The video lesson "Simplifying Trigonometric Expressions" is designed to develop students' skills in solving trigonometric problems using basic trigonometric identities. In the course of the video lesson, the types of trigonometric identities, examples of solving problems using them, are considered. By using the visual aid, it is easier for the teacher to achieve lesson objectives. A vivid presentation of the material helps to remember important points. The use of animation effects and dubbing make it possible 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, valid for t ≠ πk, where kϵZ, tg t · ctg t = 1, for t ≠ πk / 2, where kϵZ, called the basic trigonometric identities. It is noted that these identities are often used in solving problems where it is necessary to prove equality or simplify an expression.
Further, examples of the application of these identities in solving problems are considered. First, it is proposed to consider the solution of problems to simplify expressions. In example 1, it is necessary to simplify the expression cos 2 t- cos 4 t + sin 4 t. To solve the example, first place the common factor cos 2 t outside the brackets. As a result of such a transformation in parentheses, 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 transforming the expression, it is obvious that one more common factor sin 2 t can be parenthesized, after which the expression takes the form sin 2 t (sin 2 t + cos 2 t). From the same basic identity, we derive the value of the expression in parentheses, equal to 1. As a result of simplification, we obtain cos 2 t- cos 4 t + sin 4 t = sin 2 t.
Example 2 also needs to simplify the expression cost / (1- sint) + cost / (1+ sint). Since the expression cost is in the numerators of both fractions, it can be bracketed as a common factor. Then the fractions in brackets are reduced to a common denominator by multiplying (1-sint) (1+ sint). After bringing such terms in the numerator remains 2, and in the denominator 1 - sin 2 t. On the right side of the screen, the basic trigonometric identity sin 2 t + cos 2 t = 1 is reminded. 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.
Further, examples of identity proofs are considered, in which the knowledge gained 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. On the right side of the screen, three identities are displayed that will be needed for the proof - tg t · ctg t = 1, ctg t = cos t / sin t and tan t = sin t / cos t with restrictions. To prove the identity, first the parentheses are expanded, 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 the cotangent, ctg 2 t is transformed. As a result of the transformations, the expression 1-cos 2 t is obtained. Using the basic identity, we find the meaning of the expression. Thus, it has been 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 calculate the expression, the right and left sides of the equality (tg t + ctg t) 2 = 6 2 are first squared. The abbreviated multiplication formula resembles 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 reminded 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 for a teacher carrying out distance learning. In order to develop skills in solving trigonometric problems.
TEXT CODE:
"Simplification of trigonometric expressions."
Equality
1) sin 2 t + cos 2 t = 1 (sine squared te plus cosine squared te equals one)
2) tgt =, for t ≠ + πk, kϵZ (the tangent te is equal to the ratio of the sine te to the cosine te when te is not equal to pi by two plus pi ka, ka belongs to zet)
3) ctgt =, for t ≠ πk, kϵZ (the cotangent te is equal to the ratio of the cosine te to the sine te when te is not equal to the peak, ka belongs to zet).
4) tgt ∙ ctgt = 1 for t ≠, kϵZ (the product of the tangent te and the cotangent te is equal to one if te is not equal to the peak, divided by two, ka belongs to z)
are called basic trigonometric identities.
They are often used to simplify and prove trigonometric expressions.
Let's look at examples of using these formulas to simplify trigonometric expressions.
EXAMPLE 1: Simplify the expression: cos 2 t - cos 4 t + sin 4 t. (the expression is a cosine squared te minus the fourth degree cosine te plus the fourth degree sine 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 squared te, in brackets we get the difference between the unit and the square of the cosine te, which is equal by the first identity to the square of the sine te. We get the sum of the sine of the fourth degree te of the product cosine square te and sine square te. in parentheses, in parentheses we get the sum of the squares of the cosine and sine, which by 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 ba is 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 the second unit plus sine te).
(Let's take the common factor cosine te out of the brackets, and in parentheses we bring it to the common denominator, which is the product of one minus sine te and 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 similar ones.
In the denominator, you can apply the formula of reduced multiplication (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 canceling by cosine te, we get the final answer: two divided by cosine te).
Let's consider examples of using these formulas in proving trigonometric expressions.
EXAMPLE 3. Prove the identity (tg 2 t - sin 2 t) ∙ ctg 2 t = sin 2 t (the product of the difference between the squares of the tangent te and sine te and the square of the cotangent te is equal to the square of the sine 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 te and the cotangent te is equal to one. Recall that the cotangent te is equal to the ratio of the cosine te to the sine te, which means that the square of the cotangent is the ratio of the square of the cosine te and the square of the sine te.
After canceling the square te by sine, we get the difference between the unit and the cosine of the square te, which is equal to the sine of the square 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 te and the cotangent 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 sides of the original equality:
(tgt + ctgt) 2 = 6 2 (the square of the sum of the tangent te and the cotangent te is six squared). Recall the formula for abbreviated multiplication: The square of the sum of two quantities is equal to the square of the first plus twice the product of the first by 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 (tangent squared te plus doubled product of tangent te and cotangent te plus cotangent squared te equals thirty-six) ...
Since the product of the tangent te and the cotangent te is equal to one, then tg 2 t + 2 + ctg 2 t = 36 (the sum of the squares of the tangent te and the cotangent te and two is thirty-six),
Lesson 1
Topic: Grade 11 (preparation for the exam)
Simplification of trigonometric expressions.
Solving the simplest trigonometric equations. (2 hours)
Goals:
- To systematize, generalize, expand the knowledge and skills of students associated with the use of trigonometry formulas and the solution of the simplest trigonometric equations.
Equipment for the lesson:
Lesson structure:
- Organizational moment
- Testing on laptops. The discussion of the results.
- Simplifying trigonometric expressions
- Solving the simplest trigonometric equations
- Independent work.
- Lesson summary. Explanation of the home assignment.
1. Organizational moment. (2 minutes.)
The teacher greets the audience, announces the topic of the lesson, reminds them of the task previously given to repeat the trigonometry formulas, and sets up students 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 with a test version.
There can be as many options as you like, I will give an example of one of them:
Option I.
Simplify expressions:
a) basic trigonometric identities
1.sin 2 3y + cos 2 3y + 1;
b) addition formulas
3.sin5x - sin3x;
c) converting the product into a sum
6.2sin8y cozy;
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 see their answers opposite each formula.
The work is instantly checked by the computer. The results are displayed on a large screen for all 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 review, practice and consolidate the application of the basic trigonometry formulas. 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 learners (prepared in advance on a printed basis). The main emphasis is on the formulas of reduction and double angle, 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 of the discussion of the results of the work of the strong group.
Answers appear on the screen, and also, with the help of a video camera, the works of 5 different students are displayed (one task for each).
The weak group sees the condition and method of solution. Discussion and analysis are in progress. 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 to problem B3.
Any trigonometric equation, no matter how we solve it, leads to the simplest one.
When completing the assignment, students should be drawn to the recording of the roots of the equations of special cases and the general form and to the selection of the roots in the last equation.
Solve equations:
Write down the smallest positive root in response.
5. Independent work (10 min.)
The goal is to test the acquired skills, identify problems, errors and ways to eliminate them.
Different-level work is offered at the choice of the student.
Option for "3"
1) Find the value of an expression 
2) Simplify the expression 1 - sin 2 3α - cos 2 3α
3) Solve the equation 
Option for "4"
1) Find the value of an expression
2) Solve the equation 
Write down the smallest positive root in the answer.
Option for "5"
1) Find tgα if 
2) Find the root of the equation 
Write down the smallest positive root in your answer.
6. Lesson summary (5 min.)
The teacher sums up the fact that in the lesson the trigonometric formulas were repeated and fixed, the solution of the simplest trigonometric equations.
Homework assignment (prepared on a printed basis in advance) with spot checks in the next lesson.
Solve equations:
9) 
10) 
Indicate the smallest positive root in your answer.
Session 2
Topic: Grade 11 (preparation for the exam)
Methods for solving trigonometric equations. Selection of roots. (2 hours)
Goals:
- To generalize and systematize knowledge on solving trigonometric equations of various types.
- To promote the development of students' mathematical thinking, 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:
- Organizational moment
- Discussion d / h and samot. works of the last lesson
- Repetition of methods for solving trigonometric equations.
- Solving trigonometric equations
- Selection of roots in trigonometric equations.
- Independent work.
- Lesson summary. Homework.
1. Organizational moment (2 min.)
The teacher greets the audience, announces the topic of the lesson and the work plan.
2. a) Review of homework (5 min.)
The goal is to check execution. One work with the help of a video camera is displayed on the screen, the rest are selectively collected for the teacher's check.
b) Analysis of independent work (3 min.)
The goal is to analyze the mistakes, indicate the ways to overcome them.
On the screen, answers and solutions, students have their work pre-assigned. Analysis is progressing quickly.
3. Repetition of methods for solving trigonometric equations (5 min.)
The goal is to recall the 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 replacement,
- factorization,
- homogeneous equations,
and there are applied methods:
- by the formulas for converting a sum into a product and a product into a sum,
- by the degree reduction formulas,
- universal trigonometric substitution
- introduction of an auxiliary angle,
- multiplication by some trigonometric function.
It should also be remembered 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 the decision of C1 from the exam.
I consider it expedient to solve the equations for each method together with the students.
The student dictates the decision, the teacher writes it down on the tablet, the whole process is displayed on the screen. This will allow you to quickly and efficiently recall the previously covered material.
Solve equations:
1) change of variable 6cos 2 x + 5sinx - 7 = 0
2) factoring 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 power sin2x - sin 2 2x + sin 2 3x = 0.5
7) universal trigonometric substitution sinx + 5cosx + 5 = 0.
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, you need 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 conditions of fierce competition when entering universities, solving 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 the C1 problem from the Unified State Examination in 2011.
There are trigonometric equations in which you need to select roots when writing out an answer. This is due to some restrictions, for example: the denominator of the fraction is not zero, the expression under the even root is non-negative, the expression under the sign of the logarithm is positive, etc.
Such equations are considered equations of increased complexity and in the version of the exam are in the second part, namely C1.
Solve the equation:
The fraction is zero if then 
using the unit circle, we 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, in this case, does not lose its meaning. Then
Using the unit circle, select the roots (see Figure 2)
