Solving trigonometric functions. Basic methods for solving trigonometric equations

Trigonometric equations - the topic is not the simplest. They hurt them diverse.) For example, such:

sIN 2 X + COS3X \u003d CTG5X

sIN (5X + π / 4) \u003d CTG (2x-π / 3)

sINX + COS2X + TG3X \u003d CTG4X

Etc...

But these (and all others) trigonometric monsters have two common and mandatory features. The first - you will not believe - trigonometric functions are present in the equations.) Second: all expressions with XOM are located within these functions. And only there! If Xe appears somewhere outside, eg, sin2x + 3x \u003d 3, This will already be a mixed type equation. Such equations require an individual approach. Here we will not consider them.

Evil equations in this lesson we will not decide.) Here we will deal with simply simple trigonometric equations. Why? Yes, because the decision any Trigonometric equations consists of two stages. At the first stage, the evil equation by the most different transformations comes down to simple. On the second - this is the simplest equation. No other way.

So, if you have problems in the second stage - the first stage does not make sense.)

What do elementary trigonometric equations look like?

sINX \u003d A.

cOSX \u003d A.

tGX \u003d A.

cTGX \u003d A.

Here but Indicates any number. Any.

By the way, inside the function it may not be pure X, but some expression, like:

cOS (3X + π / 3) \u003d 1/2

etc. This complicates life, but does not affect the method of solving the trigonometric equation.

How to solve trigonometric equations?

Trigonometric equations can be solved in two ways. The first way: using logic and trigonometric circle. We will look at this way here. The second way is to use the memory and formulas - consider in the next lesson.

The first way is clear, reliable, and it is difficult to forget.) It is good for solving and trigonometric equations, and inequalities, and all sort of cunning non-standard examples. Logic stronger memory!)

We solve the equations using a trigonometric circle.

We turn on the elementary logic and the ability to use the trigonometric circle. Do not know how!? However ... difficult to you in trigonometry will have ...) But not trouble. Take a look in the lessons "trigonometric circle ...... what is it?" And "counts the corners on the trigonometric circle." Everything is simple there. Unlike textbooks ...)

Oh, you know!? And even mastered the "practical work with a trigonometric circle"!? Give congratulations. This topic will be close and understandable to you.) What is especially pleased, the trigonometric circle is indifferent to what equation you decide. Sinus, Kosinus, Tangent, Kothanns - everything is one. The principle of decision one.

Here we take any elementary trigonometric equation. At least this:

cosx \u003d 0.5

We must find X. If we speak human language, you need find angle (X), whose cosine is 0.5.

How did we previously used the circle? We painted an angle on it. In degrees or radians. And immediately videli Trigonometric functions of this angle. Now we will do on the contrary. Draw a cosine on a circle, equal to 0.5 and immediately see angle. It will only remain to write down the answer.) Yes, yes!

We draw a circle and mark the cosine equal to 0.5. On the axis of cosine, of course. Like this:

Now draw the angle that gives us this cosine. Mouse over the mouse over the picture (or tap pictures on the tablet), and see This very corner x.

Kosinus which angle is 0.5?

x \u003d π / 3

cos. 60 ° \u003d COS ( π / 3.) = 0,5

Someone skeptically chops, yes ... they say whether the circle was worthwhile, when everything is clear ... You can, of course, grind ...) But the fact is that it is an erroneous answer. Rather, insufficient. The connabilities of the circle understand that there is still a whole bunch of angles that also give a cosine equal to 0.5.

If you turn the movable side of OA on full turn, Point A will go to its original position. With the same cosine equal to 0.5. Those. the angle will change 360 ° or 2π radians, and cosine - no. The new angle of 60 ° + 360 ° \u003d 420 ° will also be the solution of our equation, because

Such complete revolutions can be coated with an infinite set ... And all these new corners will be solutions of our trigonometric equation. And they must be written in response to somehow. Everything. Otherwise the decision is not considered yes ...)

Mathematics can do it simple and elegant. In one brief answer write infinite set solutions. This is how it looks for our equation:

x \u003d π / 3 + 2π n, n ∈ Z

Decipher. Still writing meaningless It's nicer than stupidly draw some mysterious beaks, right?)

π / 3. - this is the same angle that we saw On the circle I. defended on the cosine table.

2π. - This is one full turn in radians.

n. - This is the number of complete, i.e. integers revolutions. It is clear that n. It may be equal to 0, ± 1, ± 2, ± 3 .... and so on. As indicated by a brief record:

n ∈ Z.

n. belongs ( ∈ ) set integers ( Z. ). By the way, instead of the letter n. letters can be used k, m, t etc.

This entry means you can take any integer. n. . Though -3, at least 0, though +55. What you want. If you substitute this number to write a response, get a specific angle that will definitely be the solution of our harsh equation.)

Or, in other words, x \u003d π / 3 - This is the only root of an infinite set. To get all the other roots, sufficient to π / 3 add any number of full revolutions ( n. ) in radians. Those. 2π N. radian.

Everything? Not. I am expensive to stretch. To be remembered better.) We received only part of the answers to our equation. This first part of the decision I will write this as:

x 1 \u003d π / 3 + 2π n, n ∈ Z

x 1 - Not one root, this is a whole series of roots recorded in brief form.

But there are still angles that also give cosine equal to 0.5!

Let's return to our picture, on which the answer was recorded. Here she is:

We carry the mouse to the picture and see Another corner that also gives cosine 0.5. What do you think it is equal to? Triangles are the same ... yes! It is equal to the corner h. , only postponed in the negative direction. This corner -H. But the X is already calculated. π / 3 or60 °. Therefore, you can safely write:

x 2 \u003d - π / 3

Well, of course, add all the angles that are obtained through full revs:

x 2 \u003d - π / 3 + 2π n, n ∈ Z

Now everything.) According to the trigonometric circle, we saw (who understands, of course)) everything Corners giving a cosine equal to 0.5. And they recorded these corners in a brief mathematical form. The answer turned out two endless series of roots:

x 1 \u003d π / 3 + 2π n, n ∈ Z

x 2 \u003d - π / 3 + 2π n, n ∈ Z

This is the correct answer.

I hope general principle of solving trigonometric equations Using a circle is understandable. We note on the circle of cosine (sinus, tangent, catangent) from the specified equation, we draw the corresponding angles and write the answer. Of course, you need to figure out what kind of corners we saw On a circle. Sometimes it is not so obvious. Well, I said that logic is required here.)

For example, we will analyze another trigonometric equation:

I ask you to consider that the number 0.5 is not the only possible number in the equations!) I just write it more convenient for it than roots and fractions.

We work according to the general principle. We draw a circle, mark (on the axis of sinuses, of course!) 0.5. Draw all the corners corresponding to this sinus at once. We get this picture:

First we deal with the angle h. in the first quarter. Remember the sinus table and determine the magnitude of this angle. It's a simple thing:

x \u003d π / 6

We remember about full revisions and, with a clean conscience, write down the first series of answers:

x 1 \u003d π / 6 + 2π n, n ∈ Z

Half the case is made. But now it is necessary to determine second angle ... It is a still than in cosine, yes ... but the logic will save us! How to determine the second angle through x? Yes Easy! Triangles on the picture are the same, and red angle h. equal to the corner h. . It is only counted from the angle π in the negative direction. Therefore, red.) And we need an angle, counted correctly, from the positive semi-axis, i.e. From an angle of 0 degrees.

We bring the cursor to the drawing and see everything. The first corner I removed not to complicate the picture. The angle of interest to us (painted green) will be equal to:

π - H.

Ix we know this π / 6. . It became, the second corner will be:

π - π / 6 \u003d 5π / 6

We remember again about the addition of full revolutions and write the second series of answers:

x 2 \u003d 5π / 6 + 2π n, n ∈ Z

That's all. A full-fledged response consists of two series of roots:

x 1 \u003d π / 6 + 2π n, n ∈ Z

x 2 \u003d 5π / 6 + 2π n, n ∈ Z

Equations with Tangent and Kotangent can be easily solved by the general principle of solving trigonometric equations. If, of course, you know how to draw Tangent and Cotangenes on a trigonometric circle.

In the examples above, I used the table value of the sine and cosine: 0.5. Those. one of those values \u200b\u200bthat the student know must. And now we will expand our opportunities for all other values. Decide so solve!)

So, let us need to solve such a trigonometric equation:

There is no such value of the cosine in short tables. Coldly ignore this terrible fact. We draw a circle, mark on the axis of cosine 2/3 and draw the appropriate angles. We get this picture.

We understand, to begin with, with an angle in the first quarter. To know what is equal to X, I would immediately write the answer! We do not know ... Failure!? Tranquility! Mathematics of their own in trouble is not thrown! She invented Arkkosinus in this case. Do not know? In vain. Find out, it's much easier than you think. Under this link, not a single surround spell about "reverse trigonometric functions" there is no ... is unnecessary in this topic.

If you know, it is enough to say to yourself: "X is an angle whose cosine is 2/3." And immediately, purely by definition of Arkkosinus, you can write:

We remember about additional revs and calmly write down the first series of roots of our trigonometric equation:

x 1 \u003d Arccos 2/3 + 2π n, n ∈ Z

The second series of roots, for the second angle, is written almost automatic. All the same, only X (ArcCOS 2/3) will be with a minus:

x 2 \u003d - Arccos 2/3 + 2π n, n ∈ Z

And all things! This is the correct answer. Even easier than with tabular values. I don't need to remember anything.) By the way, the most attentive notice that this picture with the decision through the Arkkosinus no, in essence, does not differ from the picture for the Cosx \u003d 0.5 equation.

Exactly! General principle on the general! I specifically painted two almost the same pictures. Circle Shows Corner h. by his cosine. Tablet is a cosine, or not - the circle is unknown. What is this angle, π / 3, or the arcsinus, what we can decide.

Sinus the same song. For example:

We again draw a circle, mark the sinus equal to 1/3, draw the corners. It turns out this picture:

And again the picture is almost the same as for the equation sINX \u003d 0.5. Night start from the corner in the first quarter. What is ix, if its sinus is 1/3? No problem!

Here is the first pack of roots:

x 1 \u003d arcsin 1/3 + 2π n, n ∈ Z

We understand with the second angle. In the example with a table value of 0.5, it was equal to:

π - H.

So here he will be exactly the same! Only X is another, Arcsin 1/3. So what!? You can safely write the second pack of the roots:

x 2 \u003d π - arcsin 1/3 + 2π n, n ∈ Z

This is absolutely the right answer. Although it looks not very familiar. But it is clear, I hope.)

This is how trigonometric equations are solved with a circle. This path is visual and understand. It is he who saves in trigonometric equations with the selection of roots at a given interval, in trigonometric inequalities - those generally solved almost always in a circle. In short, in any tasks that are slightly complicated by standard.

Apply knowledge in practice?)

Solve trigonometric equations:

First, easier, right on this lesson.

Now more comprehensive.

Tip: Here you have to reflect on the circle. Personally.)

And now outwardly simple ... they are still called private cases.

sINX. = 0

sINX. = 1

cOSX. = 0

cOSX. = -1

Tip: Here it is necessary to figure out in a circle, where two series of responses, and where one ... and how instead of two episodes of answers to write one. Yes, so that no root from the infinite amount is lost!)

Well, quite simple):

sINX. = 0,3

cOSX. = π

tGX. = 1,2

cTGX = 3,7

Tip: Here you need to know what Arksinus is, Arkkosinus? What is Arctangent, Arkkothangence? The simplest definitions. But remember any table values \u200b\u200bnot necessary!)

Answers, of course, in disarray):

x 1 \u003d arcsin0.3 + 2π n, n ∈ Z
x 2 \u003d π - arcsin0,3 + 2

Not everything works? It happens. Read the lesson again. Only thoughtful (There is such a obsolete word ...) and click on the links. The main links are about the circle. Without him in trigonometry - how the road to move with blindfolded eyes. Sometimes it turns out.)

If you like this site ...

By the way, I have another couple of interesting sites for you.)

It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)

You can get acquainted with features and derivatives.

When solving many mathematical tasksEspecially those encountered up to 10 class, the procedure for actions performed, which will lead to the goal, is definitely defined. Such objectives include, for example, linear and square equations, linear and square inequalities, fractional equations and equations that are reduced to square. The principle of successful solution of each of the mentioned tasks is as follows: It is necessary to establish how the type is the solved task relates, to recall the necessary sequence of actions that will lead to the desired result, i.e. Answer, and perform these actions.

It is obvious that the success or failure in solving one or another task depends mainly on how correctly the type of equation is defined how correctly the sequence of all stages of its solution is reproduced. Of course, it is necessary to own the skills of performing identical transformations and calculations.

Other situation is obtained with trigonometric equations. Establish the fact that the equation is trigonometric, absolutely not difficult. Difficulties appear when determining the sequence of actions that would led to the correct answer.

According to the appearance of the equation, sometimes it is difficult to determine its type. And not knowing the type of equation, it is almost impossible to choose from several dozen trigonometric formulas necessary.

To solve the trigonometric equation, you must try:

1. Create all functions included in the equation to the "same corners";
2. Create an equation to "identical functions";
3. Lay the left part of the factory equation, etc.

Consider basic methods for solving trigonometric equations.

I. Bringing to the simplest trigonometric equations

Schematic solution

Step 1. Express trigonometric function through well-known components.

Step 2. Find an argument function by formulas:

cos x \u003d a; x \u003d ± Arccos a + 2πn, n єz.

sin x \u003d a; x \u003d (-1) n arcsin a + πn, n є z.

tG X \u003d A; x \u003d arctg a + πn, n є z.

cTG X \u003d A; x \u003d arcctg a + πn, n є z.

Step 3. Find an unknown variable.

Example.

2 cos (3x - π / 4) \u003d -√2.

Decision.

1) COS (3X - π / 4) \u003d -√2 / 2.

2) 3x - π / 4 \u003d ± (π - π / 4) + 2πn, n є z;

3x - π / 4 \u003d ± 3π / 4 + 2πn, N є Z.

3) 3x \u003d ± 3π / 4 + π / 4 + 2πn, n є z;

x \u003d ± 3π / 12 + π / 12 + 2πn / 3, n є z;

x \u003d ± π / 4 + π / 12 + 2πn / 3, n є z.

Answer: ± π / 4 + π / 12 + 2πn / 3, n є z.

II. Replacing the variable

Schematic solution

Step 1. Create an equation to algebraic form relative to one of the trigonometric functions.

Step 2. Designate the resulting function of the variable T (if necessary, enter the restrictions on T).

Step 3. Record and solve the resulting algebraic equation.

Step 4. Make a replacement.

Step 5. Solve the simplest trigonometric equation.

Example.

2COS 2 (X / 2) - 5Sin (X / 2) - 5 \u003d 0.

Decision.

1) 2 (1 - sin 2 (x / 2)) - 5Sin (X / 2) - 5 \u003d 0;

2Sin 2 (X / 2) + 5Sin (X / 2) + 3 \u003d 0.

2) Let sin (x / 2) \u003d T, where | T | ≤ 1.

3) 2t 2 + 5t + 3 \u003d 0;

t \u003d 1 or E \u003d -3/2, does not satisfy the condition | T | ≤ 1.

4) sin (x / 2) \u003d 1.

5) x / 2 \u003d π / 2 + 2πn, n є z;

x \u003d π + 4πn, n є z.

Answer: x \u003d π + 4πn, n є z.

III. The method of lowering the order of the equation

Schematic solution

Step 1. Replace this linear equation using a degree reduction formula for this:

sin 2 x \u003d 1/2 · (1 - COS 2x);

cos 2 x \u003d 1/2 · (1 + cos 2x);

tG 2 X \u003d (1 - COS 2X) / (1 + COS 2x).

Step 2. Solve the obtained equation using methods I and II.

Example.

cOS 2X + COS 2 X \u003d 5/4.

Decision.

1) COS 2X + 1/2 · (1 + COS 2X) \u003d 5/4.

2) COS 2X + 1/2 + 1/2 · COS 2X \u003d 5/4;

3/2 · cos 2x \u003d 3/4;

2x \u003d ± π / 3 + 2πn, n є z;

x \u003d ± π / 6 + πn, n є z.

Answer: x \u003d ± π / 6 + πn, n є z.

IV. Uniform equations

Schematic solution

Step 1. Bring this equation to the form

a) a sin x + b cos x \u003d 0 (homogeneous equation of the first degree)

or to sight

b) a Sin 2 x + b sin x · cos x + c cos 2 x \u003d 0 (homogeneous equation of the second degree).

Step 2. Split both parts of the equation on

a) cos x ≠ 0;

b) COS 2 x ≠ 0;

and get the equation relative to TG X:

a) a TG X + B \u003d 0;

b) a TG 2 x + B arctg x + c \u003d 0.

Step 3. Solve equation by known methods.

Example.

5Sin 2 x + 3sin x · COS X - 4 \u003d 0.

Decision.

1) 5Sin 2 x + 3sin x · COS X - 4 (SIN 2 x + COS 2 x) \u003d 0;

5Sin 2 x + 3sin x · COS X - 4SIN² x - 4cos 2 x \u003d 0;

sIN 2 X + 3SIN X · COS X - 4COS 2 x \u003d 0 / COS 2 x ≠ 0.

2) TG 2 X + 3TG X - 4 \u003d 0.

3) Let TG x \u003d T, then

t 2 + 3T - 4 \u003d 0;

t \u003d 1 or t \u003d -4, then

tG x \u003d 1 or TG x \u003d -4.

From the first equation x \u003d π / 4 + πn, n є z; From the second equation x \u003d -arctg 4 + πk, k є z.

Answer: x \u003d π / 4 + πn, n є z; x \u003d -arctg 4 + πk, k є z.

V. Method of converting an equation using trigonometric formulas

Schematic solution

Step 1. Using all sorts of trigonometric formulas, lead this equation to the equation, solved methods I, II, III, IV.

Step 2. Solve the resulting equation known methods.

Example.

sIN X + SIN 2X + SIN 3X \u003d 0.

Decision.

1) (SIN X + SIN 3X) + SIN 2X \u003d 0;

2Sin 2X · COS X + SIN 2X \u003d 0.

2) sIN 2X · (2cos x + 1) \u003d 0;

sin 2x \u003d 0 or 2cos x + 1 \u003d 0;

From the first equation 2x \u003d π / 2 + πn, n є z; From the second equation COS X \u003d -1/2.

We have x \u003d π / 4 + πn / 2, n є z; From the second equation x \u003d ± (π - π / 3) + 2πk, k є z.

As a result, x \u003d π / 4 + πn / 2, n є z; x \u003d ± 2π / 3 + 2πk, k є z.

Answer: x \u003d π / 4 + πn / 2, n є z; x \u003d ± 2π / 3 + 2πk, k є z.

Skills and skills to solve trigonometric equations are very important, their development requires considerable efforts, both by the student and from the teacher.

With the solution of trigonometric equations, many challenges of stereometry, physics, and others are associated with the process of solving such tasks, as it were, concludes many knowledge and skills, which are purchased in the study of elements of trigonometry.

Trigonometric equations occupy an important place in the process of learning mathematics and personality development as a whole.

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The lesson for the integrated application of knowledge.

Objectives lesson.

1. Consider various methods for solving trigonometric equations.
2. Development of student's creative abilities by solving equations.
3. The prompting of students to self-control, interconnected, the self-analysis of its study activities.

Equipment: screen, projector, reference material.

During the classes

Introductory conversation.

The main method of solving trigonometric equations is the information of their simplest. In this case, ordinary methods are used, for example, decomposition of multipliers, as well as techniques used only to solve trigonometric equations. These techniques are quite a lot, for example, various trigonometric substitutions, conversion of angles, transforming trigonometric functions. The disorderly application of any trigonometric transformations usually does not simplify the equation, and it makes it difficult to disastrously. In order to work out in general terms, the equation solutions plan, outline the path of the equation to the simplest, must first of all analyze the angles - the arguments of the trigonometric functions included in the equation.

Today we will talk about the methods of solving trigonometric equations. The correctly selected method often allows you to significantly simplify the solution, so all methods we have learned always need to keep their attention in the zone to solve trigonometric equations the most suitable method.

II. (With the help of the projector, we repeat the methods of solving equations.)

1. The method of bringing the trigonometric equation to algebraic.

It is necessary to express all trigonometric functions through one, with the same argument. This can be done with the help of the main trigonometric identity and its consequences. We obtain equation with one trigonometric function. Having accepted it for a new unknown, we obtain an algebraic equation. We find it roots and return to the old unknown, solving the simplest trigonometric equations.

2. The method of decomposition on multipliers.

To change the corners, the formulas, sums and differences of arguments, as well as the formula for the transformation of the amount (difference) of trigonometric functions in the work and vice versa, are often useful.

sIN X + SIN 3X \u003d SIN 2X + SIN4X

3. Method of introducing an additional angle.

4. Method of using a universal substitution.

Equations of the form F (SINX, COSX, TGX) \u003d 0 are reduced to algebraic with the help of a universal trigonometric substitution

Expressing sinus, cosine and tangent through a half angle tangent. This technique can lead to high-order equation. Whose solution is difficult.

Concept of solving trigonometric equations.

• To solve the trigonometric equation, convert it into one or more of the main trigonometric equations. The solution of the trigonometric equation is ultimately reduced to solving four main trigonometric equations.

Solution of the main trigonometric equations.

• There are 4 types of basic trigonometric equations:
• sin x \u003d a; COS X \u003d A
• tG X \u003d A; CTG X \u003d A
• The solution of the main trigonometric equations implies consideration of various provisions "x" on a single circle, as well as using the conversion table (or calculator).
• Example 1. SIN X \u003d 0.866. Using the conversion table (or calculator), you will receive an answer: x \u003d π / 3. A single circle gives another answer: 2π / 3. Remember: All trigonometric functions are periodic, that is, their values \u200b\u200bare repeated. For example, the frequency of SIN X and COS X is 2πn, and the frequency of TG X and CTG x is equal to πn. Therefore, the answer is written as follows:
• x1 \u003d π / 3 + 2πn; x2 \u003d 2π / 3 + 2πn.
• Example 2. COs x \u003d -1/2. Using the conversion table (or calculator), you will receive the answer: x \u003d 2π / 3. A single circle gives another answer: -2π / 3.
• x1 \u003d 2π / 3 + 2π; x2 \u003d -2π / 3 + 2π.
• Example 3. TG (X - π / 4) \u003d 0.
• Answer: x \u003d π / 4 + πn.
• Example 4. CTG 2X \u003d 1.732.
• Answer: x \u003d π / 12 + πn.

Transformation used in solving trigonometric equations.

• To transform trigonometric equations, algebraic transformations are used (decomposition of multipliers, bringing homogeneous members, etc.) and trigonometric identities.
• Example 5. Using trigonometric identities, the equation SIN X + SIN 2X + SIN 3X \u003d 0 is converted to 4COS X * SIN equation (3X / 2) * COS (X / 2) \u003d 0. Thus, the following main trigonometric equations should be solved: cos x \u003d 0; sin (3x / 2) \u003d 0; COS (X / 2) \u003d 0.

• Finding angles according to known values \u200b\u200bof functions.

  • Before studying the methods of solving trigonometric equations, you need to learn how to find corners according to known values \u200b\u200bof functions. This can be done using the conversion or calculator table.
  • Example: COs x \u003d 0.732. Calculator will give an answer x \u003d 42.95 degrees. A single circle will give additional angles whose cosine is also equal to 0.732.

• Postulate the decision on a single circle.

  • You can postpone the solutions of the trigonometric equation on a single circle. The solutions of the trigonometric equation on a single circle are vertices of the correct polygon.
  • Example: Solutions X \u003d π / 3 + πN / 2 on a single circle are the vertices of the square.
  • Example: Solutions X \u003d π / 4 + πN / 3 on a single circle are the vertices of the correct hexagon.

• Methods for solving trigonometric equations.

  • If this trigonometric equation contains only one trigonometric function, decide this equation as a basic trigonometric equation. If this equation includes two or more trigonometric functions, then there are 2 methods for solving such an equation (depending on the possibility of its transformation).
    • Method 1.
  • Convert this equation to the equation of the form: F (x) * G (x) * h (x) \u003d 0, where f (x), G (x), h (x) is the main trigonometric equations.
  • Example 6. 2COS X + SIN 2X \u003d 0. (0< x < 2π)
  • Decision. Using the formula of a double angle SIN 2X \u003d 2 * SIN X * COs X, replace SIN 2x.
  • 2SOs x + 2 * sin x * COs x \u003d 2cos x * (sin x + 1) \u003d 0. Now decide two main trigonometric equations: COs x \u003d 0 and (Sin x + 1) \u003d 0.
  • Example 7. COS X + COS 2X + COS 3X \u003d 0. (0< x < 2π)
  • Solution: Using trigonometric identities, convert this equation to the equation of the form: COS 2X (2cos x + 1) \u003d 0. Now decide the two main trigonometric equations: COS 2X \u003d 0 and (2COS X + 1) \u003d 0.
  • Example 8. SIN X - SIN 3X \u003d COS 2X. (0.< x < 2π)
  • Solution: Using trigonometric identities, convert this equation to the equation of the form: -cos 2x * (2sin x + 1) \u003d 0. Now decide the two main trigonometric equations: COS 2X \u003d 0 and (2Sin x + 1) \u003d 0.
    • Method 2.
  • Convert this trigonometric equation to an equation containing only one trigonometric function. Then replace this trigonometric function to some unknown, for example, t (sin x \u003d t; cos x \u003d t; cos 2x \u003d T, TG x \u003d T; Tg (x / 2) \u003d t, etc.).
  • Example 9. 3Sin ^ 2 x - 2cos ^ 2 x \u003d 4sin x + 7 (0< x < 2π).
  • Decision. In this equation, replace (COS ^ 2 x) on (1 - sin ^ 2 x) (according to the identity). The transformed equation is:
  • 3Sin ^ 2 x - 2 + 2sin ^ 2 x - 4sin x - 7 \u003d 0. Replace SIN x on t. Now the equation looks like: 5t ^ 2 - 4t - 9 \u003d 0. This is a square equation having two roots: T1 \u003d -1 and T2 \u003d 9/5. The second root T2 does not satisfy the values \u200b\u200bof the function values \u200b\u200b(-1< sin x < 1). Теперь решите: t = sin х = -1; х = 3π/2.
  • Example 10. TG X + 2 TG ^ 2 x \u003d CTG X + 2
  • Decision. Replace TG X on T. Relieve the initial equation in the following form: (2t + 1) (T ^ 2 - 1) \u003d 0. Now find T, and then find x for t \u003d tg x.

Compliance with your privacy is important to us. For this reason, we have developed a privacy policy that describes how we use and store your information. Please read our privacy policy and inform us if you have any questions.

Collection and use of personal information

Under personal information is subject to data that can be used to identify a certain person or communicating with it.

You can be requested to provide your personal information at any time when you connect with us.

Below are some examples of the types of personal information that we can collect, and how we can use such information.

What personal information we collect:

• When you leave an application on the site, we can collect various information, including your name, phone number, email address, etc.

As we use your personal information:

• We collected personal information allows us to contact you and report on unique proposals, promotions and other events and nearest events.
• From time to time, we can use your personal information to send important notifications and messages.
• We can also use personalized information for internal purposes, such as auditing, data analysis and various studies in order to improve the services of our services and providing you with recommendations for our services.
• If you participate in the prizes, competition or similar stimulating event, we can use the information you provide to manage such programs.

Information disclosure to third parties

We do not reveal the information received from you to third parties.

Exceptions:

• If it is necessary - in accordance with the law, judicial procedure, in the trial, and / or on the basis of public queries or requests from state bodies in the territory of the Russian Federation - to reveal your personal information. We can also disclose information about you if we define that such disclosure is necessary or appropriate for the purpose of security, maintaining law and order, or other socially important cases.
• In the case of reorganization, mergers or sales, we can convey the personal information we collect the corresponding to the third party - a successor.

Protection of personal information

We are making precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and unscrupulous use, as well as from unauthorized access, disclosure, changes and destruction.

Compliance with your privacy at the company level

In order to make sure that your personal information is safe, we bring the norm of confidentiality and security to our employees, and strictly follow the execution of confidentiality measures.