Cos x 1 2 graph. Graph of the function y=ctg(x)

The main trigonometric functions are the functions y=sin(x), y=cos(x), y=tg(x), y=ctg(x). Let's consider each of them separately.

Y = sin(x)

Graph of the function y=sin(x).

Basic properties:

3. The function is odd.

Y = cos(x)

Graph of the function y=cos(x).

Basic properties:

1. The domain of definition is the entire numerical axis.

2. Function limited. The set of values is the segment [-1;1].

3. The function is even.

4.The function is periodic with the smallest positive period equal to 2*π.

Y = tan(x)

Graph of the function y=tg(x).

Basic properties:

1. The domain of definition is the entire numerical axis, with the exception of points of the form x=π/2 +π*k, where k is an integer.

3. The function is odd.

Y = ctg(x)

Graph of the function y=ctg(x).

Basic properties:

1. The domain of definition is the entire numerical axis, with the exception of points of the form x=π*k, where k is an integer.

2. Unlimited function. The set of values is the entire number line.

3. The function is odd.

4. The function is periodic with the smallest positive period equal to π.

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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 according to 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 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

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 page 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 computer, 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 (

Lesson and presentation on the topic: "Function y=cos(x). Definition and graph of the function"

Additional materials
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Teaching aids and simulators in the Integral online store for grade 10
Algebraic problems with parameters, grades 9–11
Software environment "1C: Mathematical Constructor 6.1"

What we will study:
1. Definition.
2. Graph of a function.
3. Properties of the function Y=cos(X).
4. Examples.

Definition of the cosine function y=cos(x)

Guys, we have already met the function Y=sin(X).

Let's remember one of the ghost formulas: sin(X + π/2) = cos(X).

Thanks to this formula, we can claim that the functions sin(X + π/2) and cos(X) are identical, and their function graphs coincide.

The graph of the function sin(X + π/2) is obtained from the graph of the function sin(X) by parallel translation π/2 units to the left. This will be the graph of the function Y=cos(X).

The graph of the function Y=cos(X) is also called a sine wave.

Properties of the function cos(x)

The domain of definition is the set of real numbers.
The function is even. Let's remember the definition of an even function. A function is called even if the equality y(-x)=y(x) holds. As we remember from the ghost formulas: cos(-x)=-cos(x), the definition is fulfilled, then cosine is an even function.
The function Y=cos(X) decreases on the segment and increases on the segment [π; 2π]. We can verify this in the graph of our function.
The function Y=cos(X) is limited from below and from above. This property follows from the fact that
-1 ≤ cos(X) ≤ 1
The smallest value of the function is -1 (at x = π + 2πk). The largest value of the function is 1 (at x = 2πk).
The function Y=cos(X) is a continuous function. Let's look at the graph and make sure that our function has no breaks, this means continuity.
Range of values: segment [- 1; 1]. This is also clearly visible from the graph.
Function Y=cos(X) is a periodic function. Let's look at the graph again and see that the function takes the same values at certain intervals.

Examples with the cos(x) function

1. Solve the equation cos(X)=(x - 2π) 2 + 1

Solution: Let's build 2 graphs of the function: y=cos(x) and y=(x - 2π) 2 + 1 (see figure).

y=(x - 2π) 2 + 1 is a parabola shifted to the right by 2π and upward by 1. Our graphs intersect at one point A(2π;1), this is the answer: x = 2π.

2. Plot the function Y=cos(X) for x ≤ 0 and Y=sin(X) for x ≥ 0

Solution: To build the required graph, let's build two graphs of the function in “pieces”. First piece: y=cos(x) for x ≤ 0. Second piece: y=sin(x)
for x ≥ 0. Let us depict both “pieces” on one graph.

3. Find the greatest and smallest value functions Y=cos(X) on the interval [π; 7π/4]

Solution: Let's build a graph of the function and consider our segment [π; 7π/4]. The graph shows that the highest and lowest values are achieved at the ends of the segment: at points π and 7π/4, respectively.
Answer: cos(π) = -1 – the smallest value, cos(7π/4) = the largest value.

4. Graph the function y=cos(π/3 - x) + 1

Solution: cos(-x)= cos(x), then the desired graph will be obtained by moving the graph of the function y=cos(x) π/3 units to the right and 1 unit up.

Problems to solve independently

1)Solve the equation: cos(x)= x – π/2.
2) Solve the equation: cos(x)= - (x – π) 2 - 1.
3) Graph the function y=cos(π/4 + x) - 2.
4) Graph the function y=cos(-2π/3 + x) + 1.
5) Find the largest and smallest value of the function y=cos(x) on the segment.
6) Find the largest and smallest value of the function y=cos(x) on the segment [- π/6; 5π/4].

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 ().