Properties and graph of the function y sin x. Sine (sin x) and cosine (cos x) - properties, graphs, formulas. Topic: Trigonometric functions
>>Mathematics: Functions y \u003d sin x, y \u003d cos x, their properties and graphs
Functions y \u003d sin x, y \u003d cos x, their properties and graphs
In this section we discuss some properties of the functions y = sin x,y= cos x and plot their graphs.
1. Function y \u003d sin X.
Above, in § 20, we formulated a rule that allows each number t to be associated with the number cos t, i.e. characterized the function y = sin t. We note some of its properties.
Properties of the function u = sint.
The domain of definition is the set K of real numbers.
This follows from the fact that any number 2 corresponds to a point M(1) on the number circle, which has a well-defined ordinate; this ordinate is cos t.
u = sin t is an odd function.
This follows from the fact that, as was proved in § 19, for any t the equality
This means that the graph of the function and \u003d sin t, like the graph of any odd function, is symmetric with respect to the origin in the rectangular coordinate system tOi.
The function u = sin t increases on the segment 
This follows from the fact that when the point moves along the first quarter number circle the ordinate gradually increases (from 0 to 1 - see Fig. 115), and when the point moves along the second quarter of the numerical circle, the ordinate gradually decreases (from 1 to 0 - see Fig. 116).
The function u = sin t is bounded both from below and from above. This follows from the fact that, as we saw in § 19, for any t the inequality
(the function reaches this value at any point of the form 
(the function reaches this value at any point of the form
Using the obtained properties, we construct a graph of the function of interest to us. But (attention!) instead of u - sin t, we will write y \u003d sin x (after all, we are more accustomed to writing y \u003d f (x), and not u \u003d f (t)). This means that we will build a graph in the usual coordinate system хОу (and not tOy).
Let's make a table of function values \u200b\u200by - sin x:
Comment.
Here is one of the versions of the origin of the term "sine". In Latin, sinus means bend (bowstring).
The constructed graph to some extent justifies this terminology. 
The line that serves as a graph of the function y \u003d sin x is called a sinusoid. That part of the sinusoid, which is shown in Fig. 118 or 119, is called a sinusoid wave, and that part of the sinusoid, which is shown in fig. 117 is called a half-wave or arch of a sine wave.
2. Function y = cos x.
The study of the function y \u003d cos x could be carried out approximately according to the same scheme that was used above for the function y \u003d sin x. But we will choose the path that leads to the goal faster. First, we will prove two formulas that are important in themselves (you will see this in high school), but so far have only an auxiliary value for our purposes.
For any value of t, the equalities
Proof. Let the number t correspond to the point M of the numerical n circle, and the number * + - to the point P (Fig. 124; for the sake of simplicity, we took the point M in the first quarter). The arcs AM and BP are equal, respectively, and right-angled triangles OKM and OBP are also equal. Hence, O K = Ob, MK = Pb. From these equalities and from the location of the triangles OKM and OLR in the coordinate system, we draw two conclusions:
1) the ordinate of the point P both in absolute value and in sign coincides with the abscissa of the point M; it means that
2) the abscissa of the point P is equal in absolute value to the ordinate of the point M, but differs from it in sign; it means that
Approximately the same reasoning is carried out in cases where the point M does not belong to the first quarter.
Let's use the formula 
(this is the formula proved above, only instead of the variable t we use the variable x). What does this formula give us? It allows us to assert that the functions
are identical, so their graphs are the same.
Let's plot the function 
To do this, let's move on to an auxiliary coordinate system with the origin at a point (the dotted line is drawn in Fig. 125). Associate the function y \u003d sin x to new system coordinates - this will be the graph of the function 
(Fig. 125), i.e. graph of the function y - cos x. It, like the graph of the function y \u003d sin x, is called a sinusoid (which is quite natural).
Properties of the function y = cos x.
y = cos x is an even function.
The stages of construction are shown in fig. 126:
1) we build a graph of the function y \u003d cos x (more precisely, one half-wave);
2) by stretching the constructed graph from the x-axis with a coefficient of 0.5, we get one half-wave of the required graph;
3) using the resulting half-wave, we build the entire graph of the function y \u003d 0.5 cos x.
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standing water rots or freezes in the cold,
and the human mind, not finding a use for itself, languishes.
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Used technologies: problem-based learning, critical thinking, communicative communication.
Goals:
- Development of cognitive interest in learning.
- Studying the properties of the function y \u003d sin x.
- Formation of practical skills for constructing a graph of the function y \u003d sin x based on the studied theoretical material.
Tasks:
1. Use the existing potential of knowledge about the properties of the function y \u003d sin x in specific situations.
2. Apply the conscious establishment of links between the analytical and geometric models of the function y \u003d sin x.
Develop initiative, a certain readiness and interest in finding a solution; the ability to make decisions, not to stop there, to defend one's point of view.
To educate students in cognitive activity, a sense of responsibility, respect for each other, mutual understanding, mutual support, self-confidence; culture of communication.
During the classes
Stage 1. Actualization of basic knowledge, motivation for learning new material
"Lesson Entry"
There are 3 statements written on the board:
- The trigonometric equation sin t = a always has solutions.
- An odd function can be graphed using a symmetry transformation about the y-axis.
- A trigonometric function can be graphed using one main half wave.
Students discuss in pairs: Are the statements true? (1 minute). The results of the initial discussion (yes, no) are then entered into the table in the "Before" column.
The teacher sets the goals and objectives of the lesson.
2. Updating knowledge (frontally on the trigonometric circle model).
We have already met with the function s = sin t.
1) What values can the variable t take. What is the scope of this function?
2) In what interval are the values of the expression sin t. Find the largest and smallest values of the function s = sin t.
3) Solve the equation sin t = 0.
4) What happens to the ordinate of the point as it moves along the first quarter? (the ordinate increases). What happens to the ordinate of a point as it moves along the second quarter? (the ordinate gradually decreases). How does this relate to the monotonicity of the function? (the function s = sin t increases on the segment and decreases on the segment ).
5) Let's write the function s = sin t in the usual form for us y = sin x (we will build in the usual xOy coordinate system) and compile a table of values for this function.
| X | 0 | ||||||
| at | 0 | 1 | 0 |
Stage 2. Perception, comprehension, primary consolidation, involuntary memorization
Stage 4. Primary systematization of knowledge and methods of activity, their transfer and application in new situations
6. No. 10.18 (b, c)
Stage 5 Final control, correction, assessment and self-assessment
7. We return to the statements (the beginning of the lesson), discuss using the properties of the trigonometric function y \u003d sin x, and fill in the "After" column in the table.
8. D / z: item 10, Nos. 10.7(a), 10.8(b), 10.11(b), 10.16(a)
In this lesson, we will consider in detail the function y \u003d sin x, its main properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y \u003d sin t on the coordinate circle and consider the graph of the function on the circle and the 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 some simple problems using the graph of the function and its properties.
Topic: Trigonometric functions
Lesson: Function y=sinx, its main properties and graph
When considering a function, it is important to associate a single value of the function with each value of the argument. 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. The point has a single ordinate, which is called the sine of the number (Fig. 1).
Each argument value is assigned a single function value.
Obvious properties follow from the definition of the sine.
The figure shows that 
because is the ordinate of a point on the unit circle.
Consider the function graph. Let us recall the geometric interpretation of the argument. The argument is the central angle measured in radians. On the axis we will lay off real numbers or angles in radians along the axis of the corresponding function values.
For example, the angle on the unit circle corresponds to a point on the graph (Fig. 2)
We got the graph of the function on the site. But knowing the period of the sine, we can depict the graph of the function on 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 continue to the entire domain of definition.
Consider the properties of the function:
1) Domain of definition:
2) Range of values: 
3) Function odd:
4) The smallest positive period:
5) Coordinates of the points of intersection of the graph with the x-axis: 
6) Coordinates of the point of intersection of the graph with the y-axis:
7) Intervals on which the function takes positive values:
8) Intervals at which the function takes negative values:
9) Increasing intervals:
10) Descending intervals:
11) Low points: 
12) Minimum features:
13) High points: 
14) Maximum features:
We have considered the properties of a function and its graph. Properties will be repeatedly used in solving problems.
Bibliography
1. Algebra and the beginning of analysis, grade 10 (in two parts). Tutorial for educational institutions (profile level) ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and the beginning of analysis, grade 10 (in two parts). Task 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 ( tutorial for students of schools and classes with in-depth study of mathematics).-M .: Education, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. An in-depth study of algebra and mathematical analysis.-M .: Education, 1997.
5. Collection of problems in mathematics for applicants to technical universities (under the editorship of M.I.Skanavi).-M.: Higher school, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic trainer.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Tasks in Algebra and the Beginnings of Analysis (a manual for students in grades 10-11 of general educational institutions).-M .: Education, 2003.
8. Karp A.P. Collection of problems in algebra and the beginnings of analysis: textbook. allowance for 10-11 cells. with a deep study mathematics.-M.: Education, 2006.
Homework
Algebra and the Beginnings of Analysis, Grade 10 (in two parts). Task 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 to prepare for exams ().
Functiony = sinx
The graph of the function is a sinusoid.
The complete non-repeating part of a sine wave is called a sine wave.
A half wave of a sine wave is called a half wave of a sine wave (or arch).
Function Propertiesy =
sinx:
3) This is an odd function. 4) This continuous function.
6) On the segment [-π/2; π/2] the function is increasing, on the interval [π/2; 3π/2] is decreasing. 7) On intervals, the function takes positive values. 8) Intervals of increasing function: [-π/2 + 2πn; π/2 + 2πn]. 9) Minimum points of the function: -π/2 + 2πn. |
To plot a function y= sin x It is convenient to use the following scales:
On a sheet in a cell, we take the length of two cells as a unit of segment.
on axle x let's measure the length π. At the same time, for convenience, 3.14 will be represented as 3 - that is, without a fraction. Then on a sheet in a cell π will be 6 cells (three times 2 cells). And each cell will receive its natural name (from the first to the sixth): π/6, π/3, π/2, 2π/3, 5π/6, π. These are the values x.
On the y-axis, mark 1, which includes two cells.
Let's make a table of function values using our values x:
√3 | √3 |
Next, let's make a chart. You will get a half-wave, the highest point of which is (π / 2; 1). This is the graph of the function y= sin x on the segment. Let's add a symmetrical half-wave to the constructed graph (symmetrical about the origin, that is, on the segment -π). The crest of this half-wave is under the x-axis with coordinates (-1; -1). The result is a wave. This is the graph of the function y= sin x on the segment [-π; π].
It is possible to continue the wave by constructing it on the interval [π; 3π], [π; 5π], [π; 7π], etc. On all these segments, the graph of the function will look the same as on the segment [-π; π]. You will get a continuous wavy line with the same waves.
Functiony = cosx.
The graph of the function is a sine wave (sometimes called a cosine wave).
Function Propertiesy = cosx:
1) The domain of the function is the set of real numbers. 2) The range of function values is the segment [–1; one] 3) This is an even function. 4) This is a continuous function. 5) Coordinates of the points of intersection of the graph: 6) The function decreases on the interval, on the interval [π; 2π] - increases. 7) On intervals [-π/2 + 2πn; π/2 + 2πn] the function takes positive values. 8) Increase intervals: [-π + 2πn; 2πn]. 9) Minimum points of the function: π + 2πn. 10) The function is limited from above and below. Lowest value functions -1, 11) This is a periodic function with a period of 2π (T = 2π) |
Functiony = mf(x).
Take the previous function y= cos x. As you already know, its graph is a sine wave. If we multiply the cosine of this function by a certain number m, then the wave will stretch from the axis x(or shrink, depending on the value of m).
This new wave will be the graph of the function y = mf(x), where m is any real number.
Thus, the function y = mf(x) is the usual function y = f(x) multiplied by m.
If am< 1, то синусоида сжимается к оси x by coefficientm. If am > 1, then the sinusoid is stretched from the axisx by coefficientm.
Performing stretching or compression, you can first build only one half-wave of the sinusoid, and then complete the entire graph.
Functiony= f(kx).
If the function y=mf(x) leads to stretching of the sinusoid from the axis x or compression to the axis x, then the function y = f(kx) leads to expansion from the axis y or compression to the axis y.
And k is any real number.
At 0< k< 1 синусоида растягивается от оси y by coefficientk. If ak > 1, then the sinusoid is compressed to the axisy by coefficientk.
When composing a graph of this function, you can first build one half-wave of a sinusoid, and then complete the entire graph using it.
Functiony = tgx.
Function Graph y=tg x is the tangentoid.
It is enough to build a part of the graph on the interval from 0 to π/2, and then you can continue it symmetrically on the interval from 0 to 3π/2.
Function Propertiesy = tgx:
Functiony = ctgx
Function Graph y=ctg x is also a tangentoid (it is sometimes called a cotangentoid).
Function Propertiesy = ctgx:
"Yoshkar-Ola College of Service Technologies"
Construction and study of the graph of the trigonometric function y=sinx in spreadsheetMS excel
/methodological development/
Yoshkar - Ola
Topic. Construction and study of the graph of a trigonometric functiony = sinx in spreadsheet MS Excel
Lesson type– integrated (acquisition of new knowledge)
Goals:
Didactic purpose - explore the behavior of graphs of a trigonometric functiony= sinxdepending on the coefficients using a computer
Tutorials:
1. Find out the change in the graph of the trigonometric function y= sin x depending on coefficients
2. Show the introduction of computer technology in teaching mathematics, the integration of two subjects: algebra and computer science.
3. To form the skills of using computer technology in mathematics lessons
4. Strengthen the skills of researching functions and plotting their graphs
Developing:
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 the increase general level student development
educators :
1. Cultivate independence, accuracy, diligence
2. Foster a culture of dialogue
Forms of work in the lesson - combined
Didactic equipment and equipment:
1. Computers
2. Multimedia projector
4. Handout
5. Presentation slides
During the classes
I. Organization of the beginning of the lesson
Greeting students and guests
· Prepare for the lesson
II. Goal-setting and actualization of the topic
It takes a lot of time to study a function and build its graph, you have to perform a lot of cumbersome calculations, this is not convenient, computer technologies come to the rescue.
Today we will learn how to build graphs of trigonometric functions in the MS Excel 2007 spreadsheet environment.
The topic of our lesson is “Construction and study of the graph of a trigonometric function y= sinx in a spreadsheet"
From the course of algebra, we know the scheme for studying a function and constructing its graph. Let's remember how to do it.
slide 2
Function Study Scheme
1. Function domain (D(f))
2. Value area of the function Е(f)
3. Definition of parity
4. Periodicity
5. Function zeros (y=0)
6. Intervals of constant sign (y>0, y<0)
7. Intervals of monotonicity
8. Function extremes
III. Primary assimilation of new educational material
Open MS Excel 2007.
Let's plot the function y=sin x
Plotting in a spreadsheetMS excel 2007
The graph of this function will be built on the segment xЄ [-2π; 2π]
We will take the values of the argument with a step , 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 the handout).
1. Find the value of the function at the point x \u003d -2P. For the rest, the editor automatically calculates the corresponding function values for the corresponding values of the argument.
2. Now we have a table with argument and function values. With this data, we have to plot this function using the Chart Wizard.
3. To build a graph, you need to select the desired data range, rows with argument values and functions
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We write the conclusions in a notebook (Slide 5)
Conclusion. The graph of the function of the form y=sinx+k is obtained from the graph of the function y=sinx using parallel translation along the y-axis by k units
If k >0, then the graph is shifted up by k units
If k<0, то график смещается вниз на k единиц
Construction and study of the view functiony=k*sinx,k- const
Task 2. At work Sheet2 plot functions in one coordinate system y= sinx y=2* sinx, y= * sinx, on the interval (-2π; 2π) and see how 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 obtained graphs. We analyze together with the students the behavior of the graph of the 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 see how the graph changes.
We compare the obtained graphs. We analyze together with the students the behavior of the graph of the 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 the conclusions in a notebook (Slide 11)
Conclusion. The graph of the function of the form y \u003d sin (x + k) is obtained from the graph of the function y \u003d sinx using parallel translation along the OX axis by k units
If k >1, then the graph is shifted to the right along the OX axis
If 0 IV. Primary consolidation of acquired knowledge Differentiated cards with a task to build and study a function using a graph Y=6*sin(x) Y=1-2
sinX Y=-
sin(3x+)
1.
Domain 2.
Scope of value 3.
Parity 4.
Periodicity 5.
Constancy intervals 6.
gapsmonotony Function rises Function decreases 7.
Function extremes Minimum Maximum V. Homework organization Plot the function y=-2*sinх+1 , investigate and check the correctness of the construction in the Microsoft Excel spreadsheet environment. (Slide 12) VI. Reflection
