Trigonometric formulas how to solve. Protection of personal information

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Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

The following are some examples of the types of personal information we may collect and how we may use such information.

What personal information we collect:

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How we use your personal information:

The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
From time to time, we may use your personal information to send you important notices and messages.
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Disclosure to third parties

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In the event that it is necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from state bodies in the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest purposes.
In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.

Protection of personal information

Maintaining your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.

It is no secret that success or failure in the process of solving almost any problem mainly depends on the correctness of determining the type of a given equation, as well as on the correctness of reproducing the sequence of all stages of its solution. However, in the case of trigonometric equations, it is not at all difficult to determine the fact that the equation is trigonometric. But in the process of determining the sequence of actions that should lead us to the correct answer, we may encounter certain difficulties. Let's figure out how to solve trigonometric equations correctly from the very beginning.

Solving trigonometric equations

In order to solve the trigonometric equation, you need to try to perform the following points:

We bring all the functions that are included in our equation to "the same angles";
It is necessary to bring the given equation to "identical functions";
We decompose the left side of the given equation into factors or other necessary components.

Methods

Method 1. It is necessary to solve such equations in two stages. First, we transform the equation in order to obtain its simplest (simplified) form. Equation: Cosx = a, Sinx = a and the like are called the simplest trigonometric equations. The second step is to solve the resulting simple equation. It should be noted that the simplest equation can be solved by the algebraic method, which is well known to us from the school algebra course. It is also called the substitution and variable substitution method. With the help of reduction formulas, you first need to convert, then make a replacement and then find the roots.

Next, you need to decompose our equation into possible factors, for this you need to move all the terms to the left and then you can decompose into factors. Now you need to bring this equation to a homogeneous one, in which all terms are equal to the same degree, and the cosine and sine have the same angle.

Before solving trigonometric equations, you need to transfer its terms to the left side, taking them from the right side, and then we take out all the common denominators in brackets. We equate our brackets and factors to zero. Our equated brackets are a reduced degree homogeneous equation to be divided by sin(cos) to the highest power. Now we solve the algebraic equation that was obtained in relation to tan.

Method 2. Another method by which you can solve the trigonometric equation is the transition to a half angle. For example, we solve the equation: 3sinx-5cosx=7.

We need to go to half angle, in our case it is: 6sin(x/2)*cos(x/2)- 5cos²(x/2)+5sin²(x/2) = 7sin²(x/2)+7cos²(x / 2). And after that, we reduce all the terms into one part (for convenience, it is better to choose the right one) and proceed to solve the equation.

If necessary, you can enter an auxiliary angle. This is done when you need to replace the integer value sin (a) or cos (a) and the sign “a” just acts as an auxiliary angle.

product to sum

How to solve trigonometric equations using sum product? The method known as product-to-sum conversion can also be used to solve such equations. In this case, it is necessary to use the formulas corresponding to the equation.

For example, we have an equation: 2sinx * sin3x= cos4x

We need to solve this problem by converting the left side into a sum, namely:

cos 4x –cos8x=cos4x ,

x = p/16 + pk/8.

If the above methods are not suitable, and you still do not know how to solve the simplest trigonometric equations, you can use another method - universal substitution. With it, you can transform the expression and make a replacement. For example: Cos(x/2)=u. Now we can solve the equation with the given parameter u. And having received the desired result, do not forget to translate this value into the opposite.

Many "experienced" students are advised to turn to people online for solving equations. How to solve a trigonometric equation online, you ask. To solve the problem online, you can turn to the forums of the relevant topics, where you can be helped with advice or in solving the problem. But the best thing is to try to manage on your own.

Skills and abilities in solving trigonometric equations are very important and useful. Their development will require a lot of effort from you. Many problems in physics, stereometry, etc. are associated with the solution of such equations. And the very process of solving such problems implies the presence of skills and knowledge that can be acquired while studying the elements of trigonometry.

Learn trigonometric formulas

In the process of solving an equation, you may encounter the need to use any formula from trigonometry. You can, of course, start looking for it in your textbooks and cheat sheets. And if these formulas are put in your head, you will not only save your nerves, but also make your task much easier, without wasting time searching for the necessary information. Thus, you will have the opportunity to think through the most rational way to solve the problem.

Trigonometric equations are not the easiest topic. Painfully they are diverse.) For example, these:

sin2x + cos3x = ctg5x

sin(5x+π /4) = ctg(2x-π /3)

sinx + cos2x + tg3x = ctg4x

Etc...

But these (and all other) trigonometric monsters have two common and obligatory features. First - you won't believe it - there are trigonometric functions in the equations.) Second: all expressions with x are within these same functions. And only there! If x appears somewhere outside, For example, sin2x + 3x = 3, this will be a mixed type equation. Such equations require an individual approach. Here we will not consider them.

We will not solve evil equations in this lesson either.) Here we will deal with the simplest trigonometric equations. Why? Yes, because the decision any trigonometric equations consists of two stages. At the first stage, the evil equation is reduced to a simple one by various transformations. On the second - this simplest equation is solved. No other way.

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

What do elementary trigonometric equations look like?

sinx = a

cosx = a

tgx = a

ctgx = a

Here a stands for any number. Any.

By the way, inside the function there may be not a pure x, but some kind of expression, such as:

cos(3x+π /3) = 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 a trigonometric circle. We will explore this path here. The second way - using memory and formulas - will be considered in the next lesson.

The first way is clear, reliable, and hard to forget.) It is good for solving trigonometric equations, inequalities, and all sorts of tricky non-standard examples. Logic is stronger than memory!

We solve equations using a trigonometric circle.

We include elementary logic and the ability to use a trigonometric circle. Can't you!? However... It will be difficult for you in trigonometry...) But it doesn't matter. Take a look at the lessons "Trigonometric circle ...... What is it?" and "Counting angles on a trigonometric circle." Everything is simple there. Unlike textbooks...)

Ah, you know!? And even mastered "Practical work with a trigonometric circle"!? Accept congratulations. This topic will be close and understandable to you.) What is especially pleasing is that the trigonometric circle does not care which equation you solve. Sine, cosine, tangent, cotangent - everything is the same for him. The solution principle is the same.

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

cosx = 0.5

I need to find X. Speaking in human language, you need find the angle (x) whose cosine is 0.5.

How did we use the circle before? We drew a corner on it. In degrees or radians. And immediately seen trigonometric functions of this angle. Now let's do the opposite. Draw a cosine equal to 0.5 on the circle and immediately we'll see injection. It remains only to write down the answer.) Yes, yes!

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

Now let's draw the angle that this cosine gives us. Hover your mouse over the picture (or touch the picture on a tablet), and see this same corner X.

Which angle has a cosine of 0.5?

x \u003d π / 3

cos 60°= cos( π /3) = 0,5

Some people will grunt skeptically, yes... They say, was it worth it to fence the circle, when everything is clear anyway... You can, of course, grunt...) But the fact is that this is an erroneous answer. Or rather, inadequate. Connoisseurs of the circle understand that there are still a whole bunch of angles that also give a cosine equal to 0.5.

If you turn the movable side OA for a full turn, point A will return to its original position. With the same cosine equal to 0.5. Those. the angle will change 360° or 2π radians, and cosine is not. The new angle 60° + 360° = 420° will also be a solution to our equation, because

There are an infinite number of such full rotations... And all these new angles will be solutions to our trigonometric equation. And they all need to be written down somehow. All. Otherwise, the decision is not considered, yes ...)

Mathematics can do this simply and elegantly. In one short answer, write down infinite set solutions. Here's what it looks like for our equation:

x = π /3 + 2π n, n ∈ Z

I will decipher. Still write meaningfully nicer than stupidly drawing some mysterious letters, right?)

π /3 is the same angle that we saw on the circle and determined according to the table of cosines.

2π is one full turn in radians.

n - this is the number of complete, i.e. whole revolutions. It is clear that n can be 0, ±1, ±2, ±3.... and so on. As indicated by the short entry:

n ∈ Z

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

This notation means that you can take any integer n . At least -3, at least 0, at least +55. What do you want. If you plug that number into your answer, you get a specific angle, which is sure to be the solution to our harsh equation.)

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

Everything? No. I specifically stretch the pleasure. To remember better.) We received only a part of the answers to our equation. I will write this first part of the solution as follows:

x 1 = π /3 + 2π n, n ∈ Z

x 1 - not one root, it is a whole series of roots, written in short form.

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

Let's return to our picture, according to which we wrote down the answer. Here she is:

Move the mouse over the image and see another corner that also gives a cosine of 0.5. What do you think it equals? The triangles are the same... Yes! It is equal to the angle X , only plotted in the negative direction. This is the corner -X. But we have already calculated x. π /3 or 60°. Therefore, we can safely write:

x 2 \u003d - π / 3

And, of course, we add all the angles that are obtained through full turns:

x 2 = - π /3 + 2π n, n ∈ Z

That's all now.) In a trigonometric circle, we saw(who understands, of course)) all angles that give a cosine equal to 0.5. And they wrote down these angles in a short mathematical form. The answer is two infinite series of roots:

x 1 = π /3 + 2π n, n ∈ Z

x 2 = - π /3 + 2π n, n ∈ Z

This is the correct answer.

Hope, general principle for solving trigonometric equations with the help of a circle is understandable. We mark the cosine (sine, tangent, cotangent) from the given equation on the circle, draw the corresponding angles and write down the answer. Of course, you need to figure out what kind of corners we are saw on the circle. Sometimes it's not so obvious. Well, as I said, logic is required here.)

For example, let's analyze another trigonometric equation:

Please note that the number 0.5 is not the only possible number in the equations!) It's just more convenient for me to write it than roots and fractions.

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

Let's deal with the angle first. X in the first quarter. We recall the table of sines and determine the value of this angle. The matter is simple:

x \u003d π / 6

We recall full turns and, with a clear conscience, write down the first series of answers:

x 1 = π /6 + 2π n, n ∈ Z

Half the job is done. Now we need to define second corner... This is trickier than in cosines, yes ... But logic will save us! How to determine the second angle through x? Yes Easy! The triangles in the picture are the same, and the red corner X equal to the angle X . Only it is counted from the angle π in the negative direction. That's why it's red.) And for the answer, we need an angle measured correctly from the positive semiaxis OX, i.e. from an angle of 0 degrees.

Hover the cursor over the picture and see everything. I removed the first corner so as not to complicate the picture. The angle of interest to us (drawn in green) will be equal to:

π - x

x we know it π /6 . So the second angle will be:

π - π /6 = 5π /6

Again, we recall the addition of full revolutions and write down the second series of answers:

x 2 = 5π /6 + 2π n, n ∈ Z

That's all. A complete answer consists of two series of roots:

x 1 = π /6 + 2π n, n ∈ Z

x 2 = 5π /6 + 2π n, n ∈ Z

Equations with tangent and cotangent can be easily solved using the same general principle for solving trigonometric equations. Unless, of course, you know how to draw the tangent and cotangent on a trigonometric circle.

In the examples above, I used the tabular value of sine and cosine: 0.5. Those. one of those meanings that the student knows must. Now let's expand our capabilities to all other values. Decide, so decide!)

So, let's say we need to solve the following trigonometric equation:

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

We understand, for starters, with an angle in the first quarter. To know what x is equal to, they would immediately write down the answer! We don't know... Failure!? Calm! Mathematics does not leave its own in trouble! She invented arc cosines for this case. Do not know? In vain. Find out. It's a lot easier than you think. According to this link, there is not a single tricky spell about "inverse trigonometric functions" ... It's superfluous in this topic.

If you're in the know, just say to yourself, "X is an angle whose cosine is 2/3." And immediately, purely by definition of the arccosine, we can write:

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

x 1 = arccos 2/3 + 2π n, n ∈ Z

The second series of roots is also written almost automatically, for the second angle. Everything is the same, only x (arccos 2/3) will be with a minus:

x 2 = - arccos 2/3 + 2π n, n ∈ Z

And all things! This is the correct answer. Even easier than with tabular values. You don’t need to remember anything.) By the way, the most attentive will notice that this picture with the solution through the arc cosine is essentially no different from the picture for the equation cosx = 0.5.

Exactly! The general principle on that and the general! I specifically drew two almost identical pictures. The circle shows us the angle X by its cosine. It is a tabular cosine, or not - the circle does not know. What kind of angle is this, π / 3, or what kind of arc cosine is up to us to decide.

With a sine the same song. For example:

Again we draw a circle, mark the sine 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 = 0.5. Again we start from the corner in the first quarter. What is x equal to if its sine is 1/3? No problem!

So the first pack of roots is ready:

x 1 = arcsin 1/3 + 2π n, n ∈ Z

Let's take a look at the second angle. In the example with a table value of 0.5, it was equal to:

π - x

So here it will be exactly the same! Only x is different, arcsin 1/3. So what!? You can safely write the second pack of roots:

x 2 = π - arcsin 1/3 + 2π n, n ∈ Z

This is a completely correct answer. Although it does not look very familiar. But it's understandable, I hope.)

This is how trigonometric equations are solved using a circle. This path is clear and understandable. It is he who saves in trigonometric equations with the selection of roots on a given interval, in trigonometric inequalities - they are generally solved almost always in a circle. In short, in any tasks that are a little more complicated than standard ones.

Putting knowledge into practice?

Solve trigonometric equations:

At first it is simpler, directly on this lesson.

Now it's more difficult.

Hint: here you have to think about the circle. Personally.)

And now outwardly unpretentious ... They are also called special cases.

sinx = 0

sinx = 1

cosx = 0

cosx = -1

Hint: here you need to figure out in a circle where there are two series of answers, and where there is one ... And how to write down one instead of two series of answers. Yes, so that not a single root from an infinite number is lost!)

Well, quite simple):

sinx = 0,3

cosx = π

tgx = 1,2

ctgx = 3,7

Hint: here you need to know what is the arcsine, arccosine? What is arc tangent, arc tangent? The simplest definitions. But you don’t need to remember any tabular values!)

The answers are, of course, in disarray):

x 1= arcsin0,3 + 2πn, n ∈ Z
x 2= π - arcsin0.3 + 2

Not everything works out? It happens. Read the lesson again. Only thoughtfully(there is such an obsolete word...) And follow the links. The main links are about the circle. Without it in trigonometry - how to cross the road blindfolded. Sometimes it works.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

The concept of solving trigonometric equations.

To solve a trigonometric equation, convert it to one or more basic trigonometric equations. Solving the trigonometric equation ultimately comes down to solving the four basic trigonometric equations.

Solution of basic trigonometric equations.

There are 4 types of basic trigonometric equations:
sin x = a; cos x = a
tan x = a; ctg x = a
Solving basic trigonometric equations involves looking at the different x positions on the unit circle, as well as using a conversion table (or calculator).
Example 1. sin x = 0.866. Using a conversion table (or calculator), you get the answer: x = π/3. The unit circle gives another answer: 2π/3. Remember: all trigonometric functions are periodic, that is, their values are repeated. For example, the periodicity of sin x and cos x is 2πn, and the periodicity of tg x and ctg x is πn. So the answer is written like this:
x1 = π/3 + 2πn; x2 = 2π/3 + 2πn.
Example 2 cos x = -1/2. Using a conversion table (or calculator), you get the answer: x = 2π/3. The unit circle gives another answer: -2π/3.
x1 = 2π/3 + 2π; x2 = -2π/3 + 2π.
Example 3. tg (x - π/4) = 0.
Answer: x \u003d π / 4 + πn.
Example 4. ctg 2x = 1.732.
Answer: x \u003d π / 12 + πn.

Transformations used in solving trigonometric equations.

To transform trigonometric equations, algebraic transformations (factorization, reduction of homogeneous terms, etc.) and trigonometric identities are used.
Example 5. Using trigonometric identities, the equation sin x + sin 2x + sin 3x = 0 is converted to the equation 4cos x*sin (3x/2)*cos (x/2) = 0. Thus, the following basic trigonometric equations need to be solved: cos x = 0; sin(3x/2) = 0; cos(x/2) = 0.
Finding angles from known values of functions.
- Before learning how to solve trigonometric equations, you need to learn how to find angles from known values of functions. This can be done using a conversion table or calculator.
- Example: cos x = 0.732. The calculator will give the answer x = 42.95 degrees. The unit circle will give additional angles, the cosine of which is also equal to 0.732.
Set aside the solution on the unit circle.
- You can put solutions to the trigonometric equation on the unit circle. The solutions of the trigonometric equation on the unit circle are the vertices of a regular polygon.
- Example: The solutions x = π/3 + πn/2 on the unit circle are the vertices of the square.
- Example: The solutions x = π/4 + πn/3 on the unit circle are the vertices of a regular hexagon.
Methods for solving trigonometric equations.
- If the given trigonometric equation contains only one trigonometric function, solve 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
- Transform this equation into an equation of the form: f(x)*g(x)*h(x) = 0, where f(x), g(x), h(x) are the basic trigonometric equations.
- Example 6. 2cos x + sin 2x = 0. (0< x < 2π)
- Decision. Using the double angle formula sin 2x = 2*sin x*cos x, replace sin 2x.
- 2cos x + 2*sin x*cos x = 2cos x*(sin x + 1) = 0. Now solve two basic trigonometric equations: cos x = 0 and (sin x + 1) = 0.
- Example 7 cos x + cos 2x + cos 3x = 0. (0< x < 2π)
- Solution: Using trigonometric identities, transform this equation into an equation of the form: cos 2x(2cos x + 1) = 0. Now solve two basic trigonometric equations: cos 2x = 0 and (2cos x + 1) = 0.
- Example 8. sin x - sin 3x \u003d cos 2x. (0< x < 2π)
- Solution: Using trigonometric identities, transform this equation into an equation of the form: -cos 2x*(2sin x + 1) = 0. Now solve two basic trigonometric equations: cos 2x = 0 and (2sin x + 1) = 0.
  - Method 2
- Convert the given trigonometric equation to an equation containing only one trigonometric function. Then replace this trigonometric function with some unknown, for example, t (sin x = t; cos x = t; cos 2x = t, tg x = t; tg (x/2) = t, etc.).
- Example 9. 3sin^2 x - 2cos^2 x = 4sin x + 7 (0< x < 2π).
- Decision. In this equation, replace (cos^2 x) with (1 - sin^2 x) (according to the identity). The transformed equation looks like:
- 3sin^2 x - 2 + 2sin^2 x - 4sin x - 7 = 0. Replace sin x with t. Now the equation looks like: 5t^2 - 4t - 9 = 0. This is a quadratic equation with two roots: t1 = -1 and t2 = 9/5. The second root t2 does not satisfy the range of the function (-1< sin x < 1). Теперь решите: t = sin х = -1; х = 3π/2.
- Example 10. tg x + 2 tg^2 x = ctg x + 2
- Decision. Replace tg x with t. Rewrite the original equation as follows: (2t + 1)(t^2 - 1) = 0. Now find t and then find x for t = tg x.