Now we will look at the question of how to plot trigonometric functions of multiple angles ωx, Where ω - some positive number.

To graph a function y = sin ωx Let's compare this function with the function we have already studied y = sin x. Let's assume that when x = x 0 function y = sin x takes the value equal to 0. Then

y 0 = sin x 0 .

Let us transform this relationship as follows:

Therefore, the function y = sin ωx at X = x 0 / ω takes the same value at 0 , which is the same as the function y = sin x at x = x 0 . This means that the function y = sin ωx repeats its meanings in ω times more often than the function y = sin x. Therefore, the graph of the function y = sin ωx obtained by "compressing" the graph of the function y = sin x V ω times along the x axis.

For example, the graph of a function y = sin 2x obtained by “compressing” a sinusoid y = sin x twice along the x-axis.

Graph of a function y = sin x / 2 is obtained by “stretching” the sinusoid y = sin x twice (or “compressing” it by 1 / 2 times) along the x axis.

Since the function y = sin ωx repeats its meanings in ω times more often than the function
y = sin x, then its period is ω times less than the period of the function y = sin x. For example, the period of the function y = sin 2x equals 2π/2 = π , and the period of the function y = sin x / 2 equals π / x/ 2 = 4π .

It is interesting to study the behavior of the function y = sin ax using the example of animation, which can be very easily created in the program Maple:

Graphs of other trigonometric functions of multiple angles are constructed in a similar way. The figure shows the graph of the function y = cos 2x, which is obtained by “compressing” the cosine wave y = cos x twice along the x-axis.

Graph of a function y = cos x / 2 obtained by “stretching” the cosine wave y = cos x doubled along the x axis.

In the figure you see the graph of the function y = tan 2x, obtained by “compressing” the tangentsoids y = tan x twice along the x-axis.

Graph of a function y = tg x/ 2 , obtained by “stretching” the tangentsoids y = tan x doubled along the x axis.

And finally, the animation performed by the program Maple:

Exercises

1. Construct graphs of these functions and indicate the coordinates of the points of intersection of these graphs with the coordinate axes. Determine the periods of these functions.

A). y = sin 4x/ 3 G). y = tan 5x/ 6 and). y = cos 2x/ 3

b). y=cos 5x/ 3 d). y = ctg 5x/ 3 h). y=ctg x/ 3

V). y = tan 4x/ 3 e). y = sin 2x/ 3

2. Determine the periods of functions y = sin (πх) And y = tg (πх/2).

3. Give two examples of functions that take all values ​​from -1 to +1 (including these two numbers) and change periodically with period 10.

4 *. Give two examples of functions that take all values ​​from 0 to 1 (including these two numbers) and change periodically with a period π/2.

5. Give two examples of functions that take all real values ​​and vary periodically with period 1.

6 *. Give two examples of functions that accept all negative values and zero, but not accepted positive values and change periodically with a period of 5.

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
Solving 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 numeric 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 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 segment [- 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) – 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 – highest value, sin(5π/4) = 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]

"Yoshkar-Ola College of Service Technologies"

Construction and study of the graph 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 implementation computer technology in teaching mathematics, integrating 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

Lesson progress

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 into 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

4..jpg" width="667" height="236 src=">

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)

https://pandia.ru/text/78/510/images/image008_35.jpg" width="649" height="281 src=">

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 of definition

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

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.