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 the geometric and physical quantities of the world around us, as well as to carry out such operations as extracting a root, calculating logarithms, and solving algebraic equations. If natural numbers arose in the process of counting, rational numbers - from the need to operate with parts of a whole, then real numbers are intended for measuring continuous quantities. Thus, the expansion of the stock of numbers under consideration has led to the set of real numbers, which, in addition to rational numbers, also includes other elements called irrational numbers .
Absolute error and its limit.
Let there be some numerical value, and the numerical value assigned to it is considered to be exact, then under the error of the approximate value of the 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. The simplest quantitative measure of error is 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. The use of relative errors is 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 of complex numbers in mathematical research. In addition, the use of complex numbers makes it possible to conveniently and compactly formulate many mathematical models used in mathematical physics and 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, a number function is a function whose domains and values are subsets of number sets—generally the set of real numbers or the set of complex numbers.
Verbal: Using natural language, Y equals the integer part of 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 argument values x, and the ordinates are the corresponding values of the function y .
Straight line- graph of a 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- graph of the square trinomial function y \u003d ax 2 + bx + c. It has a vertical axis of 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 of 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 in which the larger value of the argument from this interval corresponds to the 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). The graph of an even function is 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). The graph of an odd function is 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). This smallest number is 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 sum of periodic functions: The sum of 2 functions with commensurate (even basic) periods T 1 and T 2 is a function with period LCM ( T 1 ,T 2). The sum of 2 continuous functions with incommensurable (even basic) periods is a non-periodic function. There are no periodic functions that are not 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 the power function becomes a constant: 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 exponent 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 and are positive numbers, and are any real numbers, then the following properties are true:
.
.
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 the product of a finite number of instances of the identity mapping 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 argument of the function is a real exponent.
In the theory of complex functions, a more general case is considered, when an arbitrary complex number can be an argument and an exponent.
In the most general way - 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 solve an 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
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 monotonicity property 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 "arc-" (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 the 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 the given point to any point on the 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- a 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 a dihedral angle on each 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.
Publication date: 2016-03-23
Short description: ...
EXAMPLES OF SOLVING EQUATIONS USING SOME ORIGINAL TECHNIQUES.
1
. Solution of irrational equations.
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, which means that the original equation also has no solution.
Answer: no solutions.
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
Solve the equation: .
ODZ: .
Consider the right side of the equation.
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.
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 can be easily found 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 of the equation cos 5 x= -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.
1.1 Irrational equations
Irrational equations are often encountered at entrance exams in mathematics, since with their help knowledge of such concepts as equivalent transformations, domain of definition, and others is easily diagnosed. 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 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 the 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 a high degree of ingenuity is required to cope with the task. 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. The solution of this 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.
