Video lesson “Function y = cos x, its properties and graph” presents visual material to study this topic. The manual presents the features of the function, its properties, as well as descriptions of solving problems in which knowledge about the properties of the cosine is applied. With the help of a video lesson, it is easier for a teacher to provide the required knowledge and develop the skills of students. Visual material can help improve lesson effectiveness by providing deeper understanding of the material and better memorization, as well as freeing up lesson time for individual work.
Using a video lesson gives the teacher an advantage to present the material more effectively. The manual can be used only for clarity, accompanying the teacher’s explanation or as an independent part of the lesson, giving the teacher the opportunity to improve individual work with students. The demonstrated plotting of graphs and transformations using animation effects become more understandable for students and help them master problem solving skills using of this material. Highlighting and voicing the properties of a function using video tutorial tools helps you remember them better.
The demo begins by introducing the theme name. To construct a graph of the function y = cos x, students are reminded of the formula for reducing cos x = sin (x + π/2), which indicates that the graphs of the functions y = cos x and y = sin (x + π/2) are identically equal . To plot a graph of the function y= sin (x+π/2), a coordinate plane is used, on the abscissa axis of which the point -π/2 is marked. If we take this point as the origin of coordinates for constructing sin graphics x, then this graph is also a graph of the function y = sin (x + π/2) for the origin. That is, the graph of the function y = cos x is shifted by π/2 along the abscissa axis of the graph of the function y = sin x. It is obvious that the graph of the function y = cos x is also a sinusoid. Its location allows us to draw conclusions about the properties of the function.
The first property of a function is about the domain of definition. Obviously, the domain of definition of the function will be the entire number line, that is, D(f)=(- ∞;+∞).
The second property of a function indicates the parity of the function. Students are reminded of the material studied in grade 9, in which the condition for the parity of a function was indicated. For even function the equality f(-x)=f(x) is true. Speaking about the parity of the cosine function, it should be noted that the graph of this function is symmetrical about the ordinate axis. You can demonstrate the properties of the function in the figure, which shows coordinate plane unit circle. In the first and fourth quarters, points are marked that are symmetrical relative to the abscissa axis. The cosine is determined by the abscissa of the point, so for two points L(t) and N(-t) the abscissas are the same. Therefore cos (-t)= cos t.
The third property marks the intervals of decrease and increase of a function. The property states that the function decreases on the segment , and on the segment [π;2π] the cosine increases. The figure shows a graph of the function, which clearly shows the area of decreasing and increasing functions.
It is obvious that the function y = cos x increases on each segment [π+2πk;2π+2πk]. Descending segments in general view look like this, where k is an integer.
The fourth property notes that the cosine function is bounded above and below. Similar to sine, we can note the limited values of cosine -1<= cos х<=1. Поэтому функция является ограниченной.
The fifth property specifies the smallest and largest values of the function. In this case, the smallest value -1 is achieved at any point x=π+2πk, and the largest value 1 is achieved at any point x=2πk.
The sixth property indicates the continuity of the function y = cos x. The figure showing the graph shows that this function has no breaks throughout the entire domain of definition.
The seventh property of the function states that the set of values y = cos x is located on the segment [-1;1].
Next, examples are considered in which it is necessary to use knowledge about the properties of the function y = cos x. In the first example it is necessary to solve the equation cos x=1-2. The solution to this equation will be the intersection points of the function graphs, which are represented by the expressions of the right and left sides of the equation, that is, y = cos x and y = 1-x 2. Obviously, the graph of the first equation is the sinusoid demonstrated earlier in the topic. The graph of the second function is a parabola, the vertex of which is located at the point (0;1). Having plotted the graphs of each function, the figure for this problem shows that the only point of intersection of the two graphs will be point B(0;1).
In the second example, you need to build and read a graph of a function that is defined on the segment x<π/2 выражением sinx, а на отрезке х>=π/2 by expression cosx. In the figure accompanying the solution to the example, a graph of the function у=sinx is plotted on the segment [-3π/2; π/2]. In this case, at the point π/2 the function does not take on a value. On the segment [π/2; 3π/2] a fragment of the function y = cos x is constructed. Obviously, the constructed fragments will be repeated throughout the entire definition domain. The following describes how the function is read. It is noted that this means to describe its properties. The properties of this function are listed - the domain of definition (-∞;+∞), the absence of signs of even or odd for the entire domain of definition, the function being bounded both above and below. The largest value of the function will be 1, and the smallest -1. It is also noted that there is a discontinuity at the point x=π/2, a set of function values (-1;1).
The video lesson “Function y = cos x, its properties and graph” is used in a mathematics lesson on this topic as visual material. Also, this video can be useful for teachers who teach remotely to develop the necessary skills in students. The material can be recommended for independent review by students who have not mastered the topic well enough and require additional training.
TEXT DECODING:
Before constructing a graph of the function y = cos x, remember the reduction formula, according to which cos x = sin(x + 14ПЂ2) "> (the cosine of the argument x is equal to the sine of the argument x plus pi by two). This means that the functions y = cos x And
y = sin(x +14ПЂ2)"> are identically equal, therefore their graphs coincide.
To graph the function y = sin(x +14ПЂ2)"> we will need an auxiliary coordinate system with the origin at point B(-14ПЂ2"> ; 0) (at the point BE with coordinates minus pi by two, zero). If we plot the function y = sin x in the new coordinate system, we get a graph of the function
y = sin(x +14ПЂ2)"> or the graph of the function y = cos x, since their graphs coincide (see Fig. 1).
Since the graph of the function y = cos x is obtained from the sine graph using parallel translation over a distance14ПЂ2"> in the negative direction, then the graph of this function is also a sinusoid.
The graph of the function y = cos x gives a clear idea of the properties of this function.
PROPERTY 1. Domain is the set of all real numbers or D (f) = (-14в€ћ"> ; +14в€ћ">) (de from ef is equal to the interval from minus infinity to plus infinity).
PROPERTY 2. The function y = cos x is even.
In the 9th grade lessons, we learned that the function y = f (x), x ϵX (the y is equal to eff of x, where x belongs to the set x is large) is called even if for any value x from the set X the equality
f (- x) = f (x) (eff from minus x is equal to ef from x).
PROPERTY 3.On the interval [ 0 ; π ] (from zero to pi) the function decreases and increases on the segment [ π ; 2π ] (from pi to two pi) and so on.
We can draw a general conclusion: the function y = cos x increases on the segment
[π 14+2ПЂk ">;142ПЂ+2ПЂk "> ] (from pi plus two pi ka to two pi plus two pi ka), and decreases on the segment [14 2ПЂk">;14ПЂ+2ПЂk]"> (from two peaks to pi plus two peaks), where (ka belongs to the set of integers).
PROPERTY 4. The function is limited above and below.
PROPERTY 5. The smallest value of the function is equal to minus one and is achieved at any point of the form x =14ПЂ+2ПЂk"> (or you can write y name = - 1); the largest value is 1 and is achieved at any point of the form x =142ПЂk">
(or you can write y max. = 1).
PROPERTY 6. The function y = cos x is continuous.
PROPERTY 7. The set of values of a function is a segment from minus one to one (or you can write E(f) = [ - 1; 1]).
Let's look at examples.
EXAMPLE 1.Solve the equation cos x= 1 - x 2 (cosine x is equal to one minus x squared).
Solution. Let's solve this equation graphically. In one coordinate system we will construct two graphs of functions: y = cos x and y = 1 - x 2. Function graph
y = 1 - x 2 is a parabola whose branches are directed downward, since the coefficient of x squared is negative. (see Fig. 2) The constructed graphs have only one common point - this is point B(0; 1) (be with coordinates zero, one).
Solution. We will build the schedule “piece by piece”. First, let's plot part of the graph of the function y = sin x on the open beam (-14в€ћ"> ;14ПЂ2">), then in the same coordinate system on the ray [14 ПЂ2"> ; +14в€ћ">) we will construct part of the graph of the function y = cos x. We will obtain the graph of the function y = f(x).
Let's read the graph of this function (this means listing the properties of the function):
- The domain of definition is the set of all real numbers, i.e.
D(f) = (-14в€ћ; + в€ћ)"> (i.e. de from ef is equal to the interval from minus infinity to plus infinity).
- The function is neither even nor odd.
- The function is limited both below and above.
- The smallest value of the function is equal to minus one (there are infinitely many such points), the largest value of the function is equal to one (there are also infinitely many such points).
- The function has a discontinuity at the point x =14ПЂ 2"> .
- The set of function values is the segment from minus one to one.
Back forward
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Lesson topic: “Function y=cosx”
Lesson #1
Lesson objectives: To familiarize students with the properties of a function
Lesson objectives.
Educational – the formation of functional concepts using visual material, the formation of skills in constructing graphs of the function y=cosx, the formation of skills in fluent reading of graphs, the ability to reflect the properties of a function on a graph.
During the classes
| № | Lesson stage | Slide show | Time |
| 1 | Organizing time. Greetings | ||
| 2 | Announcing the topic and purpose of the lesson | ||
| 3 | Updating of reference knowledge Performing oral exercises. |
Frontal survey |
|
| 4 | Presentation of new material The task of constructing a graph of y = cosx on a segment Discussion of the properties of the function y =cosx on an interval The task of constructing a sketch of a graph of the function y = cosх Discussion of the properties of the function y = cosx |
Entering properties into a table |
|
| 5 | Solving problems according to textbook No. 708, No. 709 |
The solution is accompanied by slide No. 4 | |
| 6 | The task is to construct a graph of a function with a shift along the ordinate axis and along the abscissa axis. Discussion of function properties |
||
| 7 | Independent work using the textbook | №710 (1;3), №711 (1;3), №711 (1;3) |
|
| Summarizing. Lesson summary. Grading. |
|||
| 9 | Homework | §40 No. 710(2;4), No. 711(2;4), No. 711(2;4). Construct graphs of functions y =cosx on and describe the properties of this function. Additional No. 717 (1) |
Purpose of the lesson: To familiarize students with the properties of the function y=cosx, learning to build a graph of the function y=cosx, read this graph, use the properties and graph of the function when solving equations and inequalities.
2. The announcement of the topic and purpose of the lesson is accompanied by slide No. 2
3. Updating basic knowledge
Performing oral exercises.
- Review the definition of trigonometric functions and the signs of the values of these functions.
- Draw students' attention to the fact that for any real number you can indicate the corresponding point on the unit circle, and therefore its abscissa and ordinate, i.e. cosine and sine of a number x: y = cosx and y = sinx, the domain of which is all real numbers.
Then students answer the questions:
- For what values of x does the function y=cosx take on the value 0? 1? -1?
- Can the function y=cosx take a value greater than 1 or less than -1?
- At what values of x does the function y=cosx take on the largest (smallest) value?
- What is the set of values of the function y=cosx?
The answers to these and the following questions are accompanied by an illustration on the unit circle.
Having repeated the signs of the values of trigonometric functions in each quarter of the coordinate plane, students are asked to show several points on the unit circle corresponding to numbers whose cosine is a positive (negative) number. Then answer the questions:
1) What sign does the function y=cosx have if x=, x=,
0<х<, 0<х<, <х<, <х<2.5?
2) Indicate several values of x at which the values of the function y = cosx are positive and negative.
3) Is it possible to name all the values of a number whose cosine is positive or negative?
4) Is it possible to name all the values of the argument x for which the values of the function y = cosx are positive and negative?
5) Even or odd function y = cosx.
6) What is the period of this function?
4. Presentation of new material.
Generalization and concretization of knowledge acquired earlier: the study of the domain of definition, set of values, parity, periodicity allows you to construct a graph first on a segment, then on a segment, and then on the entire number line. The explanation is accompanied by slide number 3.
Then students learn to draw a sketch of the graph of the function y = cosx using points (0;1), (;0),
(:-1), (;0), (;1) and summarize the properties of the function, recording them in a table.
Let's check using slide number 4.
(At this stage, supporting notes are issued (Appendix 1))
5. Consolidation of primary knowledge.
Using a sketch of the graph of the function y=cosx, students answer questions No. 708, using a table of properties of the function y=cosx, answer questions No. 709
6. The task of constructing a graph of a function with a shift along the ordinate axis and along the abscissa axis.
1. Slide No. 5, 6
During the conversation, the properties of these functions are discussed.
7. Independent work using the textbook
№710(1;3), №711(1;3), №711(1;3), №710
Divide this segment into two segments so that on one of them the function y = cosx increases, and on the other it decreases:
Descending; - increases
Descending; - increases
Using the increasing or decreasing property of the function y = cosx, compare the numbers:
On the segment the function y = cosx decreases; , hence, .
On the segment the function y = cosx increases;
<, следовательно, cos < cos
Find all roots of the equation belonging to the segment:
1) cosx = x = ±+2 n, n Z
Answer: ; ; .
2) cosx = - x = ± 
8. Summing up.
Grading.
During the lesson we learned how to build a graph of the function y = cosx, read the properties of this graph, build a sketch of the graph, and solve problems related to the use of the graph and properties of the function y = cosx.
9. Homework.
§40 No. 710(2;4), No. 711(2;4), No. 711(2;4). Construct graphs of functions y =cosx on and describe the properties of this function.
Additional No. 717(1).
Topic: “Function y=cosx”
Lesson #2
Lesson objectives: Review the rules for constructing a graph of the function у=cosx, learn how to transform a graph, read this graph, use the properties and graph of a function when solving equations and inequalities.
Lesson objectives.
Educational – the formation of functional representations using visual material, the formation of skills in plotting graphs of the function y=cosx under various transformations, the formation of skills in fluent reading of graphs, the ability to reflect the properties of a function on a graph.
Developmental – developing the ability to analyze and generalize acquired knowledge. Formation of logical thinking.
Educational - to intensify interest in acquiring new knowledge, fostering a graphic culture, developing precision and accuracy when making drawings.
Equipped with: multimedia projector, screen, Microsoft Windows 98/Me/2000/XP operating system, MS Office 2003 program: Power Point, Microsoft Word, Microsoft Excel.
During the classes
| № | Lesson stage | Slide show | Time |
| 1 | Organizing time. Greetings | 1 | |
| 2 | Announcing the topic and purpose of the lesson | 2 | |
| 3 | Checking homework | No. 717(1), Slide No. 7 |
5 |
| 4 | Presentation of new material The task of constructing a graph by squeezing and stretching to the OX axis Discussion of the properties of the function y =k cosx for k>1 and 0 The task of constructing a graph by squeezing and stretching an ori op-amp Discussion of the properties of the function y = cos(k x) for k>1 and 0 |
Slide No. 8, 9 |
12 |
| 5 | Consolidation of primary knowledge. Solving problems according to the textbook №713(1;3), №715(1) №716(1) |
No. 717(2) textbook p. 208. When solving No. 715(1), No. 716(1), use the constructed graph of the function y = cos2x. Slide No. 10 | 5 |
| 6 | The task is to construct a graph of a function that is symmetrical about the abscissa axis. 1. Organizational moment. Greetings. 2. The announcement of the topic and purpose of the lesson is accompanied by slide No. 2. 3. Checking homework 4. Presentation of new material 1. The task of constructing a graph by squeezing and stretching to the OX axis. Discussion of the properties of the function y =k cosx for k>1 and 0 Slide number 8 2. The task of constructing a graph by squeezing and stretching to the axis of the op-amp. Discussion of the properties of the function y = cos(kx) for k>1 and 0 Slide number 9 5. Consolidation of primary knowledge Solving problems according to textbook No. 713(1;3), No. 715(1) No. 716(1) We check task No. 715(1) No. 716(1) using slide No. 10 6. The task of constructing a graph of a function symmetrical about the abscissa axis Discussion of function properties .
Slide No. 11 (use supporting summary (Appendix 1)) 7. Independent work Solving test problems .
(Half of the students solve tests in XL (Appendix 2), at the computers, the other half on handouts (Appendix 3). Then the students change places.) 8. Lesson summary. As a result of studying the topic, students learned to build a graph of the function y = cosх, read the properties of a function, build graphs of a function using various transformations, read the properties of graphs with transformations, solve simple problems using graphs and properties of the function y = cosx. Grading. 9. Homework. §40 No. 717(3), No. 713(4), No. 715(4), No. 716(2). Additional No. 719(2) (Check slide No. 13) At the beginning of the next lesson, you can invite students to complete the work of constructing graphs on ready-made handouts ( |
In this lesson we will look in detail at the function y = cos x, its main properties and graph. At the beginning of the lesson we will give the definition of the trigonometric function y = cost on the coordinate circle and consider the graph of the function on the circle and line. Let's show the periodicity of this function on the graph and consider the main properties of the function. At the end of the lesson, we will solve several simple problems using the graph of a function and its properties.
Topic: Trigonometric functions
Lesson: Function y=cost, its basic properties and graph
A function is a law according to which each value of an independent argument is associated with a single value of the function.
Let's remember function definition Let t- any real number. There is only one point corresponding to it M on the number circle. At the point M there is a single abscissa. It is called the cosine of the number t. Each argument value t only one function value corresponds (Fig. 1).
The central angle is numerically equal to the arc value in radians, i.e. number Therefore, the argument can be either a real number or an angle in radians.
If we can determine for each value, then we can build a graph of the function
You can get the graph of a function in another way. According to reduction formulas 
so the cosine graph is a sine wave shifted along the axis x to the left (Fig. 2).
Function properties
1) Scope of definition:
2) Range of values: 
3) Even function:
4) Smallest positive period:
5) Coordinates of the points of intersection with the abscissa axis:
6) Coordinates of the point of intersection with the ordinate axis:
7) Intervals at which the function takes positive values:
8) Intervals at which the function takes negative values:
9) Increasing intervals:
10) Decreasing intervals:
11) Minimum points: 
12) Minimum function: .
13) Maximum points: 
14) Maximum functions:
We have looked at the basic properties and graph of the function. Next, they will be used to solve problems.
Bibliography
1. Algebra and beginning of analysis, grade 10 (in two parts). Textbook for general education institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
3. Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I. Algebra and mathematical analysis for grade 10 (textbook for students of schools and classes with in-depth study of mathematics). - M.: Prosveshchenie, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. In-depth study of algebra and mathematical analysis.-M.: Education, 1997.
5. Collection of problems in mathematics for applicants to higher educational institutions (edited by M.I. Skanavi). - M.: Higher School, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic simulator.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Problems on algebra and principles of analysis (a manual for students in grades 10-11 of general education institutions). - M.: Prosveshchenie, 2003.
8. Karp A.P. Collection of problems on algebra and principles of analysis: textbook. allowance for 10-11 grades. with depth studied Mathematics.-M.: Education, 2006.
Homework
Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
№№ 16.6, 16.7, 16.9.
Additional web resources
3. Educational portal for exam preparation ().