Let us recall the basic information from trigonometry that is necessary for what follows.

Trigonometric functions are considered initially as functions of angle, since numerical value each of them (if it makes sense) is determined by specifying an angle. One-to-one correspondence between arcs of a circle and central angles allows you to consider trigonometric functions as arc functions. So, for example, the function argument sinφ we have the opportunity to interpret it as an angle or as an arc at will. Thus, initially the argument of the trigonometric function acts as a geometric object - an angle or an arc. However, both in mathematics itself and in its applications, there is a need to consider trigonometric functions as functions of numeric argument. Even in school math The argument of a trigonometric function is not always considered an angle. So, for example, harmonic oscillatory motion is given using the equation: s = A sin at. Here the argument t is time, not angle (coefficient a is a number characterizing the oscillation frequency).

The process of measuring angles (or arcs) assigns each angle (arc) a certain number as its measure. As a result of measuring the angle (arc), you may get any real number, since we can consider directed angles (arcs) of any size. By choosing a certain unit of measurement for angles (arcs), you can associate each angle (arc) with a number that measures it, and, conversely, each number can associate an angle (arc) measured by a given number. This allows the argument of a trigonometric function to be interpreted as a number. Let's consider some trigonometric function, for example, sine. Let x be any real number, this number corresponds completely certain angle(arc), measured by the number x, and the resulting angle (arc) corresponds to a very specific sine value, sin x. Ultimately, a correspondence between numbers is obtained: each real number x corresponds to a well-defined real number y = sin x. Therefore, sin x can be interpreted as a function numeric argument. When considering trigonometric functions as functions of a numerical argument, it was agreed upon to take arcs and angles as a unit of measurement radian. By virtue of this convention, the symbols sin x, cos x, tgx and ctg x should be interpreted as sine, cosine, tangent and cotangent of an angle (arc), the radian measure of which is expressed by the number x. So, for example, sin 2 is the sine of an arc measured in two radians *.

* (Note that in some manuals the radian measure is extremely unsuccessfully called abstract, in contrast to the degree measure. Between both measurement methods no fundamental difference, only different units of measurement are selected. Unfortunately, to this day this question sometimes gives rise to pseudoscientific, harmful “methodological” idle talk.)

Selecting the unit of measurement for arcs and angles doesn't have of fundamental importance. Radian selection not dictated necessity. The radian turns out to be only the most convenient unit, since in the radian measurement the formulas mathematical analysis, related to trigonometric functions, take the simplest form *.

* (This simplification is explained by the fact that in radian measure, let us take, for example, a degree as a unit of measurement for angles. Let t and x be degree and radian measures, respectively given angle, then we have:

The law of correspondence between the values of an argument and a trigonometric function is not established by direct indication mathematical operations(formula), which must be performed on the argument, and geometrically *. However, in order to be able to talk about a function, it is necessary to have a law of correspondence, by virtue of which each acceptable value argument corresponds to a specific function value, but not essential how this law is established.

* (By means elementary mathematics it is impossible to construct formulas expressing the values of trigonometric functions using algebraic operations over the argument. Formulas known from higher mathematics, expressing the values of trigonometric functions directly through the value of the argument,

The functions sin x and cos x make sense for any real values of x, and therefore their domain of definition is the set of all real numbers.

The function tg x is defined for all real values of x, different from numbers of the form π / 2 + kπ.

The function ctg x is defined for all real values of x, different from numbers of the form kπ.

So, the argument of a trigonometric function, at our discretion, can be interpreted as an angle, or as an arc, or, finally, as a number. When calling an argument an arc (or an angle), we can mean by it not the arc (or angle) itself, but the number that measures it. Preserving geometric terminology, we allow ourselves instead of, for example, the following phrase: “sine of the number π / 2” to say: “sine of the arc π / 2”.

Geometric terminology is convenient because it reminds us of the corresponding geometric images.

One of the most important properties trigonometric functions is their periodicity. The functions sin x and cos x have a period of 2π. This means that for any value of x the equalities hold:

sin x = sin (x + 2π) = sin (x + 4π) = ... = sin (x + 2kπ);

cos x = cos (x + 2π) = cos (x + 4π) = ... = cos (x + 2kπ),

Where k- any integer.

Strictly speaking, the functions sin x and cos x have infinite set periods:

±2π, ±4π, ±6π, ... ±2kπ,

the number 2tr, which is the smallest positive period, is usually called simply a period.

The property of periodicity has the following geometric interpretation: the meaning of trigonometric functions sin x And cos x does not change if an integer number of circles is added (or subtracted) to the arc x. If the function sin x or cos x has any property for the value of the argument x = a, then it has the same property for any of the values a + 2kπ.

The functions tg x and ctg x are also periodic; their period (the smallest positive) is the number π.

When studying properties periodic function it is enough to consider it in some interval, equal in magnitude to the period.

Let us list the main properties of trigonometric functions.

1°. sin function x on the segment (I and I negative quarters) increases. The sine values at the ends of the segment, i.e. at x = π / 2 and at x = - π / 2 are equal to 1 and -1, respectively.

2°. Whatever the real number k is, according to absolute value no more than 1, on the segment - π / 2 ≤x≤ π / 2 there is a single arc x = x 1, the sine of which is equal to k. In other words, on the segment the sine has, for one single value of the argument x = x 1, an arbitrary set value, not exceeding 1 in absolute value.

In fact, according to given value sine is possible in the I and I negative quarters trigonometric circle(we will always assume the radius of the trigonometric circle to be equal to 1) construct the corresponding arc. It is enough to plot a segment of size k on the vertical diameter (up for k>0 and down for k

Properties 1° and 2° are usually combined in the form of the following conditional statement.

On the segment - π / 2 ≤x≤ π / 2, the sine increases from -1 to 1.

Using similar geometric reasoning, or using the formula casts sin(π - x) = sin x, it is easy to establish that in the segment π / 2 ≤x≤ 3π / 2 (i.e. in the II and III quarters) the sine decreases from 1 to -1. Segments - π/2 ≤x≤ π/2 and π/2 ≤x≤ 3π/2 together make up full circle, i.e. cover the full period of the sine. Further study of the sine becomes unnecessary, and we can claim that on any segment [- π / 2 +2kπ, π / 2 +2kπ] the sine increases from -1 to 1, and on any segment [π / 2 +2kπ, 3π / 2 +2kπ] sine decreases from 1 to -1. The sine graph is shown in Figure 11.

The study of cosine is carried out in a similar way. The main properties of cosine are:

The cos x function on a segment (i.e. in the 1st and 2nd quarters) decreases from 1 to -1. In the segment [π, 2π] (i.e. in the III and IV quarters) the cosine increases from -1 to 1. Due to periodicity, the cosine decreases from 1 to -1 on segments and increases from -1 to 1 on segments [(2k-1)π, 2kπ] (Fig. 12).

Consider the function y = tan x in the interval (- π / 2, π / 2).

The limit values ±π/2 should be excluded because tg(±π/2) does not exist.

1°. In the interval (- π / 2, π / 2) the function tg x increases.

2°. Whatever the real number k is, in the interval - - π / 2

It is easy to verify the existence and uniqueness of the arc x 1 from geometric construction, presented in drawing 13.

So, in the interval (- π / 2, π / 2) the tangent increases and, with a single value of the argument, has an arbitrary given real value. Properties 1° and 2° are briefly formulated as the following statement:

in the interval (- π / 2, π / 2) the tangent increases from -∞ to ∞.

Whatever the given (as large as you like) positive number N, tangent values are greater than N for all values of x less than π/2 and sufficiently close to π/2. Symbolically this statement is written as follows:

For values of x greater than - π / 2 and sufficiently close to - π / 2 y values of tg x

* (Often they write tan π / 2 = ∞ and say that the value of tangent π / 2 is equal to ∞. This statement in a course of elementary mathematics can only lead to ridiculous anti-scientific ideas. The symbol ∞ is not a number and cannot be the value of a function. Exact meaning, in which the symbols ±∞ should be used, is explained in the text.)

Further study of the tangent is unnecessary, because the value of the interval (- π / 2, π / 2) is equal to π, i.e. full period tangent Consequently, in any interval (- π / 2 + π, π / 2 + π) the tangent increases from -∞ to ∞, and at points x = (2k+1)π / 2 it makes sense. The tangent graph is presented in Figure 14.

The function ctg x in the interval (0, π), as well as in each of the intervals (kπ, (k+1)π) decreases from ∞ to -∞, and at the points x = kπ the cotangent has no meaning. The cotangent graph is presented in Figure 15.

Trigonometric functions of a numeric argument. Properties and graphs of trigonometric functions.

Definition1: Numeric function, given by the formula y=sin x is called sine.

This curve is called - sine wave.

Properties of the function y=sin x

2. Function value range: E(y)=[-1; 1]

3. Parity function:

y=sin x – odd,.

4. Periodicity: sin(x+2πn)=sin x, where n is an integer.

This function after a certain period accepts same values. This property of a function is called frequency. The interval is the period of the function.

For the function y=sin x the period is 2π.

The function y=sin x is periodic, with period Т=2πn, n is an integer.

Least positive period T=2π.

Mathematically, this can be written as follows: sin(x+2πn)=sin x, where n is an integer.

Definition2: The numerical function given by the formula y=cosx is called cosine.

Properties of the function y=cos x

1. Function domain: D(y)=R

2. Function value area: E(y)=[-1;1]

3. Parity function:

y=cos x – even.

4. Periodicity: cos(x+2πn)=cos x, where n is an integer.

The function y=cos x is periodic, with period Т=2π.

Definition 3: The numerical function given by the formula y=tan x is called tangent.

Properties of the function y=tg x

1. Domain of the function: D(y) - all real numbers except π/2+πk, k – integer. Because at these points the tangent is not defined.

2. Function range: E(y)=R.

3. Parity function:

y=tg x – odd.

4. Periodicity: tg(x+πk)=tg x, where k is an integer.

The function y=tg x is periodic with period π.

Definition 4: The numerical function given by the formula y=ctg x is called cotangent.

Properties of the function y=ctg x

1. Domain of definition of the function: D(y) - all real numbers except πk, k is an integer. Because at these points the cotangent is not defined.

In this lesson we will look at basic trigonometric functions, their properties and graphs, and also list main types trigonometric equations and systems. In addition, we indicate general solutions of the simplest trigonometric equations and their special cases.

This lesson will help you prepare for one of the types of tasks B5 and C1.

Preparation for the Unified State Exam in mathematics

Experiment

Lesson 10. Trigonometric functions. Trigonometric equations and their systems.

Theory

Lesson summary

We have already used the term “trigonometric function” many times. Back in the first lesson of this topic, we identified them using right triangle and single trigonometric circle. Using these methods of specifying trigonometric functions, we can already conclude that for them one value of the argument (or angle) corresponds to exactly one value of the function, i.e. we have the right to call sine, cosine, tangent and cotangent functions.

In this lesson, it's time to try to abstract from the previously discussed methods of calculating the values of trigonometric functions. Today we will move on to the usual algebraic approach working with functions, we will look at their properties and draw graphs.

As for the properties of trigonometric functions, then special attention should be noted:

The domain of definition and the range of values, because for sine and cosine there are restrictions on the range of values, and for tangent and cotangent there are restrictions on the range of definition;

The periodicity of all trigonometric functions, because We have already noted the presence of the smallest non-zero argument, the addition of which does not change the value of the function. This argument is called the period of the function and is denoted by the letter . For sine/cosine and tangent/cotangent these periods are different.

Consider the function:

1) Scope of definition;

2) Value range ;

3) The function is odd ;

Let's build a graph of the function. In this case, it is convenient to begin the construction with an image of the area that limits the graph from above by the number 1 and from below by the number , which is associated with the range of values of the function. In addition, for construction it is useful to remember the values of the sines of several basic table angles, for example, that This will allow you to build the first full “wave” of the chart and then redraw it to the right and left, taking advantage of the fact that the picture will be repeated with an offset for a period, i.e. on .

Now let's look at the function:

The main properties of this function:

1) Scope of definition;

2) Value range ;

3) Even function This implies that the graph of the function is symmetrical about the ordinate;

4) The function is not monotonic throughout its entire domain of definition;

Let's build a graph of the function. As when constructing a sine, it is convenient to start with an image of the area that limits the graph at the top with the number 1 and at the bottom with the number , which is associated with the range of values of the function. We will also plot the coordinates of several points on the graph, for which we need to remember the values of the cosines of several main table angles, for example, that with the help of these points we can build the first complete “wave” of the graph and then redraw it to the right and left, taking advantage of the fact that the picture will repeat with a period shift, i.e. on .

Let's move on to the function:

The main properties of this function:

1) Domain except , where . We have already indicated in previous lessons that it does not exist. This statement can be generalized by considering the tangent period;

2) Range of values, i.e. tangent values are not limited;

3) The function is odd ;

4) The function increases monotonically within its so-called tangent branches, which we will now see in the figure;

5) The function is periodic with a period

Let's build a graph of the function. In this case, it is convenient to start building from an image vertical asymptotes graphics at points that are not included in the definition area, i.e. etc. Next, we depict the branches of the tangent inside each of the strips formed by the asymptotes, pressing them to the left asymptote and to the right one. At the same time, do not forget that each branch increases monotonically. We depict all branches the same way, because the function has a period equal to . This can be seen from the fact that each branch is obtained by shifting the neighboring one along the abscissa axis.

And we finish with a look at the function:

The main properties of this function:

1) Domain except , where . From the table of values of trigonometric functions, we already know that it does not exist. This statement can be generalized by considering the cotangent period;

2) Range of values, i.e. cotangent values are not limited;

3) The function is odd ;

4) The function decreases monotonically within its branches, which are similar to the tangent branches;

5) The function is periodic with a period

Let's build a graph of the function. In this case, as for the tangent, it is convenient to begin the construction by depicting the vertical asymptotes of the graph at points that are not included in the definition domain, i.e. etc. Next, we depict the branches of the cotangent inside each of the stripes formed by the asymptotes, pressing them to the left asymptote and to the right one. In this case, we take into account that each branch decreases monotonically. We depict all branches similarly to the tangent in the same way, because the function has a period equal to .

Separately, it should be noted that trigonometric functions with complex arguments may have a non-standard period. It's about about functions of the form:

Their period is equal. And about the functions:

Their period is equal.

As you can see, to calculate a new period, the standard period is simply divided by the factor in the argument. It does not depend on other modifications of the function.

You can understand in more detail and understand where these formulas come from in the lesson about constructing and transforming graphs of functions.

We have come to one of the most important parts of the topic “Trigonometry”, which we will devote to solving trigonometric equations. The ability to solve such equations is important, for example, when describing oscillatory processes in physics. Let’s imagine that you have driven a few laps in a go-kart in a sports car; solving a trigonometric equation will help you determine how long you have been racing depending on the position of the car on the track.

Let's write the simplest trigonometric equation:

The solution to such an equation is the arguments whose sine is equal to . But we already know that due to the periodicity of the sine, there is an infinite number of such arguments. Thus, the solution to this equation will be, etc. The same applies to solving any other simple trigonometric equation, there will be infinite number.

Trigonometric equations are divided into several main types. Separately, we should dwell on the simplest ones, because everything else comes down to them. There are four such equations (according to the number of basic trigonometric functions). General solutions are known for them; they must be remembered.

The simplest trigonometric equations and their general solutions look like this:

Please note that the values of sine and cosine must take into account the limitations known to us. If, for example, then the equation has no solutions and the specified formula should not be applied.

In addition, the specified root formulas contain a parameter in the form of an arbitrary integer. IN school curriculum This is the only case when the solution to an equation without a parameter contains a parameter. This arbitrary integer shows that it is possible to write down an infinite number of roots of any of the above equations simply by substituting all the integers in turn.

You can get acquainted with the detailed derivation of these formulas by repeating the chapter “Trigonometric Equations” in the 10th grade algebra program.

Separately, it is necessary to pay attention to solving special cases of the simplest equations with sine and cosine. These equations look like:

Finding formulas should not be applied to them general solutions. Such equations are most conveniently solved using the trigonometric circle, which gives a simpler result than general solution formulas.

For example, the solution to the equation is . Try to get this answer yourself and solve the remaining equations indicated.

In addition to the most common type of trigonometric equations indicated, there are several more standard ones. We list them taking into account those that we have already indicated:

1) Protozoa, For example, ;

2) Special cases of the simplest equations, For example, ;

3) Equations with complex argument, For example, ;

4) Equations reduced to the simplest by derivation common multiplier , For example, ;

5) Equations reduced to their simplest by transforming trigonometric functions, For example, ;

6) Equations reduced to their simplest by substitution, For example, ;

7) Homogeneous equations , For example, ;

8) Equations that can be solved using the properties of functions, For example, . Don’t be alarmed by the fact that there are two variables in this equation; it solves itself;

As well as equations that can be solved using various methods.

In addition to solving trigonometric equations, you must be able to solve their systems.

The most common types of systems are:

1) In which one of the equations is power, For example, ;

2) Systems of simple trigonometric equations, For example, .

In today's lesson we looked at the basic trigonometric functions, their properties and graphs. We also met general formulas solutions of the simplest trigonometric equations, indicated the main types of such equations and their systems.

In the practical part of the lesson, we will examine methods for solving trigonometric equations and their systems.

Box 1.Solving special cases of the simplest trigonometric equations.

As we already said in the main part of the lesson, special cases of trigonometric equations with sine and cosine of the form:

have more simple solutions, what the formulas for general solutions give.

A trigonometric circle is used for this. Let us analyze the method for solving them using the example of the equation.

Let us depict on the trigonometric circle the point at which the cosine value is zero, which is also the coordinate along the abscissa axis. As you can see, there are two such points. Our task is to indicate what angle is equal, which corresponds to these points on the circle.

We start counting from the positive direction of the abscissa axis (cosine axis) and when setting the angle we get to the first depicted point, i.e. one solution would be this angle value. But we are still satisfied with the angle that corresponds to the second point. How to get into it?

In this lesson we will look at basic trigonometric functions, their properties and graphs, and also list basic types of trigonometric equations and systems. In addition, we indicate general solutions of the simplest trigonometric equations and their special cases.

This lesson will help you prepare for one of the types of tasks B5 and C1.

Preparation for the Unified State Exam in mathematics

Experiment

Lesson 10. Trigonometric functions. Trigonometric equations and their systems.

Theory

Lesson summary

We have already used the term “trigonometric function” many times. Back in the first lesson of this topic, we defined them using a right triangle and a unit trigonometric circle. Using these methods of specifying trigonometric functions, we can already conclude that for them one value of the argument (or angle) corresponds to exactly one value of the function, i.e. we have the right to call sine, cosine, tangent and cotangent functions.

In this lesson, it's time to try to abstract from the previously discussed methods of calculating the values of trigonometric functions. Today we will move on to the usual algebraic approach to working with functions, we will look at their properties and depict graphs.

Regarding the properties of trigonometric functions, special attention should be paid to:

Consider the function:

1) Scope of definition;

2) Value range ;

3) The function is odd ;

Let's build a graph of the function. In this case, it is convenient to begin the construction with an image of the area that limits the graph from above by the number 1 and from below by the number , which is associated with the range of values of the function. In addition, for construction it is useful to remember the values of the sines of several main table angles, for example, that this will allow you to build the first full “wave” of the graph and then redraw it to the right and left, taking advantage of the fact that the picture will be repeated with an offset by a period, i.e. on .

Now let's look at the function:

The main properties of this function:

1) Scope of definition;

2) Value range ;

3) Even function This implies that the graph of the function is symmetrical about the ordinate;

4) The function is not monotonic throughout its entire domain of definition;

Let's move on to the function:

The main properties of this function:

1) Domain except , where . We have already indicated in previous lessons that it does not exist. This statement can be generalized by considering the tangent period;

2) Range of values, i.e. tangent values are not limited;

3) The function is odd ;

4) The function increases monotonically within its so-called tangent branches, which we will now see in the figure;

5) The function is periodic with a period

Let's build a graph of the function. In this case, it is convenient to begin the construction by depicting the vertical asymptotes of the graph at points that are not included in the definition domain, i.e. etc. Next, we depict the branches of the tangent inside each of the strips formed by the asymptotes, pressing them to the left asymptote and to the right one. At the same time, do not forget that each branch increases monotonically. We depict all branches the same way, because the function has a period equal to . This can be seen from the fact that each branch is obtained by shifting the neighboring one along the abscissa axis.

And we finish with a look at the function:

The main properties of this function:

1) Domain except , where . From the table of values of trigonometric functions, we already know that it does not exist. This statement can be generalized by considering the cotangent period;

2) Range of values, i.e. cotangent values are not limited;

3) The function is odd ;

4) The function decreases monotonically within its branches, which are similar to the tangent branches;

5) The function is periodic with a period

Separately, it should be noted that trigonometric functions with complex arguments may have a non-standard period. We are talking about functions of the form:

Their period is equal. And about the functions:

Their period is equal.

As you can see, to calculate a new period, the standard period is simply divided by the factor in the argument. It does not depend on other modifications of the function.

You can understand in more detail and understand where these formulas come from in the lesson about constructing and transforming graphs of functions.

Let's write the simplest trigonometric equation:

The simplest trigonometric equations and their general solutions look like this:

Please note that the values of sine and cosine must take into account the limitations known to us. If, for example, then the equation has no solutions and the specified formula should not be applied.

In addition, the specified root formulas contain a parameter in the form of an arbitrary integer. In the school curriculum, this is the only case when the solution to an equation without a parameter contains a parameter. This arbitrary integer shows that it is possible to write down an infinite number of roots of any of the above equations simply by substituting all the integers in turn.

You can get acquainted with the detailed derivation of these formulas by repeating the chapter “Trigonometric Equations” in the 10th grade algebra program.

Separately, it is necessary to pay attention to solving special cases of the simplest equations with sine and cosine. These equations look like:

Formulas for finding general solutions should not be applied to them. Such equations are most conveniently solved using the trigonometric circle, which gives a simpler result than general solution formulas.

For example, the solution to the equation is . Try to get this answer yourself and solve the remaining equations indicated.

In addition to the most common type of trigonometric equations indicated, there are several more standard ones. We list them taking into account those that we have already indicated:

1) Protozoa, For example, ;

2) Special cases of the simplest equations, For example, ;

3) Equations with complex argument, For example, ;

4) Equations reduced to their simplest by taking out a common factor, For example, ;

5) Equations reduced to their simplest by transforming trigonometric functions, For example, ;

6) Equations reduced to their simplest by substitution, For example, ;

7) Homogeneous equations, For example, ;

8) Equations that can be solved using the properties of functions, For example, . Don’t be alarmed by the fact that there are two variables in this equation; it solves itself;

As well as equations that are solved using various methods.

In addition to solving trigonometric equations, you must be able to solve their systems.

The most common types of systems are:

1) In which one of the equations is power, For example, ;

2) Systems of simple trigonometric equations, For example, .

In today's lesson we looked at the basic trigonometric functions, their properties and graphs. We also got acquainted with the general formulas for solving the simplest trigonometric equations, indicated the main types of such equations and their systems.

In the practical part of the lesson, we will examine methods for solving trigonometric equations and their systems.

Box 1.Solving special cases of the simplest trigonometric equations.

As we already said in the main part of the lesson, special cases of trigonometric equations with sine and cosine of the form:

have simpler solutions than those given by the general solution formulas.

A trigonometric circle is used for this. Let us analyze the method for solving them using the example of the equation.

Let us depict on the trigonometric circle the point at which the cosine value is zero, which is also the coordinate along the abscissa axis. As you can see, there are two such points. Our task is to indicate what the angle that corresponds to these points on the circle is equal to.

Graphing trigonometric functions in 11th grade

Math teacher first qualification category MAOU "Gymnasium No. 37", Kazan

Spiridonova L.V.

Trigonometric functions of numeric argument
y=sin(x)+m And y=cos(x)+m
Plotting graphs of functions of the form y=sin(x+t) And y=cos(x+t)
Plotting graphs of functions of the form y=A · sin(x) And y=A · cos(x)
Examples

Trigonometric functions numeric argument.

y=sin(x)

y=cos(x)

Graphing a Function y = sin x .

Properties of the function y = sin ( x ) .

all real numbers ( R )

2. Area of changes (Area of values) ,E(y)= [ - 1; 1 ] .

3. Function y = sin ( x) odd, because sin(-x ) = - sin x

π .

sin(x+2 π ) = sin(x).

5. Continuous function

Descending: [ π /2; 3 π /2 ] .

6. Increasing: [ - π /2; π /2 ] .

Graphing a Function y = cos x .

Graph of the function y = cos x obtained by transfer

graph of function y = sin x left by π /2.

Properties of the function y = co s ( x ) .

1. The domain of definition of a function is the set

all real numbers ( R )

2. Area of change (Area of values), E(y)= [ - 1; 1 ] .

3. Function y = cos (X) even, because cos(- X ) = cos (X)

The function is periodic, with main period 2 π .

cos( X + 2 π ) = cos (X) .

5. Continuous function

Descending: [ 0 ; π ] .

6. Increasing: [ π ; 2 π ] .

Construction

graphs functions of the form

y = sin ( x ) +m

And

y = cos (X) + m.

0 , or down if m " width="640"

Parallel transfer of the graph along the Oy axis

Graph of a function y=f(x) + m it turns out parallel transfer function graphics y=f(x) , up on m units if m 0 ,

or down if m .

0 y m 1 x" width="640"

Conversion: y= sin ( x ) +m

Shift y= sin ( x ) along the axis y up if m 0

0 y m 1 x" width="640"

Conversion: y= cos ( x ) +m

Shift y= cos ( x ) along the axis y up , If m 0

Conversion: y=sin ( x ) +m

Shift y= sin ( x ) along the axis y down, If m 0

Conversion: y=cos ( x ) +m

Shift y= cos ( x ) along the axis y down if m 0

Construction

graphs functions of the form

y = sin ( x + t )

And

y = cos ( X +t )

0 and to the right if t 0." width="640"

Parallel transfer of the graph along the Ox axis

Graph of a function y = f(x + t) obtained by parallel transfer of the graph of the function y=f(x) along the axis X on |t| scale units left, If t 0

And right , If t 0.

0 y 1 x t" width="640"

Conversion: y = sin(x + t)

shift y= f(x) along the axis X left, If t 0

0 y 1 x t" width="640"

Conversion: y= cos(x + t)

shift y= f(x) along the axis X left, If t 0

Conversion: y=sin(x+t)

shift y= f(x) along the axis X right, If t 0

Conversion: y= cos(x + t)

shift y= f(x) along the axis X right, If t 0

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Plotting graphs of functions of the form y = A · sin ( x ) And y= A · cos ( x ) , at a 1 and 0 A 1

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Compression and stretching along the Ox axis

Graph of a function y=A · f(x ) we obtain by stretching the graph of the function y= f(x) with coefficient A along the Ox axis, if A 1 And compression to the Ox axis with a coefficient of 0 A .