Graph of the function y sinx 3. Construction and study of the graph of the trigonometric function y=sinx in the MS Excel spreadsheet processor. Sine problems for independent solution

"Yoshkar-Ola College of Service Technologies"

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)

https://pandia.ru/text/78/510/images/image005_66.gif" width="16" height="41 src=">x , on the interval (-2π; 2π) and watch how the appearance of the graph changes.

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

	Y=6*sin(x)	Y=1-2 sinX	Y=- sin(3x+)
1. Domain
2. Range of value
3. Parity
4. Periodicity
5. Intervals of sign constancy
6. Gapsmonotony
Function increases
Function decreases
7. Extrema of the function
Minimum
Maximum

V. Homework organization

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

VI. Reflection

Lesson and presentation on the topic: "Function y=sin(x). Definitions and properties"

Additional materials
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Manuals and simulators in the Integral online store for grade 10 from 1C
We solve problems in geometry. Interactive construction tasks for grades 7-10
Software environment "1C: Mathematical Constructor 6.1"

What we will study:

• Properties of the function Y=sin(X).
• Function graph.
• How to build a graph and its scale.
• Examples.

Properties of sine. Y=sin(X)

Guys, we have already become acquainted with trigonometric functions of a numerical argument. Do you remember them?

Let's take a closer look at the function Y=sin(X)

Let's write down some properties of this function:
1) The domain of definition is the set of real numbers.
2) The function is odd. Let's remember the definition of an odd function. A function is called odd if the equality holds: y(-x)=-y(x). As we remember from the ghost formulas: sin(-x)=-sin(x). The definition is fulfilled, which means Y=sin(X) is an odd function.
3) The function Y=sin(X) increases on the segment and decreases on the segment [π/2; π]. When we move along the first quarter (counterclockwise), the ordinate increases, and when we move through the second quarter it decreases.

4) The function Y=sin(X) is limited from below and from above. This property follows from the fact that
-1 ≤ sin(X) ≤ 1
5) The smallest value of the function is -1 (at x = - π/2+ πk). The largest value of the function is 1 (at x = π/2+ πk).

Let's use properties 1-5 to plot the function Y=sin(X). We will build our graph sequentially, applying our properties. Let's start building a graph on the segment.

Particular attention should be paid to the scale. On the ordinate axis it is more convenient to take a unit segment equal to 2 cells, and on the abscissa axis it is more convenient to take a unit segment (two cells) equal to π/3 (see figure).

Plotting the sine function x, y=sin(x)

Let's calculate the values ​​of the function on our segment:

Let's build a graph using our points, taking into account the third property.

Conversion table for ghost formulas

Let's use the second property, which says that our function is odd, which means that it can be reflected symmetrically with respect to the origin:

We know that sin(x+ 2π) = sin(x). This means that on the interval [- π; π] the graph looks the same as on the segment [π; 3π] or or [-3π; - π] and so on. All we have to do is carefully redraw the graph in the previous figure along the entire x-axis.

The graph of the function Y=sin(X) is called a sinusoid.

Let's write a few more properties according to the constructed graph:
6) The function Y=sin(X) increases on any segment of the form: [- π/2+ 2πk; π/2+ 2πk], k is an integer and decreases on any segment of the form: [π/2+ 2πk; 3π/2+ 2πk], k – integer.
7) Function Y=sin(X) is a continuous function. Let's look at the graph of the function and make sure that our function has no breaks, this means continuity.
8) Range of values: segment [- 1; 1]. This is also clearly visible from the graph of the function.
9) Function Y=sin(X) - periodic function. Let's look at the graph again and see that the function takes the same values ​​at certain intervals.

Examples of problems with sine

1. Solve the equation sin(x)= x-π

Solution: Let's build 2 graphs of the function: y=sin(x) and y=x-π (see figure).
Our graphs intersect at one point A(π;0), this is the answer: x = π

2. Graph the function y=sin(π/6+x)-1

Solution: The desired graph will be obtained by moving the graph of the function y=sin(x) π/6 units to the left and 1 unit down.

Solution: Let's plot the function and consider our segment [π/2; 5π/4].
The graph of the function shows that the largest and smallest values ​​are achieved at the ends of the segment, at points π/2 and 5π/4, respectively.
Answer: sin(π/2) = 1 – the largest value, sin(5π/4) = the smallest value.

Sine problems for independent solution

• Solve the equation: sin(x)= x+3π, sin(x)= x-5π
• Graph the function y=sin(π/3+x)-2
• Graph the function y=sin(-2π/3+x)+1
• Find the largest and smallest value of the function y=sin(x) on the segment
• Find the largest and smallest value of the function y=sin(x) on the interval [- π/3; 5π/6]

Stretching the graph y=sinx along the y axis. Given the function y=3sinx. To build its graph, you need to stretch the graph y=sinx so that E(y): (-3; 3).

Picture 7 from the presentation “Build a graph of a function” for algebra lessons on the topic “Graph of a function”

Dimensions: 960 x 720 pixels, format: jpg. To download a free picture for an algebra lesson, right-click on the image and click “Save image as...”. To display pictures in the lesson, you can also download for free the entire presentation “Build a graph of a function.ppt” with all the pictures in a zip archive. The archive size is 327 KB.

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Graph of a function

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There are a total of 25 presentations in the topic

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 sinusoid is symmetrical about the origin).

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

5) In 2n intervals 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").

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, let’s 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.