a laptop on his desk in which there is a test option.

There can be any number of options, I will give an example of one of them:

I option.

Simplify expressions:

a) basic trigonometric identities

1. sin 2 3y + cos 2 3y + 1;

b) addition formulas

3. sin5x - sin3x;

c) converting a product to a sum

6. 2sin8y cos3y;

d) double angle formulas

7.2sin5x cos5x;

e) half angle formulas

f) triple angle formulas

g) universal substitution

h) lowering the degree

16. cos 2 (3x/7);

Students on a laptop in front of each formula see their answers.

The work is instantly checked by the computer. The results are displayed on a large screen for everyone to see.

Also, after the end of the work, the correct answers are shown on the students' laptops. Each student sees where the mistake was made and what formulas he needs to repeat.

3. Simplification of trigonometric expressions. (25 min.)

The goal is to repeat, work out and consolidate the application of the basic formulas of trigonometry. Solving problems B7 from the exam.

At this stage, it is advisable to divide the class into groups of strong (work independently with subsequent verification) and weak students who work with the teacher.

Assignment for strong students (prepared in advance on a printed basis). The main emphasis is on the reduction and double angle formulas, according to the USE 2011.

Simplify expressions (for strong learners):

In parallel, the teacher works with weak students, discussing and solving tasks on the screen under the dictation of the students.

Calculate:

5) sin(270º - α) + cos(270º + α)

6)

Simplify:

It was the turn to discuss the results of the work of the strong group.

Answers appear on the screen, and also, with the help of a video camera, the work of 5 different students is displayed (one task for each).

The weak group sees the condition and the solution method. There is discussion and analysis. With the use of technical means, this happens quickly.

4. Solution of the simplest trigonometric equations. (30 minutes.)

The goal is to repeat, systematize and generalize the solution of the simplest trigonometric equations, recording their roots. Solution of problem B3.

Any trigonometric equation, no matter how we solve it, leads to the simplest.

When completing the assignment, students should pay attention to writing the roots of equations of particular cases and general form and to the selection of roots in the last equation.

Solve Equations:

Write down the smallest positive root of the answer.

5. Independent work (10 min.)

The goal is to test the acquired skills, identify problems, errors and ways to eliminate them.

A variety of work is offered at the student's choice.

Option for "3"

1) Find the value of the expression

2) Simplify the expression 1 - sin 2 3α - cos 2 3α

3) Solve the equation

Option for "4"

1) Find the value of the expression

2) Solve the equation Write down the smallest positive root of your answer.

Option for "5"

1) Find tgα if

2) Find the root of the equation Write down the smallest positive root of your answer.

6. Summary of the lesson (5 min.)

The teacher sums up the fact that the lesson repeated and consolidated the trigonometric formulas, the solution of the simplest trigonometric equations.

Homework is assigned (prepared on a printed basis in advance) with a spot check in the next lesson.

Solve Equations:

9)

10) Give your answer as the smallest positive root.

Lesson 2

Topic: Grade 11 (preparation for the exam)

Methods for solving trigonometric equations. Root selection. (2 hours)

Goals:

  • Generalize and systematize knowledge on solving trigonometric equations of various types.
  • To promote the development of mathematical thinking of students, the ability to observe, compare, generalize, classify.
  • Encourage students to overcome difficulties in the process of mental activity, to self-control, introspection of their activities.

Equipment for the lesson: KRMu, laptops for each student.

Lesson structure:

  1. Orgmoment
  2. Discussion d / s and samot. the work of the last lesson
  3. Repetition of methods for solving trigonometric equations.
  4. Solving trigonometric equations
  5. Selection of roots in trigonometric equations.
  6. Independent work.
  7. Summary of the lesson. Homework.

1. Organizing moment (2 min.)

The teacher greets the audience, announces the topic of the lesson and the work plan.

2. a) Analysis of homework (5 min.)

The goal is to check performance. One work is shown on the screen with the help of a video camera, the rest are selectively collected for the teacher to check.

b) Analysis of independent work (3 min.)

The goal is to sort out the mistakes, indicate ways to overcome them.

On the screen are the answers and solutions, the students have pre-issued their work. The analysis is going fast.

3. Repetition of methods for solving trigonometric equations (5 min.)

The goal is to recall methods for solving trigonometric equations.

Ask students what methods of solving trigonometric equations they know. Emphasize that there are so-called basic (frequently used) methods:

  • variable substitution,
  • factorization,
  • homogeneous equations,

and there are applied methods:

  • according to the formulas for converting a sum to a product and a product to a sum,
  • by the reduction formulas,
  • universal trigonometric substitution
  • introduction of an auxiliary angle,
  • multiplication by some trigonometric function.

It should also be recalled that one equation can be solved in different ways.

4. Solving trigonometric equations (30 min.)

The goal is to generalize and consolidate knowledge and skills on this topic, to prepare for solving C1 from the USE.

I consider it expedient to solve equations for each method together with students.

The student dictates the solution, the teacher writes down on the tablet, the whole process is displayed on the screen. This will allow you to quickly and efficiently restore previously covered material in your memory.

Solve Equations:

1) variable change 6cos 2 x + 5sinx - 7 = 0

2) factorization 3cos(x/3) + 4cos 2 (x/3) = 0

3) homogeneous equations sin 2 x + 3cos 2 x - 2sin2x = 0

4) converting the sum to the product cos5x + cos7x = cos(π + 6x)

5) converting the product to the sum 2sinx sin2x + cos3x = 0

6) lowering the degree of sin2x - sin 2 2x + sin 2 3x \u003d 0.5

7) universal trigonometric substitution sinx + 5cosx + 5 = 0.

When solving this equation, it should be noted that the use of this method leads to a narrowing of the domain of definition, since the sine and cosine are replaced by tg(x/2). Therefore, before writing out the answer, it is necessary to check whether the numbers from the set π + 2πn, n Z are horses of this equation.

8) introduction of an auxiliary angle √3sinx + cosx - √2 = 0

9) multiplication by some trigonometric function cosx cos2x cos4x = 1/8.

5. Selection of roots of trigonometric equations (20 min.)

Since in the conditions of fierce competition when entering universities, the solution of one first part of the exam is not enough, most students should pay attention to the tasks of the second part (C1, C2, C3).

Therefore, the purpose of this stage of the lesson is to recall the previously studied material, to prepare for solving problem C1 from the USE in 2011.

There are trigonometric equations in which you need to select the roots when writing out the answer. This is due to some restrictions, for example: the denominator of a fraction is not equal to zero, the expression under the root of an even degree is non-negative, the expression under the sign of the logarithm is positive, etc.

Such equations are considered to be equations of increased complexity and in the USE version they are in the second part, namely C1.

Solve the equation:

The fraction is zero if then using the unit circle, we will select the roots (see Figure 1)

Picture 1.

we get x = π + 2πn, n Z

Answer: π + 2πn, n Z

On the screen, the selection of roots is shown on a circle in a color image.

The product is equal to zero when at least one of the factors is equal to zero, and the arc, at the same time, does not lose its meaning. Then

Using the unit circle, select the roots (see Figure 2)

Voronkova Olga Ivanovna

MBOU "Secondary school

No. 18"

Engels, Saratov region.

Mathematic teacher.

"Trigonometric expressions and their transformations"

Introduction …………………………………………………………………………....3

Chapter 1 Classification of tasks for the use of transformations of trigonometric expressions ………………………….……………………...5

1.1. Calculation tasks values ​​of trigonometric expressions……….5

1.2.Tasks for simplifying trigonometric expressions .... 7

1.3. Tasks for the conversion of numerical trigonometric expressions ... ..7

1.4 Mixed tasks…………………………………………………….....9

Chapter 2

2.1 Thematic repetition in grade 10………………………………………...11

Test 1……………………………………………………………………………..12

Test 2………………………………………………………………………………..13

Test 3………………………………………………………………………………..14

2.2 Final repetition in grade 11……………………………………………...15

Test 1………………………………………………………………………………..17

Test 2………………………………………………………………………………..17

Test 3………………………………………………………………………………..18

Conclusion.……………………………………………………………………......19

List of used literature………………………………………..…….20

Introduction.

In today's conditions, the most important question is: "How can we help eliminate some gaps in the knowledge of students and warn them against possible mistakes in the exam?" To solve this issue, it is necessary to achieve from students not a formal assimilation of the program material, but its deep and conscious understanding, the development of the speed of oral calculations and transformations, as well as the development of skills for solving the simplest problems “in the mind”. It is necessary to convince students that only in the presence of an active position, in the study of mathematics, subject to the acquisition of practical skills, skills and their use, one can count on real success. It is necessary to use every opportunity to prepare for the exam, including elective subjects in grades 10-11, regularly analyze complex tasks with students, choosing the most rational way to solve them in the classroom and extra classes.positive result inthe area of ​​solving typical problems can be achieved if mathematics teachers, by creatinggood basic training of students, to look for new ways to solve the problems that have opened before us, to actively experiment, to apply modern pedagogical technologies, methods, techniques that create favorable conditions for effective self-realization and self-determination of students in new social conditions.

Trigonometry is an integral part of the school mathematics course. Good knowledge and strong skills in trigonometry are evidence of a sufficient level of mathematical culture, an indispensable condition for the successful study of mathematics, physics, and a number of technical disciplines.

The relevance of the work. A significant part of school graduates shows from year to year very poor preparation in this important section of mathematics, as evidenced by the results of past years (percentage of completion in 2011-48.41%, 2012-51.05%), since the analysis of passing the unified state exam showed that students make many mistakes when completing assignments of this particular section or do not undertake such assignments at all. In One State exam questions in trigonometry are found in almost three types of tasks. This is the solution of the simplest trigonometric equations in task B5, and work with trigonometric expressions in task B7, and the study of trigonometric functions in task B14, as well as tasks B12, in which there are formulas describing physical phenomena and containing trigonometric functions. And this is only part of the tasks B! But there are also favorite trigonometric equations with the selection of roots C1, and “not very favorite” geometric tasks C2 and C4.

Objective. Analyze the material of the USE tasks B7, devoted to the transformation of trigonometric expressions and classify the tasks according to the form of their presentation in the tests.

The work consists of two chapters, introduction and conclusion. The introduction emphasizes the relevance of the work. The first chapter provides a classification of tasks for the use of transformations of trigonometric expressions in test tasks for the USE (2012).

In the second chapter, the organization of the repetition of the topic "Conversion of trigonometric expressions" in grades 10, 11 is considered and tests on this topic are developed.

The list of references includes 17 sources.

Chapter 1. Classification of tasks for the use of transformations of trigonometric expressions.

In accordance with the standard of secondary (complete) education and the requirements for the level of training of students, tasks for knowledge of the basics of trigonometry are included in the codifier of requirements.

Learning the basics of trigonometry will be most effective when:

    students will be positively motivated to repeat previously studied material;

    a student-centered approach will be implemented in the educational process;

    a system of tasks will be applied that contributes to the expansion, deepening, systematization of students' knowledge;

    advanced pedagogical technologies will be used.

After analyzing the literature and Internet resources for preparing for the exam, we have proposed one of the possible classifications of tasks B7 (KIM USE 2012-trigonometry): tasks for calculatingvalues ​​of trigonometric expressions; assignments forconversion of numerical trigonometric expressions; assignments for the transformation of literal trigonometric expressions; mixed tasks.

1.1. Calculation tasks values ​​of trigonometric expressions.

One of the most common types of simple trigonometry problems is the calculation of the values ​​of trigonometric functions by the value of one of them:

a) Use of the basic trigonometric identity and its corollaries.

Example 1 . Find if
and
.

Solution.
,
,

Because , then
.

Answer.

Example 2 . Find
, if
and .

Solution.
,
,
.

Because , then
.

Answer. .

b) Use of double angle formulas.

Example 3 . Find
, if
.

Solution. , .

Answer.
.

Example 4 . Find the value of an expression
.

Solution. .

Answer.
.

1. Find , if
and
. Answer. -0.2

2. Find , if
and
. Answer. 0.4

3. Find
, if . Answer. -12.884. Find
, if
. Answer. -0.845. Find the value of the expression:
. Answer. 66. Find the value of an expression
.Answer. -19

1.2.Tasks for simplifying trigonometric expressions. The reduction formulas should be well mastered by students, as they will be further used in the lessons of geometry, physics and other related disciplines.

Example 5 . Simplify Expressions
.

Solution. .

Answer.
.

Tasks for independent solution:

1. Simplify the expression
. Answer. 0.62. Find
, if
and. Answer. 10.563. Find the value of an expression
, if
. Answer. 2

1.3. Tasks for the transformation of numerical trigonometric expressions.

When developing the skills and abilities of tasks for converting numerical trigonometric expressions, attention should be paid to knowledge of the table of values ​​of trigonometric functions, the properties of parity and periodicity of trigonometric functions.

a) Using exact values ​​of trigonometric functions for some angles.

Example 6 . Calculate
.

Solution.
.

Answer.
.

b) Using the properties of parity trigonometric functions.

Example 7 . Calculate
.

Solution. .

Answer.

in) Using Periodicity Propertiestrigonometric functions.

Example 8 . Find the value of an expression
.

Solution. .

Answer.
.

Tasks for independent solution:

1. Find the value of an expression
. Answer. -40.52. Find the value of the expression
. Answer. 17

3. Find the value of an expression
. Answer. 6


. Answer. -24
Answer. -64

1.4 Mixed tasks.

The test form of certification has very significant features, so it is important to pay attention to the tasks associated with the use of several trigonometric formulas at the same time.

Example 9 Find
, if
.

Solution.
.

Answer.
.

Example 10 . Find
, if
and
.

Solution. .

Because , then
.

Answer.
.

Example 11. Find
, if .

Solution. , ,
,
,
,
,
.

Answer.

Example 12 Calculate
.

Solution. .

Answer.
.

Example 13 Find the value of an expression
, if
.

Solution. .

Answer.
.

Tasks for independent solution:

1. Find
, if
. Answer. -1.75
2. Find
, if
. Answer. 33. Find
, if .Answer. 0.254. Find the value of the expression
, if
. Answer. 0.35. Find the value of the expression
, if
. Answer. 5

Chapter 2. Methodological aspects organization of the final repetition of the topic "Transformation of trigonometric expressions."

One of the most important issues contributing to the further improvement of academic performance, the achievement of deep and solid knowledge among students is the issue of repeating previously studied material. Practice shows that in the 10th grade it is more expedient to organize a thematic repetition; in 11th grade - the final repetition.

2.1. Thematic repetition in 10th grade.

In the process of working on mathematical material, the repetition of each completed topic or an entire section of the course becomes especially important.

With thematic repetition, students' knowledge on the topic is systematized at the final stage of its passage or after a break.

For thematic repetition, special lessons are allocated, on which the material of one particular topic is concentrated and generalized.

Repetition in the lesson is carried out through a conversation with the wide involvement of students in this conversation. After that, students are given the task to repeat a certain topic and are warned that there will be credit work on tests.

A test on a topic should include all of its main questions. After the work is completed, characteristic errors are analyzed and a repetition is organized to eliminate them.

For lessons of thematic repetition, we offer developed test papers on the topic "Conversion of trigonometric expressions".

Test #1

Test #2

Test #3

Answer table

Test

2.2. Final repetition in 11th grade.

The final repetition is carried out at the final stage of studying the main issues of the mathematics course and is carried out in logical connection with the study of educational material for this section or the course as a whole.

The final repetition of the educational material has the following goals:

1. Activation of the material of the entire training course to clarify its logical structure and build a system within subject and inter subject relationships.

2. Deepening and, if possible, expanding students' knowledge on the main issues of the course in the process of repetition.

In the context of the compulsory examination in mathematics for all graduates, the gradual introduction of the USE makes teachers take a new approach to preparing and conducting lessons, taking into account the need to ensure that all students master the educational material at a basic level, as well as the opportunity for motivated students interested in getting high scores for admission to a university, dynamic advancement in mastering the material at an increased and high level.

In the lessons of the final repetition, you can consider the following tasks:

Example 1 . Calculate the value of the expression .Solution. =
= =
=
=
=
=0,5. Answer. 0.5. Example 2 Specify the largest integer value that the expression can take
.

Solution. Because
can take any value belonging to the interval [–1; 1], then
takes any value of the segment [–0.4; 0.4], therefore . The integer value of the expression is one - the number 4.

Answer: 4 Example 3 . Simplify the expression
.

Solution: Let's use the formula for factoring the sum of cubes: . We have

We have:
.

Answer: 1

Example 4 Calculate
.

Solution. .

Answer: 0.28

For the lessons of the final repetition, we offer developed tests on the topic "Conversion of trigonometric expressions".

Specify the largest integer not exceeding 1

Conclusion.

Having worked through the relevant methodological literature on this topic, we can conclude that the ability and skills to solve tasks related to trigonometric transformations in the school mathematics course is very important.

In the course of the work done, the classification of tasks B7 was carried out. The trigonometric formulas most frequently used in CMMs of 2012 are considered. Examples of tasks with solutions are given. Differentiable tests have been developed to organize the repetition and systematization of knowledge in preparation for the exam.

It is advisable to continue the work begun, considering solution of the simplest trigonometric equations in task B5, the study of trigonometric functions in task B14, task B12, in which there are formulas describing physical phenomena and containing trigonometric functions.

In conclusion, I would like to note that the effectiveness of passing the exam is largely determined by how effectively the preparation process is organized at all levels of education, with all categories of students. And if we manage to form students' independence, responsibility and readiness to continue learning throughout their subsequent lives, then we will not only fulfill the order of the state and society, but also increase our own self-esteem.

Repetition of educational material requires creative work from the teacher. He must provide a clear connection between the types of repetition, implement a deeply thought-out system of repetition. Mastering the art of organizing repetition is the task of the teacher. The strength of students' knowledge largely depends on its solution.

Literature.

    Vygodsky Ya.Ya., Handbook of elementary mathematics. -M.: Nauka, 1970.

    Tasks of increased difficulty in algebra and the beginnings of analysis: Textbook for 10-11 grades of secondary school / B.M. Ivlev, A.M. Abramov, Yu.P. Dudnitsyn, S.I. Schwarzburd. – M.: Enlightenment, 1990.

    Application of basic trigonometric formulas to the transformation of expressions (grade 10) // Festival of Pedagogical Ideas. 2012-2013.

    Koryanov A.G. , Prokofiev A.A. We prepare good students and excellent students for the exam. - M.: Pedagogical University "First of September", 2012.- 103 p.

    Kuznetsova E.N. Simplification of trigonometric expressions. Solving trigonometric equations by various methods (preparation for the exam). 11th grade. 2012-2013.

    Kulanin E.D. 3000 competitive problems in mathematics. 4th id., correct. and additional – M.: Rolf, 2000.

    Mordkovich A.G. Methodological problems of studying trigonometry in a general education school // Mathematics at school. 2002. No. 6.

    Pichurin L.F. About trigonometry and not only about it: -M. Enlightenment, 1985

    Reshetnikov N.N. Trigonometry at school: -M. : Pedagogical University "First of September", 2006, lk 1.

    Shabunin M.I., Prokofiev A.A. Maths. Algebra. Beginnings of mathematical analysis. Profile level: textbook for grade 10 - M .: BINOM. Knowledge Lab, 2007.

    Educational portal for preparing for the exam.

    Preparing for the exam in mathematics "Oh, this trigonometry! http://festival.1september.ru/articles/621971/

    Project "Mathematics? Easy!!!" http://www.resolventa.ru/

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