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 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.
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 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 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 full “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 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 tangent branches 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 area, 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. 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.
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 in the race 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 an infinite number of them.
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 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 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 the angle that corresponds to these points on the circle is equal to.
 |
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:
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 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 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 full “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 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 tangent branches 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 area, 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. 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.
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 in the race 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 an infinite number of them.
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 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.
|
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?
1. Trigonometric functions represent elementary functions, whose argument is corner. Trigonometric functions describe the relationships between sides and acute angles in a right triangle. The areas of application of trigonometric functions are extremely diverse. For example, any periodic processes can be represented as a sum of trigonometric functions (Fourier series). These functions often appear when solving differential and functional equations.
2. Trigonometric functions include the following 6 functions: sinus, cosine, tangent,cotangent, secant And cosecant. For each specified functions there is an inverse trigonometric function.
3. It is convenient to introduce the geometric definition of trigonometric functions using unit circle. The figure below shows a circle with radius r=1. The point M(x,y) is marked on the circle. The angle between the radius vector OM and the positive direction of the Ox axis is equal to α.
4. Sinus angle α is the ratio of the ordinate y of the point M(x,y) to the radius r:
sinα=y/r.
Since r=1, then the sine is equal to the ordinate of the point M(x,y).
5. Cosine angle α is the ratio of the abscissa x of the point M(x,y) to the radius r:
cosα=x/r
6. Tangent angle α is the ratio of the ordinate y of a point M(x,y) to its abscissa x:
tanα=y/x,x≠0
7. Cotangent angle α is the ratio of the abscissa x of a point M(x,y) to its ordinate y:
cotα=x/y,y≠0
8. Secant angle α is the ratio of the radius r to the abscissa x of the point M(x,y):
secα=r/x=1/x,x≠0
9. Cosecant angle α is the ratio of the radius r to the ordinate y of the point M(x,y):
cscα=r/y=1/y,y≠0
10. In the unit circle, the projections x, y, the points M(x,y) and the radius r form a right triangle, in which x,y are the legs, and r is the hypotenuse. Therefore, the above definitions of trigonometric functions in the appendix to right triangle are formulated as follows:
Sinus angle α is the ratio of the opposite side to the hypotenuse.
Cosine angle α is the ratio of the adjacent leg to the hypotenuse.
Tangent angle α is called the opposite leg to the adjacent one.
Cotangent angle α is called the adjacent side to the opposite side.
Secant angle α is the ratio of the hypotenuse to the adjacent leg.
Cosecant angle α is the ratio of the hypotenuse to the opposite leg.
11. Graph of the sine function
y=sinx, domain of definition: x∈R, range of values: −1≤sinx≤1
12. Graph of the cosine function
y=cosx, domain: x∈R, range: −1≤cosx≤1
13. Graph of the tangent function 14. Graph of the cotangent function 15. Graph of the secant function
y=tanx, range of definition: x∈R,x≠(2k+1)π/2, range of values: −∞ 
y=cotx, domain: x∈R,x≠kπ, range: −∞ 
y=secx, domain: x∈R,x≠(2k+1)π/2, range: secx∈(−∞,−1]∪∪)
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