Function y=sinx, its main properties and graph. Functions y = sin x, y = cos x, their properties and graphs - Knowledge Hypermarket The graph of the function y is equal to sine x

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Construction and study of the graph of the trigonometric function y=sinx in a spreadsheetMS Excel

/methodological development/

Yoshkar – Ola

Subject. Construction and study of the graph of a trigonometric functiony = sinx in MS Excel spreadsheet

Lesson type– integrated (gaining new knowledge)

Goals:

Didactic purpose - explore the behavior of trigonometric function graphsy= sinxdepending on odds using a computer

Educational:

1. Find out the change in the graph of a trigonometric function y= sin x depending on odds

2. Show the introduction of computer technology in teaching mathematics, the integration of two subjects: algebra and computer science.

3. Develop skills in using computer technology in mathematics lessons

4. Strengthen the skills of studying functions and constructing their graphs

Educational:

1. To develop students’ cognitive interest in academic disciplines and the ability to apply their knowledge in practical situations

2. Develop the ability to analyze, compare, highlight the main thing

3. Contribute to improving the overall level of student development

Educating :

1. Foster independence, accuracy, and hard work

2. Foster a culture of dialogue

Forms of work in the lesson - combined

Didactic facilities and equipment:

1. Computers

2. Multimedia projector

4. Handouts

5. Presentation slides

During the classes

I. Organization of the beginning of the lesson

· Greeting students and guests

· Mood for the lesson

II. Goal setting and topic updating

It takes a lot of time to study a function and build its graph, you have to perform a lot of cumbersome calculations, it’s not convenient, computer technology comes to the rescue.

Today we will learn how to build graphs of trigonometric functions in the spreadsheet environment of MS Excel 2007.

The topic of our lesson is “Construction and study of the graph of a trigonometric function y= sinx in a table processor"

From the algebra course we know the scheme for studying a function and constructing its graph. Let's remember how to do this.

Slide 2

Function study scheme

1. Domain of the function (D(f))

2. Range of function E(f)

3. Determination of parity

4. Frequency

5. Zeros of the function (y=0)

6. Intervals of constant sign (y>0, y<0)

7. Periods of monotony

8. Extrema of the function

III. Primary assimilation of new educational material

Open MS Excel 2007.

Let's plot the function y=sin x

Building graphs in a spreadsheet processorMS Excel 2007

We will plot the graph of this function on the segment xЄ [-2π; 2π]

We will take the values ​​of the argument in steps , to make the graph more accurate.

Since the editor works with numbers, let’s convert radians to numbers, knowing that P ≈ 3.14 . (translation table in handout).

1. Find the value of the function at the point x=-2P. For the rest, the editor calculates the corresponding function values ​​automatically.

2. Now we have a table with the values ​​of the argument and function. With this data, we have to plot this function using the Chart Wizard.

3. To build a graph, you need to select the required data range, lines with argument and function values

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We write down the conclusions in a notebook (Slide 5)

Conclusion. The graph of a function of the form y=sinx+k is obtained from the graph of the function y=sinx using parallel translation along the axis of the op-amp by k units

If k >0, then the graph shifts up by k units

If k<0, то график смещается вниз на k единиц

Construction and study of a function of the formy=k*sinx,k- const

Task 2. At work Sheet2 draw graphs of functions in one coordinate system y= sinx y=2* sinx, y= * sinx, on the interval (-2π; 2π) and watch how the appearance of the graph changes.

(In order not to re-set the value of the argument, let's copy the existing values. Now you need to set the formula and build a graph using the resulting table.)

We compare the resulting graphs. Together with students, we analyze the behavior of the graph of a trigonometric function depending on the coefficients. (Slide 6)

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We compare the resulting graphs. Together with students, we analyze the behavior of the graph of a trigonometric function depending on the coefficients. (Slide 8)

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We write down the conclusions in a notebook (Slide 11)

Conclusion. The graph of a function of the form y=sin(x+k) is obtained from the graph of the function y=sinx using parallel translation along the OX axis by k units

If k >1, then the graph shifts to the right along the OX axis

If 0

IV. Primary consolidation of acquired knowledge

Differentiated cards with a task to construct and study a function using a graph

V. Homework organization

Plot a graph of the function y=-2*sinх+1, examine and check the correctness of the construction in a Microsoft Excel spreadsheet environment. (Slide 12)

VI. Reflection

In this lesson we will take a detailed look at the function y = sin x, its basic properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y = sin t 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=sinx, its basic properties and graph

When considering a function, it is important to associate each argument value with a single function value. This law of correspondence and is called a function.

Let us define the correspondence law for .

Any real number corresponds to a single point on the unit circle. A point has a single ordinate, which is called the sine of the number (Fig. 1).

Each argument value is associated with a single function value.

Obvious properties follow from the definition of sine.

The figure shows that because is the ordinate of a point on the unit circle.

Consider the graph of the function. Let us recall the geometric interpretation of the argument. The argument is the central angle, measured in radians. Along the axis we will plot real numbers or angles in radians, along the axis the corresponding values ​​of the function.

For example, an angle on the unit circle corresponds to a point on the graph (Fig. 2)

We have obtained a graph of the function in the area. But knowing the period of the sine, we can depict the graph of the function over the entire domain of definition (Fig. 3).

The main period of the function is This means that the graph can be obtained on a segment and then continued throughout the entire domain of definition.

Consider the properties of the function:

1) Scope of definition:

2) Range of values:

3) Odd function:

4) Smallest positive period:

5) Coordinates of the points of intersection of the graph with the abscissa axis:

6) Coordinates of the point of intersection of the graph 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 functions:

13) Maximum points:

14) Maximum functions:

We looked at the properties of the function and its graph. The properties will be used repeatedly when solving 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 advanced 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.

№№ 16.4, 16.5, 16.8.

Additional web resources

3. Educational portal for preparing for exams ().

How to graph the function y=sin x? First, let's look at the sine graph on the interval.

We take a single segment 2 cells long in the notebook. On the Oy axis we mark one.

For convenience, we round the number π/2 to 1.5 (and not to 1.6, as required by the rounding rules). In this case, a segment of length π/2 corresponds to 3 cells.

On the Ox axis we mark not single segments, but segments of length π/2 (every 3 cells). Accordingly, a segment of length π corresponds to 6 cells, and a segment of length π/6 corresponds to 1 cell.

With this choice of a unit segment, the graph depicted on a sheet of notebook in a box corresponds as much as possible to the graph of the function y=sin x.

Let's make a table of sine values ​​on the interval:

We mark the resulting points on the coordinate plane:

Since y=sin x is an odd function, the sine graph is symmetrical with respect to the origin - point O(0;0). Taking this fact into account, we continue plotting the graph to the left, then the points -π:

The function y=sin x is periodic with period T=2π. Therefore, the graph of a function taken on the interval [-π;π] is repeated an infinite number of times to the right and to the left.

In this lesson we will take a detailed look at the function y = sin x, its basic properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y = sin t 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=sinx, its basic properties and graph

When considering a function, it is important to associate each argument value with a single function value. This law of correspondence and is called a function.

Let us define the correspondence law for .

Any real number corresponds to a single point on the unit circle. A point has a single ordinate, which is called the sine of the number (Fig. 1).

Each argument value is associated with a single function value.

Obvious properties follow from the definition of sine.

The figure shows that because is the ordinate of a point on the unit circle.

Consider the graph of the function. Let us recall the geometric interpretation of the argument. The argument is the central angle, measured in radians. Along the axis we will plot real numbers or angles in radians, along the axis the corresponding values ​​of the function.

For example, an angle on the unit circle corresponds to a point on the graph (Fig. 2)

We have obtained a graph of the function in the area. But knowing the period of the sine, we can depict the graph of the function over the entire domain of definition (Fig. 3).

The main period of the function is This means that the graph can be obtained on a segment and then continued throughout the entire domain of definition.

Consider the properties of the function:

1) Scope of definition:

2) Range of values:

3) Odd function:

4) Smallest positive period:

5) Coordinates of the points of intersection of the graph with the abscissa axis:

6) Coordinates of the point of intersection of the graph 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 functions:

13) Maximum points:

14) Maximum functions:

We looked at the properties of the function and its graph. The properties will be used repeatedly when solving 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 advanced 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.

№№ 16.4, 16.5, 16.8.

Additional web resources

3. Educational portal for preparing for exams ().

We found out that the behavior of trigonometric functions, and the functions y = sin x in particular, on the entire number line (or for all values ​​of the argument X) is completely determined by its behavior in the interval 0 < X < π / 2 .

Therefore, first of all, we will plot the function y = sin x exactly in this interval.

Let's make the following table of values ​​of our function;

By marking the corresponding points on the coordinate plane and connecting them with a smooth line, we obtain the curve shown in the figure

The resulting curve could also be constructed geometrically, without compiling a table of function values y = sin x .

1. Divide the first quarter of a circle of radius 1 into 8 equal parts. The ordinates of the dividing points of the circle are the sines of the corresponding angles.

2.The first quarter of the circle corresponds to angles from 0 to π / 2 . Therefore, on the axis X Let's take a segment and divide it into 8 equal parts.

3. Let's draw straight lines parallel to the axes X, and from the division points we construct perpendiculars until they intersect with horizontal lines.

4. Connect the intersection points with a smooth line.

Now let's look at the interval π / 2 < X < π .
Each argument value X from this interval can be represented as

x = π / 2 + φ

Where 0 < φ < π / 2 . According to reduction formulas

sin( π / 2 + φ ) = cos φ = sin ( π / 2 - φ ).

Axis points X with abscissas π / 2 + φ And π / 2 - φ symmetrical to each other about the axis point X with abscissa π / 2 , and the sines at these points are the same. This allows us to obtain a graph of the function y = sin x in the interval [ π / 2 , π ] by simply symmetrically displaying the graph of this function in the interval relative to the straight line X = π / 2 .

Now using the property odd parity function y = sin x,

sin(- X) = - sin X,

it is easy to plot this function in the interval [- π , 0].

The function y = sin x is periodic with a period of 2π ;. Therefore, to construct the entire graph of this function, it is enough to continue the curve shown in the figure to the left and right periodically with a period 2π .

The resulting curve is called sinusoid . It represents the graph of the function y = sin x.

The figure illustrates well all the properties of the function y = sin x , which we have previously proven. Let us recall these properties.

1) Function y = sin x defined for all values X , so its domain is the set of all real numbers.

2) Function y = sin x limited. All the values ​​it accepts are between -1 and 1, including these two numbers. Consequently, the range of variation of this function is determined by the inequality -1 < at < 1. When X = π / 2 + 2k π the function takes the largest values ​​equal to 1, and for x = - π / 2 + 2k π - the smallest values ​​equal to - 1.

3) Function y = sin x is odd (the sine wave is symmetrical about the origin).

4) Function y = sin x periodic with period 2 π .

5) In intervals 2n π < x < π + 2n π (n is any integer) it is positive, and in intervals π + 2k π < X < 2π + 2k π (k is any integer) it is negative. At x = k π the function goes to zero. Therefore, these values ​​of the argument x (0; ± π ; ±2 π ; ...) are called function zeros y = sin x

6) At intervals - π / 2 + 2n π < X < π / 2 + 2n π function y = sin x increases monotonically, and in intervals π / 2 + 2k π < X < 3π / 2 + 2k π it decreases monotonically.

You should pay special attention to the behavior of the function y = sin x near the point X = 0 .

For example, sin 0.012 ≈ 0.012; sin(-0.05) ≈ -0,05;

sin 2° = sin π 2 / 180 = sin π / 90 ≈ 0,03 ≈ 0,03.

At the same time, it should be noted that for any values ​​of x

| sin x| < | x | . (1)

Indeed, let the radius of the circle shown in the figure be equal to 1,
a / AOB = X.

Then sin x= AC. But AC< АВ, а АВ, в свою очередь, меньше длины дуги АВ, на которую опирается угол X. The length of this arc is obviously equal to X, since the radius of the circle is 1. So, at 0< X < π / 2

sin x< х.

Hence, due to the oddness of the function y = sin x it is easy to show that when - π / 2 < X < 0

| sin x| < | x | .

Finally, when x = 0

| sin x | = | x |.

Thus, for | X | < π / 2 inequality (1) has been proven. In fact, this inequality is also true for | x | > π / 2 due to the fact that | sin X | < 1, a π / 2 > 1

Exercises

1.According to the graph of the function y = sin x determine: a) sin 2; b) sin 4; c) sin (-3).

2.According to the function graph y = sin x determine which number from the interval
[ - π / 2 , π / 2 ] has a sine equal to: a) 0.6; b) -0.8.

3. According to the graph of the function y = sin x determine which numbers have a sine,
equal to 1/2.

4. Find approximately (without using tables): a) sin 1°; b) sin 0.03;
c) sin (-0.015); d) sin (-2°30").