Publication date: 2016-03-23

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EXAMPLES OF SOLVING EQUATIONS USING SOME ORIGINAL TECHNIQUES.

1
. Decision irrational equations.

1. Substitution method.

1.1.1 Solve the equation .

Note that the signs of x under the radical are different. We introduce the notation

, .

Then,

Let's perform a term-by-term addition of both parts of the equation.

And we have a system of equations

Because a + b = 4, then

Z reads: 9 - x \u003d 8  x \u003d 1. Answer: x \u003d 1.

1.1.2. Solve the Equation .

We introduce the notation: , ; , .

Means:

Adding term by term the left and right sides of the equations, we have .

And we have a system of equations

a + b = 2, , , ,

Let's return to the system of equations:

, .

Having solved the equation for (ab), we have ab = 9, ab = -1 (-1 extraneous root, because , .).

This system has no solutions, so the original equation also has no solution.

Answer: no solutions.

1. 1. Solve the equation: .

We introduce the notation , where . Then , .

, ,

Consider three cases:

1) . 2) . 3) .

A + 1 - a + 2 \u003d 1, a - 1 - a + 2 \u003d 1, a - 1 + a - 2 \u003d 1, a \u003d 1, 1  [ 0; 1). [ one ; 2). a = 2.

Solution: [ 1 ; 2].

If a , then , , .

Answer: .

1.2. Method for evaluating the left and right parts (the majorant method).

The majorant method is a method for finding boundedness of a function.

Majorization - finding the points of restriction of the function. M is the majorant.

If we have f(x) = g(x) and the ODZ is known, and if

, , then

1. 1. Solve the equation: .

ODZ: .

Consider right side equations.

Let's introduce a function . The graph is a parabola with vertex A(3 ; 2).

The smallest value of the function y(3) = 2, i.e. .

Consider the left side of the equation.

Let's introduce a function . Using the derivative, it is easy to find the maximum of a function that is differentiable on x  (2 ; 4).

At ,

X=3.

G` + -

2 3 4

g(3) = 2.

We have .

As a result, , then

Let us compose a system of equations based on the above conditions:

Solving the first equation of the system, we have x = 3. By substituting this value into the second equation, we make sure that x = 3 is the solution to the system.

Answer: x = 3.

1.3. Application of function monotonicity.

1.3.1. Solve the equation:

About DZ: , because  .

It is known that the sum of increasing functions is an increasing function.

The left side is an increasing function. The right side is a linear function (k=0). Graphical interpretation suggests that the root is unique. We find it by selection, we have x = 1.

Proof:

Suppose there is a root x 1 greater than 1, then

Because x 1 >1,

.We conclude that there are no roots greater than one.

Similarly, one can prove that there are no roots less than one.

So x=1 is the only root.

Answer: x = 1.

1.3.2. Solve the equation:

About DZ: [ 0.5 ; + ), because those. .

Let's transform the equation,

The left side is an increasing function (the product of increasing functions), the right side is a linear function (k = 0). The geometric interpretation shows that the original equation must have a single root that can be found by fitting, x = 7.

Examination:

It can be proved that there are no other roots (see the example above).

Answer: x = 7.

2. Logarithmic equations.

1. Method for estimating the left and right parts.

2.1.1. Solve the equation: log 2 (2x - x 2 + 15) = x 2 - 2x + 5.

Let us estimate the left side of the equation.

2x - x 2 + 15 \u003d - (x 2 - 2x - 15) \u003d - ((x 2 - 2x + 1) - 1 - 15) \u003d - (x - 1) 2 + 16  16.

Then log 2 (2x - x 2 + 15)  4.

Let us estimate the right side of the equation.

x 2 - 2x + 5 \u003d (x 2 - 2x + 1) - 1 + 5 \u003d (x - 1) 2 + 4  4.

The original equation can only have a solution if both sides are equal to four.

Means

Answer: x = 1.

For independent work.

2.1.2. log 4 (6x - x 2 + 7) \u003d x 2 - 6x + 11 Answer: x \u003d 3.

2.1.3. log 5 (8x - x 2 + 9) \u003d x 2 - 8x + 18 Answer: x \u003d 6.

2.1.4. log 4 (2x - x 2 + 3) \u003d x 2 - 2x + 2 Answer: x \u003d 1.

2.1.5. log 2 (6x - x 2 - 5) \u003d x 2 - 6x + 11 Answer: x \u003d 3.

2.2. Using the monotonicity of the function, the selection of roots.

2.2.1. Solve the equation: log 2 (2x - x 2 + 15) = x 2 - 2x + 5.

Let's make the change 2x - x 2 + 15 = t, t>0. Then x 2 - 2x + 5 \u003d 20 - t, then

log 2 t = 20 - t .

The function y = log 2 t is increasing, and the function y = 20 - t is decreasing. The geometric interpretation makes us understand that the original equation has a single root, which is not difficult to find by selecting t = 16.

Solving the equation 2x - x 2 + 15 = 16, we find that x = 1.

Checking to make sure that the selected value is correct.

Answer: x = 1.

2.3. Some “interesting” logarithmic equations.

2.3.1. Solve the Equation .

ODZ: (x - 15) cosx > 0.

Let's move on to the equation

, , ,

Let's move on to the equivalent equation

(x - 15) (cos 2 x - 1) = 0,

x - 15 = 0, or cos 2 x = 1 ,

x = 15. cos x = 1 or cos x = -1,

x=2  k, k Z . x =  + 2 l, l Z.

Let's check the found values ​​by substituting them into the ODZ.

1) if x = 15 , then (15 - 15) cos 15 > 0,

0 > 0 is wrong.

x = 15 - is not the root of the equation.

2) if x = 2  k, k Z, then (2  k - 15) l > 0,

2 k > 15, note that 15  5 . We have

k > 2.5, k Z,

k = 3, 4, 5, … .

3) if x =  + 2 l, l Z, then ( + 2 l - 15) (- 1) > 0,

 + 2  l< 15,

2 l< 15 -  , заметим, что 15  5  .

We have: l< 2,

l = 1, 0 , -1, -2,… .

Answer: x = 2  k (k = 3,4,5,6,…); x \u003d  +2 1 (1 \u003d 1.0, -1, - 2, ...).

3. Trigonometric equations.

3.1. Method for estimating the left and right parts of the equation.

4.1.1. Solve the equation cos3x cos2x = -1.

First way..

0.5 (cos x+ cos 5 x) = -1, cos x+ cos 5 x = -2.

Because cos x - 1 , cos 5 x - 1, we conclude that cos x+ cos 5 x> -2, hence

follows the system of equations

c os x = -1,

cos 5 x = - 1.

Solving the equation cos x= -1, we get X=  + 2 k, where k Z.

These values X are also solutions cos equations 5x= -1, because

cos 5 x= cos 5 ( + 2  k) = cos ( + 4  + 10  k) = -1.

Thus, X=  + 2 k, where k Z , are all solutions of the system, and hence the original equation.

Answer: X=  (2k + 1), k Z.

The second way.

It can be shown that the set of systems follows from the original equation

cos 2 x = - 1,

cos 3 x = 1.

cos 2 x = 1,

cos 3 x = - 1.

Solving each system of equations, we find the union of the roots.

Answer: x = (2  to + 1), k Z.

For independent work.

Solve the equations:

3.1.2. 2 cos 3x + 4 sin x/2 = 7. Answer: no solutions.

3.1.3. 2 cos 3x + 4 sin x/2 = -8. Answer: no solutions.

3.1.4. 3 cos 3x + cos x = 4. Answer: x = 2 to, k Z.

3.1.5. sin x sin 3 x = -1. Answer: x = /2 +  to, k Z.

3.1.6. cos 8 x + sin 7 x = 1. Answer: x = m, m Z; x = /2 + 2  n, n Z.

Municipal educational institution

"Kudinskaya secondary school No. 2"

Ways to solve irrational equations

Completed by: Egorova Olga,

Supervisor:

Teacher

mathematics,

higher qualification

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

Section 1. Methods for solving irrational equations…………………………………6

1.1 Solving the irrational equations of part C……….….….……………………21

Section 2. Individual tasks…………………………………………….....………...24

Answers………………………………………………………………………………………….25

Bibliography…….…………………………………………………………………….26

Introduction

Mathematics education received in general education school, is an essential component general education and common culture modern man. Almost everything that surrounds a modern person is all connected in one way or another with mathematics. BUT recent achievements in physics, engineering and information technology leave no doubt that in the future the state of affairs will remain the same. Therefore, the solution of many practical problems is reduced to solving various kinds equations to learn how to solve. One of these types are irrational equations.

Irrational equations

An equation containing an unknown (or a rational algebraic expression from the unknown) under the sign of the radical, is called irrational equation. In elementary mathematics, solutions to irrational equations are sought in the set of real numbers.

Any irrational equation with the help of elementary algebraic operations (multiplication, division, raising both parts of the equation to an integer power) can be reduced to a rational algebraic equation. In this case, it should be borne in mind that the resulting rational algebraic equation may turn out to be non-equivalent to the original irrational equation, namely, it may contain "extra" roots that will not be the roots of the original ir rational equation. Therefore, having found the roots of the obtained rational algebraic equation, it is necessary to check whether all the roots of the rational equation will be the roots of the irrational equation.

In general, it is difficult to specify any universal method solution of any irrational equation, since it is desirable that as a result of transformations of the original irrational equation, not just some kind of rational algebraic equation is obtained, among the roots of which there will be the roots of this irrational equation, but a rational algebraic equation formed from polynomials of the least possible degree. The desire to obtain that rational algebraic equation formed from polynomials of the smallest possible degree is quite natural, since finding all the roots of a rational algebraic equation can in itself be a rather difficult task, which we can completely solve only in a very limited number of cases.

Types of irrational equations

Solving irrational equations of even degree always causes more problems than the solution of irrational equations of odd degree. When solving irrational equations of odd degree, the ODZ does not change. Therefore, below we will consider irrational equations, the degree of which is even. There are two kinds of irrational equations:

2..

Let's consider the first of them.

odz equation: f(x)≥ 0. In ODZ, the left side of the equation is always non-negative, so a solution can only exist when g(x)≥ 0. In this case, both sides of the equation are non-negative, and exponentiation 2 n gives equivalent equation. We get that

Let us pay attention to the fact that while ODZ is performed automatically, and you can not write it, but the conditiong(x) ≥ 0 must be checked.

Note: This is very important condition equivalence. Firstly, it frees the student from the need to investigate, and after finding solutions, check the condition f(x) ≥ 0 - the non-negativity of the root expression. Secondly, it focuses on checking the conditiong(x) ≥ 0 are the nonnegativity of the right side. After all, after squaring, the equation is solved i.e., two equations are solved at once (but on different intervals of the numerical axis!):

1. - where g(x)≥ 0 and

2. - where g(x) ≤ 0.

Meanwhile, many, according to the school habit of finding ODZ, do exactly the opposite when solving such equations:

a) check, after finding solutions, the condition f(x) ≥ 0 (which is automatically satisfied), make arithmetic errors and get an incorrect result;

b) ignore the conditiong(x) ≥ 0 - and again the answer may be wrong.

Note: The equivalence condition is especially useful when solving trigonometric equations, in which finding ODZ related to decision trigonometric inequalities, which is much more difficult than solving trigonometric equations. Checking in trigonometric equations even conditions g(x)≥ 0 is not always easy to do.

Consider the second kind of irrational equations.

. Let the equation . His ODZ:

In the ODZ, both sides are non-negative, and squaring gives the equivalent equation f(x) =g(x). Therefore, in the ODZ or

With this method of solution, it is enough to check the non-negativity of one of the functions - you can choose a simpler one.

Section 1. Methods for solving irrational equations

1 method. Liberation from radicals by successively raising both sides of the equation to the corresponding natural degree

The most commonly used method for solving irrational equations is the method of freeing from radicals by successively raising both parts of the equation to the corresponding natural degree. In this case, it should be borne in mind that when both parts of the equation are raised to even degree the resulting equation is equivalent to the original one, and when both parts of the equation are raised to an even power, the resulting equation will, generally speaking, be non-equivalent to the original equation. This can be easily verified by raising both sides of the equation to any even power. This operation results in the equation , whose set of solutions is the union of sets of solutions: https://pandia.ru/text/78/021/images/image013_50.gif" width="95" height="21 src=">. However, despite this drawback , it is the procedure for raising both parts of the equation to some (often even) power that is the most common procedure for reducing an irrational equation to a rational equation.

Solve the equation:

Where are some polynomials. By virtue of the definition of the operation of extracting the root in the set of real numbers, the admissible values ​​of the unknown https://pandia.ru/text/78/021/images/image017_32.gif" width="123 height=21" height="21">..gif " width="243" height="28 src=">.

Since both parts of the 1st equation were squared, it may turn out that not all roots of the 2nd equation will be solutions to the original equation, it is necessary to check the roots.

Solve the equation:

https://pandia.ru/text/78/021/images/image021_21.gif" width="137" height="25">

Raising both sides of the equation into a cube, we get

Given that https://pandia.ru/text/78/021/images/image024_19.gif" width="195" height="27">(The last equation may have roots that, generally speaking, are not roots of the equation ).

We raise both sides of this equation to a cube: . We rewrite the equation in the form x3 - x2 = 0 ↔ x1 = 0, x2 = 1. By checking, we establish that x1 = 0 is an extraneous root of the equation (-2 ≠ 1), and x2 = 1 satisfies the original equation.

Answer: x = 1.

2 method. Replacing an adjacent system of conditions

When solving irrational equations containing even-order radicals, the answers may appear extraneous roots which are not always easy to identify. To make it easier to identify and discard extraneous roots, in the course of solving irrational equations it is immediately replaced by an adjacent system of conditions. Additional inequalities in the system actually take into account the ODZ of the equation being solved. You can find the ODZ separately and take it into account later, but it is preferable to use mixed systems of conditions: there is less danger of forgetting something, not taking it into account in the process of solving the equation. Therefore, in some cases it is more rational to use the method of transition to mixed systems.

Solve the equation:

Answer: https://pandia.ru/text/78/021/images/image029_13.gif" width="109 height=27" height="27">

This equation is tantamount to a system

Answer: the equation has no solutions.

3 method. Using the properties of the nth root

When solving irrational equations, the properties of the root of the nth degree are used. arithmetic root n- th degrees from among a call a non-negative number, n- i whose degree is equal to a. If a n- even( 2n), then a ≥ 0, otherwise the root does not exist. If a n- odd( 2 n+1), then a is any and = - ..gif" width="45" height="19"> Then:

2.

3.

4.

5.

Applying any of these formulas, formally (without taking into account the indicated restrictions), it should be borne in mind that the ODZ of the left and right parts of each of them can be different. For example, the expression is defined with f ≥ 0 and g ≥ 0, and the expression is as in f ≥ 0 and g ≥ 0, as well as f ≤ 0 and g ≤ 0.

For each of the formulas 1-5 (without taking into account the indicated restrictions), the ODZ of its right part may be wider than the ODZ of the left. It follows from this that transformations of the equation with the formal use of formulas 1-5 "from left to right" (as they are written) lead to an equation that is a consequence of the original one. In this case, extraneous roots of the original equation may appear, so verification is a mandatory step in solving the original equation.

Transformations of equations with the formal use of formulas 1-5 “from right to left” are unacceptable, since it is possible to judge the ODZ of the original equation, and, consequently, the loss of roots.

https://pandia.ru/text/78/021/images/image041_8.gif" width="247" height="61 src=">,

which is a consequence of the original. The solution of this equation is reduced to solving the set of equations .

From the first equation of this set we find https://pandia.ru/text/78/021/images/image044_7.gif" width="89" height="27"> from where we find . Thus, the roots of this equation can only be numbers ( -1) and (-2) Verification shows that both found roots satisfy this equation.

Answer: -1,-2.

Solve the equation: .

Solution: based on the identities, replace the first term with . Note that as the sum of two non-negative numbers on the left side. “Remove” the module and, after bringing like terms, solve the equation. Since , we get the equation . Since and , then https://pandia.ru/text/78/021/images/image055_6.gif" width="89" height="27 src=">.gif" width="39" height="19 src= ">.gif" width="145" height="21 src=">

Answer: x = 4.25.

4 method. Introduction of new variables

Another example of solving irrational equations is the way in which new variables are introduced, with respect to which either a simpler irrational equation or a rational equation is obtained.

The solution of irrational equations by replacing the equation with its consequence (with subsequent checking of the roots) can be carried out as follows:

1. Find the ODZ of the original equation.

2. Go from the equation to its corollary.

3. Find the roots of the resulting equation.

4. Check if the found roots are the roots of the original equation.

The check is as follows:

A) the belonging of each found root of the ODZ to the original equation is checked. Those roots that do not belong to the ODZ are extraneous for the original equation.

B) for each root included in the ODZ of the original equation, it is checked whether they have identical signs the left and right parts of each of the equations that arise in the process of solving the original equation and are raised to an even power. Those roots for which the parts of any equation raised to an even power have different signs, are extraneous for the original equation.

C) only those roots that belong to the ODZ of the original equation and for which both parts of each of the equations that arise in the process of solving the original equation and raised to an even power have the same signs are checked by direct substitution into the original equation.

Such a solution method with the indicated verification method allows avoiding cumbersome calculations in the case of direct substitution of each of the found roots of the last equation into the original one.

Solve the irrational equation:

.

The set of admissible values ​​of this equation:

Setting , after substitution we obtain the equation

or its equivalent equation

which can be viewed as a quadratic equation for . Solving this equation, we get

.

Therefore, the solution set of the original irrational equation is the union of the solution sets of the following two equations:

, .

Cube both sides of each of these equations, and we get two rational algebraic equations:

, .

Solving these equations, we find that this irrational equation has a single root x = 2 (no verification is required, since all transformations are equivalent).

Answer: x = 2.

Solve the irrational equation:

Denote 2x2 + 5x - 2 = t. Then the original equation will take the form . By squaring both parts of the resulting equation and bringing like terms, we obtain the equation , which is a consequence of the previous one. From it we find t=16.

Returning to the unknown x, we get the equation 2x2 + 5x - 2 = 16, which is a consequence of the original one. By checking, we make sure that its roots x1 \u003d 2 and x2 \u003d - 9/2 are the roots of the original equation.

Answer: x1 = 2, x2 = -9/2.

5 method. Identity Equation Transformation

When solving irrational equations, one should not start solving an equation by raising both parts of the equations to a natural power, trying to reduce the solution of an irrational equation to solving a rational algebraic equation. First, it is necessary to see if it is possible to make some identical transformation of the equation, which can significantly simplify its solution.

Solve the equation:

The set of valid values ​​for this equation: https://pandia.ru/text/78/021/images/image074_1.gif" width="292" height="45"> Divide this equation by .

.

We get:

For a = 0, the equation will have no solutions; for , the equation can be written as

for this equation has no solutions, since for any X, belonging to the set of admissible values ​​of the equation, the expression on the left side of the equation is positive;

when the equation has a solution

Taking into account that the set of admissible solutions of the equation is determined by the condition , we finally obtain:

When solving this irrational equation, https://pandia.ru/text/78/021/images/image084_2.gif" width="60" height="19"> the solution to the equation will be . For all other values X the equation has no solutions.

EXAMPLE 10:

Solve the irrational equation: https://pandia.ru/text/78/021/images/image086_2.gif" width="381" height="51">

Decision quadratic equation The system gives two roots: x1 = 1 and x2 = 4. The first of the obtained roots does not satisfy the inequality of the system, therefore x = 4.

Notes.

1) Holding identical transformations allows you to do without checking.

2) The inequality x - 3 ≥0 refers to identical transformations, and not to the domain of the equation.

3) There is a decreasing function on the left side of the equation, and an increasing function on the right side of this equation. Graphs of decreasing and increasing functions at the intersection of their domains of definition can have no more than one common point. Obviously, in our case, x = 4 is the abscissa of the intersection point of the graphs.

Answer: x = 4.

6 method. Using the domain of definition of functions when solving equations

This method is most effective when solving equations that include functions https://pandia.ru/text/78/021/images/image088_2.gif" width="36" height="21 src="> and find its area definitions (f)..gif" width="53" height="21"> .gif" width="88" height="21 src=">, then you need to check whether the equation is true at the ends of the interval, moreover, if a< 0, а b >0, then it is necessary to check on the intervals (a;0) and . The smallest integer in E(y) is 3.

Answer: x = 3.

8 method. Application of the derivative in solving irrational equations

Most often, when solving equations using the derivative method, the estimation method is used.

EXAMPLE 15:

Solve the equation: (1)

Solution: Since https://pandia.ru/text/78/021/images/image122_1.gif" width="371" height="29">, or (2). Consider the function ..gif" width="400" height="23 src=">.gif" width="215" height="49"> at all and therefore increasing. Therefore, the equation is equivalent to an equation that has a root that is the root of the original equation.

Answer:

EXAMPLE 16:

Solve the irrational equation:

The domain of definition of the function is a segment. Find the largest and smallest value the values ​​of this function on the interval . To do this, we find the derivative of the function f(x): https://pandia.ru/text/78/021/images/image136_1.gif" width="37 height=19" height="19">. Let's find the values ​​of the function f(x) at the ends of the segment and at the point: So, But, and, therefore, equality is possible only under the condition https://pandia.ru/text/78/021/images/image136_1.gif" width="37" height="19 src=" > Verification shows that the number 3 is the root of this equation.

Answer: x = 3.

9 method. Functional

In exams, they sometimes offer to solve equations that can be written in the form , where is a certain function.

For example, some equations: 1) 2) . Indeed, in the first case , in the second case . Therefore, solve irrational equations using the following statement: if a function is strictly increasing on the set X and for any , then the equations, etc., are equivalent on the set X .

Solve the irrational equation: https://pandia.ru/text/78/021/images/image145_1.gif" width="103" height="25"> strictly increasing on the set R, and https://pandia.ru/text/78/021/images/image153_1.gif" width="45" height="24 src=">..gif" width="104" height="24 src=" > which has a unique root Therefore, the equivalent equation (1) also has a unique root

Answer: x = 3.

EXAMPLE 18:

Solve the irrational equation: (1)

By definition square root we get that if equation (1) has roots, then they belong to the set https://pandia.ru/text/78/021/images/image159_0.gif" width="163" height="47">. (2)

Consider the function https://pandia.ru/text/78/021/images/image147_1.gif" width="35" height="21"> strictly increasing on this set for any ..gif" width="100" height ="41"> which has a single root Therefore, and equivalent to it on the set X equation (1) has a single root

Answer: https://pandia.ru/text/78/021/images/image165_0.gif" width="145" height="27 src=">

Solution: This equation is equivalent to mixed system

Real numbers. Approximation of real numbers by finite decimal fractions.

A real or real number is a mathematical abstraction that arose from the need to measure geometric and physical quantities the world around, as well as performing such operations as extracting a root, calculating logarithms, solving algebraic equations. If a integers arose in the process of counting, rational - from the need to operate with parts of a whole, then real numbers are intended for measurement continuous quantities. Thus, the expansion of the stock of numbers under consideration has led to the set of real numbers, which, in addition to the rational numbers, also includes other elements called irrational numbers .

Absolute error and its limit.

Let there be some numerical value, and numerical value, which is assigned to it, is considered accurate, then under approximate value error numerical value (mistake) understand the difference between the exact and approximate value of a numerical value: . The error can take both positive and negative values. The value is called known approximation to the exact value of a numeric value - any number that is used instead of the exact value. Protozoa quantitative measure error is the absolute error. Absolute error approximate value is called the value, about which it is known that: Relative error and its limit.

The quality of the approximation essentially depends on the accepted units of measurement and scales of quantities, therefore it is advisable to correlate the error of a quantity and its value, for which the concept of relative error is introduced. Relative error An approximate value is called a value about which it is known that: . Relative error is often expressed as a percentage. Usage relative errors convenient, in particular, because they do not depend on the scales of quantities and units of measurement.

Irrational equations

An equation in which a variable is contained under the sign of the root is called irrational. When solving irrational equations, the solutions obtained require verification, because, for example, an incorrect equality when squaring can give the correct equality. Indeed, an incorrect equality when squared gives the correct equality 1 2 = (-1) 2 , 1=1. Sometimes it is more convenient to solve irrational equations using equivalent transitions.

Let's square both sides of this equation; After transformations, we arrive at a quadratic equation; and let's put it on.

Complex numbers. Actions on complex numbers.

Complex numbers - an extension of the set of real numbers, usually denoted. Any complex number can be represented as a formal sum x + iy, where x and y- real numbers, i- imaginary unit Complex numbers form an algebraically closed field - this means that the polynomial of degree n with complex coefficients has exactly n complex roots, that is, the fundamental theorem of algebra is true. This is one of the main reasons for the widespread use complex numbers in mathematical research. In addition, the use of complex numbers allows us to conveniently and compactly formulate many mathematical models applied in mathematical physics and in natural sciences- electrical engineering, hydrodynamics, cartography, quantum mechanics, the theory of oscillations and many others.

Comparison a + bi = c + di means that a = c and b = d(two complex numbers are equal if and only if their real and imaginary parts are equal).

Addition ( a + bi) + (c + di) = (a + c) + (b + d) i .

Subtraction ( a + bi) − (c + di) = (a − c) + (b − d) i .

Multiplication

Numeric function. Ways to set a function

In mathematics numeric function is a function whose domains and values ​​are subsets number sets- usually sets of real numbers or sets of complex numbers.

Verbal: Using natural language Y is equal to whole part from x. Analytical: Using an analytical formula f (x) = x !

Graphical Via graph Fragment of the function graph.

Tabular: Using a table of values

Main properties of the function

1) Function scope and function range . Function scope x(variable x) for which the function y=f(x) defined.

Function range y that the function accepts. In elementary mathematics, functions are studied only on the set of real numbers.2 ) Function zero) Monotonicity of the function . Increasing function Decreasing function . Even function X f(-x) = f(x). odd function- a function whose domain of definition is symmetric with respect to the origin and for any X f(-x) = -f(x. The function is called limited unlimited .7) Periodicity of the function. Function f(x) - periodical function period

Function graphs. The simplest transformations of graphs by a function

Function Graph- set of points whose abscissas are valid values argument x, and the ordinates are the corresponding values ​​of the function y .

Straight line- schedule linear function y=ax+b. The function y increases monotonically for a > 0 and decreases for a< 0. При b = 0 прямая линия проходит через начало координат т.0 (y = ax - прямая пропорциональность)

Parabola- function graph square trinomial y \u003d ax 2 + bx + c. It has vertical axis symmetry. If a > 0, has a minimum if a< 0 - максимум. Точки пересечения (если они есть) с осью абсцисс - корни соответствующего квадратного уравнения ax 2 + bx + c \u003d 0

Hyperbola- function graph. When a > O is located in the I and III quarters, when a< 0 - во II и IV. Асимптоты - оси координат. Ось симметрии - прямая у = х (а >0) or y - x (a< 0).

Logarithmic function y = log a x(a > 0)

trigonometric functions. When constructing trigonometric functions, we use radian measure of angles. Then the function y= sin x represented by a graph (Fig. 19). This curve is called sinusoid .

Function Graph y= cos x shown in fig. 20; it is also a sine wave resulting from moving the graph y= sin x along the axis X left by /2.

Basic properties functions. Monotonicity, evenness, oddness, periodicity of functions.

Function scope and function range . Function scope is the set of all valid valid values ​​of the argument x(variable x) for which the function y=f(x) defined.

Function range is the set of all real values y that the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.2 ) Function zero- is the value of the argument, at which the value of the function is equal to zero.3 ) Intervals of constancy of the function- those sets of argument values ​​on which the function values ​​are only positive or only negative.4 ) Monotonicity of the function .

Increasing function(in some interval) - a function for which greater value an argument from this interval corresponds to a larger value of the function.

Decreasing function(in some interval) - a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.5 ) Even (odd) functions . Even function- a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). Schedule even function symmetrical about the y-axis. odd function- a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = -f(x). Schedule odd function symmetrical about the origin.6 ) Limited and unlimited functions. The function is called limited, if there is a positive number M such that |f (x) | ≤ M for all values ​​of x. If no such number exists, then the function is unlimited .7) Periodicity of the function. Function f(x) - periodical, if there is such a non-zero number T that for any x from the domain of the function, the following holds: f (x+T) = f (x). Such smallest number called function period. All trigonometric functions are periodic. (Trigonometric formulas).

Periodic functions. Rules for finding the main period of a function.

Periodic function is a function that repeats its values ​​after some nonzero period, i.e., does not change its value when a fixed nonzero number (period) is added to the argument. All trigonometric functions are periodic. Are wrong statements about the amount periodic functions: Sum of 2 functions with comparable (even basic) periods T 1 and T 2 is a function with period LCM ( T 1 ,T 2). Amount 2 continuous functions with incommensurable (even basic) periods is a non-periodic function. There are no periodic functions equal to a constant, whose periods are incommensurable numbers.

Plotting power functions.

Power function. This is the function: y = ax n, where a,n- permanent. At n= 1 we get direct proportionality : y =ax; at n = 2 - square parabola; at n = 1 - inverse proportionality or hyperbole. Thus, these functions are special cases of a power function. We know that the zero power of any number other than zero is equal to 1, therefore, when n = 0 power function turns into constant value: y =a, i.e. its graph is a straight line parallel to the axis X, excluding the origin of coordinates (please explain why?). All these cases (with a= 1) are shown in Fig. 13 ( n 0) and Fig.14 ( n < 0). Отрицательные значения x are not considered here, because then some functions:

Inverse function

Inverse function- a function that reverses the dependence expressed by this function. The function is inverse to the function if the following identities hold: for all for all

Limit of a function at a point. Basic properties of the limit.

The root of the nth degree and its properties.

The nth root of a number a is a number whose nth power is equal to a.

Definition: The arithmetic root of the nth degree of the number a is a non-negative number, the nth power of which is equal to a.

The main properties of the roots:

Degree with arbitrary real indicator and its properties.

Let a positive number and an arbitrary real number be given. The number is called the degree, the number is the base of the degree, the number is the exponent.

By definition it is assumed:

If - positive numbers, and - any real numbers, then following properties:

.

.

Power function, its properties and graphs

Power function complex variable f (z) = z n with an integer exponent is determined using the analytic continuation of a similar function of a real argument. For this, the exponential form of writing complex numbers is used. a power function with an integer exponent is analytic in the entire complex plane, as a product finite number identity mapping instances f (z) = z. According to the uniqueness theorem, these two criteria are sufficient for the uniqueness of the resulting analytic continuation. Using this definition, we can immediately conclude that the power function of a complex variable has significant differences from its real counterpart.

This is a function of the form , . The following cases are considered:

a). If , then . Then , ; if the number is even, then the function is even (i.e. for all ); if the number is odd, then the function is odd (that is, for all).

The exponential function, its properties and graphs

Exponential function- mathematical function.

In the real case, the base of the degree is some non-negative real number, and the function argument is a real exponent.

In theory complex functions considered more general case, when the argument and exponent can be an arbitrary complex number.

In the very general view - u v, introduced by Leibniz in 1695.

The case when the number e acts as the base of the degree is especially highlighted. Such a function is called an exponent (real or complex).

Properties ; ; .

exponential equations.

Let us proceed directly to the exponential equations. In order to decide exponential equation it is necessary to use the following theorem: If the degrees are equal and the bases are equal, positive and different from one, then their exponents are also equal. Let's prove this theorem: Let a>1 and a x =a y .

Let us prove that in this case x=y. Assume the opposite of what is required to be proved, i.e. let's say that x>y or that x<у. Тогда получим по свойству показательной функции, что либо a х a y . Both of these results contradict the hypothesis of the theorem. Therefore, x=y, which is what was required to be proved.

The theorem is also proved for the case when 0 0 and a≠1.

exponential inequalities

Inequalities of the form (or less) for a(x) >0 and are solved based on the properties of the exponential function: for 0 < а (х) < 1 when comparing f(x) and g(x) the sign of the inequality changes, and when a(x) > 1- is saved. The most difficult case for a(x)< 0 . Here we can only give a general indication: to determine at what values X indicators f(x) and g(x) be integers, and choose from them those that satisfy the condition. Finally, if the original inequality holds for a(x) = 0 or a(x) = 1(for example, when the inequalities are not strict), then these cases must also be considered.

Logarithms and their properties

Logarithm of a number b by reason a (from the Greek λόγος - "word", "relation" and ἀριθμός - "number") is defined as an indicator of the degree to which the base must be raised a to get the number b. Designation: . It follows from the definition that the entries and are equivalent. Example: because . Properties

Basic logarithmic identity:

Logarithmic function, its properties and graphs.

A logarithmic function is a function of the form f (x) = log a x, defined at

Domain:

Range of value:

The graph of any logarithmic function passes through the point (1; 0)

The derivative of the logarithmic function is:

Logarithmic Equations

An equation containing a variable under the sign of the logarithm is called a logarithmic equation. The simplest example of a logarithmic equation is the equation log a x \u003d b (where a > 0, and 1). His decision x = a b .

Solving equations based on the definition of the logarithm, for example, the equation log a x \u003d b (a\u003e 0, but 1) has a solution x = a b .

potentiation method. By potentiation is meant the transition from an equality containing logarithms to an equality that does not contain them:

if log a f (x) = log a g (x), then f(x) = g(x), f(x) >0 ,g(x) >0 ,a > 0 , a 1 .

Method for reducing a logarithmic equation to a quadratic one.

The method of taking the logarithm of both parts of the equation.

Method for reducing logarithms to the same base.

Logarithmic inequalities.

An inequality containing a variable only under the sign of the logarithm is called a logarithmic one: log a f (x) > log a g (x).

When solving logarithmic inequalities, one should take into account the general properties of inequalities, the property of monotonicity of the logarithmic function and its domain of definition. Inequality log a f (x) > log a g (x) is tantamount to a system f (x) > g (x) > 0 for a > 1 and system 0 < f (x) < g (x) при 0 < а < 1 .

Radian measurement of angles and arcs. Sine, cosine, tangent, cotangent.

degree measure. Here the unit of measure is degree ( designation ) - is the rotation of the beam by 1/360 of one full revolution. Thus, a full rotation of the beam is 360. One degree is made up of 60 minutes ( their designation ‘); one minute - respectively out of 60 seconds ( marked with ").

radian measure. As we know from planimetry (see the paragraph "Arc length" in the section "Locus of points. Circle and circle"), the length of the arc l, radius r and the corresponding central angle are related by: = l / r.

This formula underlies the definition of the radian measure of angles. So if l = r, then = 1, and we say that the angle is equal to 1 radian, which is denoted: = 1 glad. Thus, we have the following definition of the radian measure:

The radian is the central angle, whose arc length and radius are equal(A m B = AO, Fig. 1). So, the radian measure of an angle is the ratio of the length of an arc drawn by an arbitrary radius and enclosed between the sides of this angle to the radius of the arc.

The trigonometric functions of acute angles can be defined as the ratio of the lengths of the sides of a right triangle.

Sinus:

Cosine:

Tangent:

Cotangent:

Trigonometric functions of a numeric argument

Definition .

The sine of x is the number equal to the sine of the angle in x radians. The cosine of a number x is the number equal to the cosine of the angle in x radians .

Other trigonometric functions of a numerical argument are defined similarly X .

Ghost formulas.

Addition formulas. Double and half argument formulas.

Double.

( ; .

Trigonometric functions and their graphs. Basic properties of trigonometric functions.

Trigonometric functions- kind of elementary functions. They are usually referred to sinus (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), Trigonometric functions are usually defined geometrically, but they can be defined analytically in terms of sums of series or as solutions to certain differential equations, which allows us to extend the domain of definition of these functions to complex numbers.

Function y sinx its properties and graph

Properties:

2. E (y) \u003d [-1; one].

3. The function y \u003d sinx is odd, since, by definition, the sine of a trigonometric angle sin(- x)= - y/R = - sinx, where R is the radius of the circle, y is the ordinate of the point (Fig.).

4. T \u003d 2n - the smallest positive period. Really,

sin(x+p) = sinx.

with Ox axis: sinx= 0; x = pn, nОZ;

with the y-axis: if x = 0, then y = 0.6. Constancy intervals:

sinx > 0, if xО (2pn; p + 2pn), nОZ;

sinx< 0 , if xО (p + 2pn; 2p+pn), nОZ.

Sine signs in quarters

y > 0 for angles a of the first and second quarters.

at< 0 для углов ее третьей и четвертой четвертей.

7. Intervals of monotonicity:

y= sinx increases on each of the intervals [-p/2 + 2pn; p/2 + 2pn],

nнz and decreases on each of the intervals , nнz.

8. Extreme points and extreme points of the function:

xmax= p/2 + 2pn, nнz; y max = 1;

ymax= - p/2 + 2pn, nнz; ymin = - 1.

Function Properties y= cosx and her schedule:

Properties:

2. E (y) \u003d [-1; one].

3. Function y= cosx- even, because by definition of the cosine of the trigonometric angle cos (-a) = x/R = cosa on the trigonometric circle (rice)

4. T \u003d 2p - the smallest positive period. Really,

cos(x+2pn) = cosx.

5. Intersection points with coordinate axes:

with the Ox axis: cosx = 0;

x = p/2 + pn, nОZ;

with the y-axis: if x = 0, then y = 1.

6. Intervals of sign constancy:

cos > 0, if xО (-p/2+2pn; p/2 + 2pn), nОZ;

cosx< 0 , if xО (p/2 + 2pn; 3p/2 + 2pn), nОZ.

This is proved on a trigonometric circle (Fig.). Cosine signs in quarters:

x > 0 for angles a of the first and fourth quadrants.

x< 0 для углов a второй и третей четвертей.

7. Intervals of monotonicity:

y= cosx increases on each of the intervals [-p + 2pn; 2pn],

nнz and decreases on each of the intervals , nнz.

Function Properties y= tgx and its plot: properties -

1. D (y) = (xОR, x ¹ p/2 + pn, nОZ).

3. Function y = tgx - odd

tgx > 0

tgx< 0 for xн (-p/2 + pn; pn), nнZ.

See the figure for the signs of the tangent in quarters.

6. Intervals of monotonicity:

y= tgx increases at each interval

(-p/2 + pn; p/2 + pn),

7. Extreme points and extreme points of the function:

8. x = p/2 + pn, nнz - vertical asymptotes

Function Properties y= ctgx and her schedule:

Properties:

1. D (y) = (xОR, x ¹ pn, nОZ). 2. E(y)=R.

3. Function y= ctgx- odd.

4. T \u003d p - the smallest positive period.

5. Intervals of sign constancy:

ctgx > 0 for xО (pn; p/2 + pn;), nОZ;

ctgx< 0 for xÎ (-p/2 + pn; pn), nÎZ.

Cotangent signs for quarters, see the figure.

6. Function at= ctgx increases on each of the intervals (pn; p + pn), nОZ.

7. Extremum points and extremums of a function y= ctgx no.

8. Function Graph y= ctgx is an tangentoid, obtained by plot shift y=tgx along the Ox axis to the left by p/2 and multiplying by (-1) (Fig)

Inverse trigonometric functions, their properties and graphs

Inverse trigonometric functions (circular functions , arcfunctions) are mathematical functions that are inverse to trigonometric functions. Inverse trigonometric functions usually include six functions: arcsine , arc cosine , arc tangent ,arccotanges. The name of the inverse trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix "ark-" (from lat. arc- arc). This is due to the fact that geometrically the value of the inverse trigonometric function can be associated with the length of the arc of a unit circle (or the angle that subtends this arc) corresponding to one or another segment. Occasionally in foreign literature they use designations like sin −1 for the arcsine, etc.; this is considered not entirely correct, since confusion with raising a function to the power of −1 is possible. Basic ratio

Function y=arcsinX, its properties and graphs.

arcsine numbers m this angle is called x for whichFunction y= sin x y= arcsin x is strictly increasing. (function is odd).

Function y=arccosX, its properties and graphs.

Arc cosine numbers m this angle is called x, for which

Function y= cos x continuous and bounded along its entire number line. Function y= arccos x is strictly decreasing. cos (arccos x) = x at arccos (cos y) = y at D(arccos x) = [− 1; 1], (domain), E(arccos x) = . (range of values). Properties of the arccos function (the function is centrally symmetric with respect to the point

Function y=arctgX, its properties and graphs.

Arctangent numbers m An angle α is called such that the Function is continuous and bounded on its entire real line. The function is strictly increasing.

at

arctg function properties

,

.

Function y=arcctg, its properties and graphs.

Arc tangent numbers m this angle is called x, for which

The function is continuous and bounded on its entire real line.

The function is strictly decreasing. at at 0< y < π Свойства функции arcctg (график функции центрально-симметричен относительно точки for any x .

.

The simplest trigonometric equations.

Definition. wada equations sin x = a ; cos x = a ; tan x = a ; ctg x = a, where x

Special cases of trigonometric equations

Definition. wada equations sin x = a ; cos x = a ; tan x = a ; ctg x = a, where x- variable, aR, are called simple trigonometric equations.

Trigonometric equations

Axioms of stereometry and consequences from them

Basic figures in space: points, lines and planes. The main properties of points, lines and planes, concerning their mutual arrangement, are expressed in axioms.

A1. Through any three points that do not lie on the same straight line, there passes a plane, and moreover, only one. A2. If two points of a line lie in a plane, then all points of the line lie in that plane.

Comment. If a line and a plane have only one common point, then they are said to intersect.

A3. If two planes have a common point, then they have a common line on which all common points of these planes lie.

	A and intersect along the line a.

Consequence 1. Through a line and a point not lying on it passes a plane, and moreover, only one. Consequence 2. A plane passes through two intersecting straight lines, and moreover, only one.

Mutual arrangement of two lines in space

Two straight lines given by equations

intersect at a point.

Parallelism of a line and a plane.

Definition 2.3 A line and a plane are called parallel if they have no common points. If the line a is parallel to the plane α, then write a || a. Theorem 2.4 Sign of parallelism of a straight line and a plane. If a line outside a plane is parallel to a line in the plane, then that line is also parallel to the plane itself. Proof Let b α, a || b and a α (drawing 2.2.1). We will prove by contradiction. Let a not be parallel to α, then the line a intersects the plane α at some point A. Moreover, A b, since a || b. According to the criterion of skew lines, lines a and b are skew. We have come to a contradiction. Theorem 2.5 If the plane β passes through the line a parallel to the plane α and intersects this plane along the line b, then b || a. Proof Indeed, the lines a and b are not skew, since they lie in the plane β. Moreover, these lines have no common points, since a || a. Definition 2.4 The line b is sometimes called the trace of the plane β on the plane α.

Crossing straight lines. Sign of intersecting lines

Lines are called intersecting if the following condition is met: If we imagine that one of the lines belongs to an arbitrary plane, then the other line will intersect this plane at a point that does not belong to the first line. In other words, two lines in three-dimensional Euclidean space intersect if there is no plane containing them. Simply put, two lines in space that do not have common points, but are not parallel.

Theorem (1): If one of two lines lies in a certain plane, and the other line intersects this plane at a point not lying on the first line, then these lines are skew.

Theorem (2): Through each of the two intersecting lines there passes a plane parallel to the other line, and moreover, only one.

Theorem (3): If the sides of two angles are respectively co-directed, then such angles are equal.

Parallelism of lines. Properties of parallel planes.

Parallel (sometimes - isosceles) straight lines called straight lines that lie in the same plane and either coincide or do not intersect. In some school definitions, coinciding lines are not considered parallel; such a definition is not considered here. Properties Parallelism is a binary equivalence relation, therefore it divides the entire set of lines into classes of lines parallel to each other. Through any given point, there can be exactly one line parallel to the given one. This is a distinctive property of Euclidean geometry, in other geometries the number 1 is replaced by others (in Lobachevsky's geometry there are at least two such lines) 2 parallel lines in space lie in the same plane. b At the intersection of 2 parallel lines by a third, called secant: The secant necessarily intersects both lines. When crossing, 8 corners are formed, some characteristic pairs of which have special names and properties: Cross lying angles are equal. Respective angles are equal. Unilateral the angles add up to 180°.

Perpendicularity of a line and a plane.

A line that intersects a plane is called perpendicular this plane if it is perpendicular to every line that lies in the given plane and passes through the point of intersection.

SIGN OF PERPENDICULARITY OF A LINE AND A PLANE.

If a line intersecting a plane is perpendicular to two lines in that plane passing through the point of intersection of the given line and the plane, then it is perpendicular to the plane.

1st PROPERTY OF PERPENDICULAR LINES AND PLANES .

If a plane is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

2nd PROPERTY OF PERPENDICULAR LINES AND PLANES .

Two lines perpendicular to the same plane are parallel.

Three perpendiculars theorem

Let be AB- perpendicular to the plane α, AC- oblique and c- a straight line in the plane α passing through the point C and perpendicular projection BC. Let's draw a straight line CK parallel to a straight line AB. Straight CK perpendicular to the plane α (because it is parallel to AB), and hence any line of this plane, therefore, CK perpendicular to the line c AB and CK plane β (parallel lines define a plane, and only one). Straight c is perpendicular to two intersecting lines lying in the plane β, this BC by condition and CK by construction, which means that it is perpendicular to any line belonging to this plane, which means that it is also perpendicular to a line AC .

Converse of the three perpendiculars theorem

If a straight line drawn in a plane through the base of an inclined line is perpendicular to the inclined line, then it is also perpendicular to its projection.

Let be AB- perpendicular to the plane a , AC- oblique and with- straight line in the plane a passing through the base of the slope With. Let's draw a straight line SC, parallel to the line AB. Straight SC perpendicular to the plane a(by this theorem, since it is parallel AB), and hence any line of this plane, therefore, SC perpendicular to the line with. Draw through parallel lines AB and SC plane b(parallel lines define a plane, and only one). Straight with perpendicular to two straight lines lying in a plane b, This AC by condition and SC by construction, it means that it is perpendicular to any line belonging to this plane, which means it is also perpendicular to a line sun. In other words, projection sun perpendicular to the line with lying in the plane a .

Perpendicular and oblique.

Perpendicular, lowered from a given point to a given plane, is called a segment connecting a given point with a point in the plane and lying on a straight line perpendicular to the plane. The end of this segment, lying in a plane, is called the base of the perpendicular .

oblique, drawn from a given point to a given plane, is any segment connecting the given point to a point in the plane that is not perpendicular to the plane. The end of a segment that lies in a plane is called the base of the inclined. The segment connecting the bases of the perpendicular of the inclined line, drawn from the same point, is called oblique projection .

Definition 1. A perpendicular to a given line is a line segment perpendicular to a given line that has one of its ends at their intersection point. The end of a segment that lies on a given line is called the base of the perpendicular.

Definition 2. An oblique line drawn from a given point to a given line is a segment connecting given point with any point of a line that is not the base of the perpendicular dropped from the same point to the given line. AB - perpendicular to the plane α.

AC - oblique, CB - projection.

C - the base of the inclined, B - the base of the perpendicular.

The angle between a line and a plane.

Angle between line and plane Any angle between a straight line and its projection onto this plane is called.

Dihedral angle.

Dihedral angle- spatial geometric figure, formed by two half-planes emanating from one straight line, as well as a part of space bounded by these half-planes. Half planes are called faces dihedral angle, and their common straight line - edge. Dihedral angles are measured by a linear angle, that is, the angle formed by the intersection of a dihedral angle with a plane perpendicular to its edge. Every polyhedron, regular or irregular, convex or concave, has dihedral angle on every edge.

Perpendicularity of two planes.

SIGN OF PLANE PERPENDICULARITY.

If a plane passes through a line perpendicular to another plane, then these planes are perpendicular.

1.1 Irrational equations

Irrational equations are often found on entrance exams in mathematics, since with their help it is easy to diagnose knowledge of such concepts as equivalent transformations, domain of definition, and others. Methods for solving irrational equations, as a rule, are based on the possibility of replacing (with the help of some transformations) an irrational equation with a rational one, which is either equivalent to the original irrational equation or is its consequence. Most often, both sides of the equation are raised to the same power. Equivalence is not violated when both parts are raised to an odd power. Otherwise, it is required to check the found solutions or estimate the sign of both parts of the equation. But there are other tricks that can be more effective in solving irrational equations. For example, the trigonometric substitution method.

Example 1: Solve the Equation

Since , then . Therefore, one can put . The equation will take the form

Let's put where, then

.

.

Answer: .

Algebraic Solution

Since then . Means, , so you can expand the module

.

Answer: .

Solving an equation in an algebraic way requires a good skill in carrying out identical transformations and competent handling of equivalent transitions. But in general, both approaches are equivalent.

Example 2: Solve the Equation

.

Solution using trigonometric substitution

The domain of the equation is given by the inequality , which is equivalent to the condition , then . Therefore, we can put . The equation will take the form

Since , then . Let's open the internal module

Let's put , then

.

The condition is satisfied by two values ​​and .

.

.

Answer: .

Algebraic Solution

.

Let us square the equation of the first set system, we obtain

Let , then . The equation will be rewritten in the form

By checking we establish that is the root, then by dividing the polynomial by the binomial we obtain the decomposition of the right side of the equation into factors

Let's move from variable to variable , we get

.

condition satisfy two values

.

Substituting these values ​​into the original equation, we get that is the root.

Solving the equation of the second system of the original population in a similar way, we find that it is also a root.

Answer: .

If in the previous example the algebraic solution and the solution using the trigonometric substitution were equivalent, then in this case the substitution solution is more profitable. When solving an equation by means of algebra, one has to solve a set of two equations, that is, to square twice. After this non-equivalent transformation, two equations of the fourth degree with irrational coefficients are obtained, which the replacement helps to get rid of. Another difficulty is the verification of the found solutions by substitution into the original equation.

Example 3. Solve the equation

.

Solution using trigonometric substitution

Since , then . Note that a negative value of the unknown cannot be a solution to the problem. Indeed, we transform the original equation to the form

.

The factor in brackets on the left side of the equation is positive, the right side of the equation is also positive, so the factor on the left side of the equation cannot be negative. That's why, then, that's why you can put The original equation will be rewritten in the form

Since , then and . The equation will take the form

Let be . Let's move from the equation to equivalent system

.

The numbers and are the roots of the quadratic equation

.

Algebraic solution Let's square both sides of the equation

We introduce the replacement , then the equation will be written in the form

The second root is redundant, so consider the equation

.

Since , then .

In this case, the algebraic solution is technically simpler, but it is necessary to consider the above solution using a trigonometric substitution. This is due, firstly, to the non-standard nature of the substitution itself, which destroys the stereotype that the use of trigonometric substitution is possible only when . It turns out that if the trigonometric substitution also finds application. Secondly, there is a certain difficulty in solving the trigonometric equation , which is reduced by introducing a change to a system of equations. In a certain sense, this replacement can also be considered non-standard, and familiarity with it allows you to enrich the arsenal of tricks and methods for solving trigonometric equations.

Example 4. Solve the equation

.

Solution using trigonometric substitution

Since a variable can take on any real value, we put . Then

,

As .

The original equation, taking into account the transformations carried out, will take the form

Since , we divide both sides of the equation by , we get

Let be , then . The equation will take the form

.

Given the substitution , we obtain a set of two equations

.

Let's solve each set equation separately.

.

Cannot be a sine value, as for any values ​​of the argument.

.

As and the right side of the original equation is positive, then . From which it follows that .

This equation has no roots, since .

So the original equation has a single root

.

Algebraic Solution

This equation can be easily "turned" into a rational equation of the eighth degree by squaring both parts of the original equation. The search for the roots of the resulting rational equation is difficult, and it is necessary to have a high degree resourcefulness to get the job done. Therefore, it is advisable to know a different way of solving, less traditional. For example, the substitution proposed by I. F. Sharygin.

Let's put , then

Let's transform the right side of the equation :

Taking into account the transformations, the equation will take the form

.

We introduce a replacement, then

.

The second root is redundant, therefore, and .

If the idea of ​​solving the equation is not known in advance , then solving in the standard way by squaring both parts of the equation is problematic, since the result is an equation of the eighth degree, whose roots are extremely difficult to find. The solution using trigonometric substitution looks cumbersome. It may be difficult to find the roots of the equation, if you do not notice that it is recurrent. Decision the specified equation occurs using the apparatus of algebra, so we can say that the proposed solution is combined. In it, information from algebra and trigonometry work together for one goal - to get a solution. Also, the solution of this equation requires careful consideration of two cases. The substitution solution is technically simpler and more beautiful than using a trigonometric substitution. It is desirable that students know this substitution method and apply it to solve problems.

We emphasize that the use of trigonometric substitution for solving problems should be conscious and justified. It is advisable to use substitution in cases where the solution in another way is more difficult or even impossible. Let us give one more example, which, unlike the previous one, is easier and faster to solve in the standard way.