Centered at a point A.
α
- angle expressed in radians.
Definition
Sine (sin α) is a trigonometric function depending on the angle α between the hypotenuse and the leg right triangle, equal to the ratio of the length of the opposite side |BC| to the length of the hypotenuse |AC|.
Cosine (cos α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the hypotenuse |AC|.
Accepted notations
;
;
.
;
;
.
Graph of the sine function, y = sin x
Graph of the cosine function, y = cos x
Properties of sine and cosine
Periodicity
Functions y = sin x and y = cos x periodic with period 2π.
Parity
The sine function is odd. The cosine function is even.
Domain of definition and values, extrema, increase, decrease
The sine and cosine functions are continuous in their domain of definition, that is, for all x (see proof of continuity). Their main properties are presented in the table (n - integer).
| y = sin x | y = cos x | |
| Scope and continuity | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Range of values | -1 ≤ y ≤ 1 | -1 ≤ y ≤ 1 |
| Increasing | ||
| Descending | ||
| Maxima, y = 1 | ||
| Minima, y = - 1 | ||
| Zeros, y = 0 | ||
| Intercept points with the ordinate axis, x = 0 | y = 0 | y = 1 |
Basic formulas
Sum of squares of sine and cosine
Formulas for sine and cosine from sum and difference
;
;
Formulas for the product of sines and cosines
Sum and difference formulas
Expressing sine through cosine
;
;
;
.
Expressing cosine through sine
;
;
;
.
Expression through tangent
; .
When , we have:
;
.
At :
;
.
Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for certain values of the argument. 
Expressions through complex variables
;
Euler's formula
Expressions through hyperbolic functions
;
;
Derivatives
; . Deriving formulas > > >
Derivatives of nth order:
{ -∞ <
x < +∞ }
Secant, cosecant
Inverse functions
Inverse functions to sine and cosine are arcsine and arccosine, respectively.
Arcsine, arcsin
Arccosine, arccos
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
>> Periodicity of functions y = sin x, y = cos x
§ 11. Periodicity of functions y = sin x, y = cos x
In the previous paragraphs we used seven properties functions: domain of definition, even or odd, monotonicity, boundedness, largest and smallest values, continuity, range of values of a function. We used these properties either to construct a graph of a function (this happened, for example, in § 9), or to read the constructed graph (this happened, for example, in § 10). Now the opportune moment has come to introduce one more (eighth) property of functions, which is clearly visible in the constructions above. graphs functions y = sin x (see Fig. 37), y = cos x (see Fig. 41).
Definition. A function is called periodic if there is a nonzero number T such that for any x in the set the double condition holds: equality:
The number T that satisfies the specified condition is called the period of the function y = f(x).
It follows that, since for any x the equalities are valid:
then the functions y = sin x, y = cos x are periodic and the number is 2 P serves as a period for both functions.
The periodicity of a function is the promised eighth property of functions.
Now look at the graph of the function y = sin x (Fig. 37). To build a sine wave, it is enough to plot one of its waves (on a segment and then shift this wave along the x axis by. As a result, using one wave we will build the entire graph.
Let's look from the same point of view at the graph of the function y = cos x (Fig. 41). We see that here, to plot a graph, it is enough to first plot one wave (for example, on the segment
And then move it along the x axis by
Summarizing, we draw the following conclusion.
If the function y = f(x) has a period T, then to build a graph of the function you must first build a branch (wave, part) of the graph on any interval of length T (most often take an interval with ends at points and then shift this branch along the x axis to the right and left to T, 2T, ZT, etc.
A periodic function has infinitely many periods: if T is a period, then 2T is a period, and ZT is a period, and -T is a period; In general, a period is any number of the form KT, where k = ±1, ±2, ± 3... Usually they try, if possible, to isolate the smallest positive period; it is called the main period.
So, any number of the form 2pk, where k = ±1, ± 2, ± 3, is the period of the functions y = sinn x, y = cos x; 2n is the main period of both functions.
Example. Find the main period of the function:
A) Let T be the main period of the function y = sin x. Let's put
For the number T to be a period of a function, the identity But, since we are talking about finding the main period, we get
b) Let T be the main period of the function y = cos 0.5x. Let's put f(x)=cos 0.5x. Then f(x + T)=cos 0.5(x + T)=cos (0.5x + 0.5T).
For the number T to be a period of the function, the identity cos (0.5x + 0.5T) = cos 0.5x must hold.
This means 0.5t = 2pp. But, since we are talking about finding the main period, we get 0.5T = 2 l, T = 4 l.
A generalization of the results obtained in the example is the following statement: main period of the function 
A.G. Mordkovich Algebra 10th grade
Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for the year guidelines discussion programs Integrated LessonsGoal: summarize and systematize students’ knowledge on the topic “Periodicity of Functions”; develop skills in applying the properties of a periodic function, finding the smallest positive period of a function, constructing graphs of periodic functions; promote interest in studying mathematics; cultivate observation and accuracy.
Equipment: computer, multimedia projector, task cards, slides, clocks, tables of ornaments, elements of folk crafts
“Mathematics is what people use to control nature and themselves.”
A.N. Kolmogorov
During the classes
I. Organizational stage.
Checking students' readiness for the lesson. Report the topic and objectives of the lesson.
II. Checking homework.
We check homework using samples and discuss the most difficult points.
III. Generalization and systematization of knowledge.
1. Oral frontal work.
Theory issues.
1) Form a definition of the period of the function
2) Name the smallest positive period of the functions y=sin(x), y=cos(x)
3). What is the smallest positive period of the functions y=tg(x), y=ctg(x)
4) Using a circle, prove the correctness of the relations:
y=sin(x) = sin(x+360º)
y=cos(x) = cos(x+360º)
y=tg(x) = tg(x+18 0º)
y=ctg(x) = ctg(x+180º)
tg(x+π n)=tgx, n € Z
ctg(x+π n)=ctgx, n € Z
sin(x+2π n)=sinx, n € Z
cos(x+2π n)=cosx, n € Z
5) How to plot a periodic function?
Oral exercises.
1) Prove the following relations
a) sin(740º) = sin(20º)
b) cos(54º) = cos(-1026º)
c) sin(-1000º) = sin(80º)
2. Prove that an angle of 540º is one of the periods of the function y= cos(2x)
3. Prove that an angle of 360º is one of the periods of the function y=tg(x)
4. Transform these expressions so that the angles included in them do not exceed 90º in absolute value.
a) tg375º
b) ctg530º
c) sin1268º
d) cos(-7363º)
5. Where did you come across the words PERIOD, PERIODICITY?
Student answers: A period in music is a structure in which a more or less complete musical thought is presented. Geological period- part of an era and is divided into epochs with a period from 35 to 90 million years.
Half-life of a radioactive substance. Periodic fraction. Periodicals are printed publications that appear within strictly defined deadlines. Periodic table Mendeleev.
6. The figures show parts of the graphs of periodic functions. Determine the period of the function. Determine the period of the function.
Answer: T=2; T=2; T=4; T=8.
7. Where in your life have you encountered the construction of repeating elements?
Student answer: Elements of ornaments, folk art.
IV. Collective problem solving.
(Solving problems on slides.)
Let's consider one of the ways to study a function for periodicity.
This method avoids the difficulties associated with proving that a particular period is the smallest, and also eliminates the need to touch upon questions about arithmetic operations on periodic functions and the periodicity of a complex function. The reasoning is based only on the definition of a periodic function and on the following fact: if T is the period of the function, then nT(n?0) is its period.
Problem 1. Find the smallest positive period of the function f(x)=1+3(x+q>5)
Solution: Assume that the T-period of this function. Then f(x+T)=f(x) for all x € D(f), i.e.
1+3(x+T+0.25)=1+3(x+0.25)
(x+T+0.25)=(x+0.25)
Let's put x=-0.25 we get
(T)=0<=>T=n, n € Z
We have obtained that all periods of the function in question (if they exist) are among the integers. Let's choose the smallest positive number among these numbers. This 1 . Let's check whether it will actually be a period 1 .
f(x+1) =3(x+1+0.25)+1
Since (T+1)=(T) for any T, then f(x+1)=3((x+0.25)+1)+1=3(x+0.25)+1=f(x ), i.e. 1 – period f. Since 1 is the smallest of all positive integers, then T=1.
Problem 2. Show that the function f(x)=cos 2 (x) is periodic and find its main period.
Problem 3. Find the main period of the function
f(x)=sin(1.5x)+5cos(0.75x)
Let us assume the T-period of the function, then for any X the ratio is valid
sin1.5(x+T)+5cos0.75(x+T)=sin(1.5x)+5cos(0.75x)
If x=0, then
sin(1.5T)+5cos(0.75T)=sin0+5cos0
sin(1.5T)+5cos(0.75T)=5
If x=-T, then
sin0+5cos0=sin(-1.5T)+5cos0.75(-T)
5= – sin(1.5T)+5cos(0.75T)
| sin(1.5T)+5cos(0.75T)=5 – sin(1.5T)+5cos(0.75T)=5 |
Adding it up, we get:
10cos(0.75T)=10
2π n, n € Z
Let us choose the smallest positive number from all the “suspicious” numbers for the period and check whether it is a period for f. This number
f(x+)=sin(1.5x+4π )+5cos(0.75x+2π )= sin(1.5x)+5cos(0.75x)=f(x)
This means that this is the main period of the function f.
Problem 4. Let’s check whether the function f(x)=sin(x) is periodic
Let T be the period of the function f. Then for any x
sin|x+Т|=sin|x|
If x=0, then sin|Т|=sin0, sin|Т|=0 Т=π n, n € Z.
Let's assume. That for some n the number π n is the period
the function under consideration π n>0. Then sin|π n+x|=sin|x|
This implies that n must be both an even and an odd number, but this is impossible. Therefore, this function is not periodic.
Task 5. Check if the function is periodic
f(x)= 
Let T be the period of f, then
, hence sinT=0, Т=π n, n € Z. Let us assume that for some n the number π n is indeed the period of this function. Then the number 2π n will be the period
Since the numerators are equal, their denominators are equal, therefore
This means that the function f is not periodic.
Work in groups.
Tasks for group 1.
Tasks for group 2.
Check if the function f is periodic and find its fundamental period (if it exists).
f(x)=cos(2x)+2sin(2x)
Tasks for group 3.
At the end of their work, the groups present their solutions.
VI. Summing up the lesson.
Reflection.
The teacher gives students cards with drawings and asks them to color part of the first drawing in accordance with the extent to which they think they have mastered the methods of studying a function for periodicity, and in part of the second drawing - in accordance with their contribution to the work in the lesson.
VII. Homework
1). Check if the function f is periodic and find its fundamental period (if it exists)
b). f(x)=x 2 -2x+4
c). f(x)=2tg(3x+5)
2). The function y=f(x) has a period T=2 and f(x)=x 2 +2x for x € [-2; 0]. Find the value of the expression -2f(-3)-4f(3.5)
Literature/
- Mordkovich A.G. Algebra and beginnings of analysis with in-depth study.
- Mathematics. Preparation for the Unified State Exam. Ed. Lysenko F.F., Kulabukhova S.Yu.
- Sheremetyeva T.G. , Tarasova E.A. Algebra and beginning analysis for grades 10-11.
The number T such that for any x F(x + T) = F(x). This number T is called the period of the function.
There may be several periods. For example, the function F = const takes the same value for any value of the argument, and therefore any number can be considered its period.
Usually you are interested in the smallest non-zero period of a function. For brevity, it is simply called a period.
A classic example of periodic functions is trigonometric: sine, cosine and tangent. Their period is the same and equal to 2π, that is, sin(x) = sin(x + 2π) = sin(x + 4π) and so on. However, of course, trigonometric functions are not the only periodic ones.
For simple, basic functions, the only way to determine whether they are periodic or non-periodic is through calculation. But for complex functions there are already several simple rules.
If F(x) is with period T, and a derivative is defined for it, then this derivative f(x) = F′(x) is also a periodic function with period T. After all, the value of the derivative at point x is equal to the tangent of the tangent angle of the graph of its antiderivative at this point to the x-axis, and since it is repeated periodically, it must be repeated. For example, the derivative of the function sin(x) is equal to cos(x), and it is periodic. Taking the derivative of cos(x) gives you –sin(x). The frequency remains unchanged.
However, the opposite is not always true. Thus, the function f(x) = const is periodic, but its antiderivative F(x) = const*x + C is not.
If F(x) is a periodic function with period T, then G(x) = a*F(kx + b), where a, b, and k are constants and k is not equal to zero - is also a periodic function, and its period is T/k. For example, sin(2x) is a periodic function, and its period is π. This can be visually represented this way: by multiplying x by some number, you seem to compress the functions horizontally exactly that many times
If F1(x) and F2(x) are periodic functions, and their periods are equal to T1 and T2, respectively, then the sum of these functions can also be periodic. However, its period will not be a simple sum of periods T1 and T2. If the result of division T1/T2 is rational number, then the sum of the functions is periodic, and its period is equal to the least common multiple (LCM) of the periods T1 and T2. For example, if the period of the first function is 12, and the period of the second is 15, then the period of their sum will be equal to LCM (12, 15) = 60.
This can be visually represented as follows: the functions come with different “step widths,” but if the ratio of their widths is rational, then sooner or (more precisely, through the LCM of steps), they will become equal again, and their sum will begin a new period.
However, if the ratio of periods is , then the total function will not be periodic at all. For example, let F1(x) = x mod 2 (the remainder when x is divided by 2), and F2(x) = sin(x). T1 here will be equal to 2, and T2 will be equal to 2π. The ratio of periods is equal to π - irrational number. Therefore, the function sin(x) + x mod 2 is not periodic.
Sources:
- Theoretical information about functions
Many mathematical functions have one feature that makes them easier to construct: periodicity, that is, the repeatability of the graph on a coordinate grid at regular intervals.
Instructions
The most famous periodic functions in mathematics are sine and cosine. These functions have a wave-like and fundamental period equal to 2P. Also a special case of a periodic function is f(x)=const. Any number is suitable for position x; this function does not have a main period, since it is a straight line.
In general, a function is periodic if there is an integer N that is non-zero and satisfies the rule f(x)=f(x+N), thus ensuring repeatability. The period of a function is the smallest number N, but not zero. That is, for example, the function sin x is equal to the function sin (x+2ПN), where N=±1, ±2, etc.
Sometimes a function may have a multiplier (for example, sin 2x), which will increase or decrease the period of the function. In order to find the period by
The video lesson “Periodicity of functions y = sin x, y = cos x” reveals the concept of periodicity of a function, considers a description of examples of solving problems in which the concept of periodicity of a function is used. This video lesson is a visual aid for explaining the topic to students. Also, this manual can become an independent part of the lesson, freeing up the teacher to conduct individual work with students.
Visibility in presenting this topic is very important. To represent the behavior of a function, plotting it, it must be visualized. It is not always possible to make constructions using a blackboard and chalk in such a way that they are understandable to all students. In the video tutorial, it is possible to highlight parts of the drawing with color when constructing, and make transformations using animation. Thus, the constructions become more understandable to most students. Also, the video lesson features contribute to better memorization of the material.
The demonstration begins by introducing the topic of the lesson, as well as reminding students of material learned in previous lessons. In particular, the list of properties that were identified in the functions y = sin x, as well as y = cos x, is summarized. Among the properties of the functions under consideration, the domain of definition, range of values, parity (oddness), other features are noted - boundedness, monotonicity, continuity, points of least (greatest) value. Students are informed that in this lesson another property of a function is studied - periodicity.
The definition of a periodic function y=f(x), where xϵX, in which the condition f(x-Т)= f(x)= f(x+Т) for some Т≠0 is presented. Otherwise, the number T is called the period of the function.
For the sine and cosine functions under consideration, the fulfillment of the condition is checked using reduction formulas. It is obvious that the form of the identity sin(x-2π)=sinx=sin(x+2π) corresponds to the form of the expression defining the condition of periodicity of the function. The same equality can be noted for cosine cos(x-2π)= cos x= cos (x+2π). This means that these trigonometric functions are periodic.
It is further noted how the property of periodicity helps to build graphs of periodic functions. The function y = sin x is considered. A coordinate plane is constructed on the screen, on which abscissas from -6π to 8π are marked with a step of π. A part of the sine graph is plotted on the plane, represented by one wave on the segment. The figure demonstrates how the graph of a function is formed over the entire definition domain by shifting the constructed fragment, resulting in a long sinusoid.
A graph of the function y = cos x is constructed using the property of its periodicity. To do this, a coordinate plane is constructed in the figure, on which a fragment of the graph is depicted. It is noted that such a fragment is usually constructed on the segment [-π/2;3π/2]. Similar to the graph of the sine function, the construction of the cosine graph is performed by shifting the fragment. As a result of the construction, a long sinusoid is formed.
Graphing a periodic function has features that can be used. Therefore they are given in a generalized form. It is noted that to construct a graph of such a function, a branch of the graph is first constructed on a certain interval of length T. Then it is necessary to shift the constructed branch to the right and left by T, 2T, 3T, etc. At the same time, another feature of the period is pointed out - for any integer k≠0, the number kT is also the period of the function. However, T is called the main period, since it is the smallest of all. For trigonometric functions For sine and cosine, the main period is 2π. However, the periods are also 4π, 6π, etc.
Next, it is proposed to consider finding the main period of the function y = cos 5x. The solution begins with the assumption that T is the period of the function. This means that the condition f(x-T)= f(x)= f(x+T) must be met. In this identity, f(x)= cos 5x, and f(x+T)=cos 5(x+T)= cos (5x+5T). In this case, cos (5x+5T)= cos 5x, therefore 5T=2πn. Now you can find T=2π/5. The problem is solved.
In the second problem, you need to find the main period of the function y=sin(2x/7). It is assumed that the main period of the T function for a given function is f(x)= sin(2x/7), and after a period f(x+T)=sin(2x/7)(x+T)= sin(2x/7 +(2/7)T). after reduction we get (2/7)Т=2πn. However, we need to find the main period, so we take the smallest value (2/7)T=2π, from which we find T=7π. The problem is solved.
At the end of the demonstration, the results of the examples are summarized to form a rule for determining the basic period of the function. It is noted that for the functions y=sinkx and y=coskx the main periods are 2π/k.
The video lesson “Periodicity of functions y = sin x, y = cos x” can be used in a traditional mathematics lesson to increase the effectiveness of the lesson. This material is also recommended for use by teachers implementing distance learning to improve the clarity of the explanation. The video can be recommended to a struggling student to deepen their understanding of the topic.
TEXT DECODING:
“Periodicity of functions y = cos x, y = sin x.”
To construct graphs of the functions y = sin x and y = cos x, the properties of the functions were used:
1 area of definition,
2 value area,
3 even or odd,
4 monotony,
5 limitation,
6 continuity,
7 highest and lowest value.
Today we will study another property: the periodicity of a function.
DEFINITION. The function y = f (x), where x ϵ X (the Greek is equal to ef of x, where x belongs to the set x), is called periodic if there is a non-zero number T such that for any x from the set X the double equality holds: f (x - T)= f (x) = f (x + T)(eff from x minus te is equal to ef from x and equal to ef from x plus te). The number T that satisfies this double equality is called the period of the function
And since sine and cosine are defined on the entire number line and for any x the equalities sin(x - 2π)= sin x= sin(x+ 2π) are satisfied (sine of x minus two pi is equal to sine of x and equal to sine of x plus two pi ) And
cos (x- 2π)= cos x = cos (x+ 2π) (the cosine of x minus two pi is equal to the cosine of x and equal to the cosine of x plus two pi), then sine and cosine are periodic functions with a period of 2π.
Periodicity allows you to quickly build a graph of a function. Indeed, in order to construct a graph of the function y = sin x, it is enough to plot one wave (most often on a segment (from zero to two pi), and then by shifting the constructed part of the graph along the x-axis to the right and left by 2π, then by 4π and so on to get a sine wave.
(show right and left shift by 2π, 4π)
Similarly for the graph of the function
y = cos x, but we build one wave most often on the segment [; ] (from minus pi over two to three pi over two).
Let us summarize the above and draw a conclusion: to construct a graph of a periodic function with a period T, you first need to construct a branch (or wave, or part) of the graph on any interval of length T (most often this is an interval with ends at points 0 and T or - and (minus te by two and te by two), and then move this branch along the x(x) axis to the right and left by T, 2T, 3T, etc.
Obviously, if the function is periodic with period T, then for any integer k0(not equal to zero) a number of the form kT(ka te) is also the period of this function. Usually they try to isolate the smallest positive period, which is called the main period.
As the period of the functions y = cos x, y = sin x, one could take - 4π, 4π, - 6π, 6π, etc. (minus four pi, four pi, minus six pi, six pi, and so on). But the number 2π is the main period of both functions.
Let's look at examples.
EXAMPLE 1. Find the main period of the function y = cos5x (the y is equal to the cosine of five x).
Solution. Let T be the main period of the function y = cos5x. Let's put
f (x) = cos5x, then f (x + T) = cos5(x + T) = cos (5x + 5T) (eff of x plus te is equal to the cosine of five multiplied by the sum of x and te is equal to the cosine of the sum of five x and five te).
cos (5x + 5T) = cos5x. Hence 5T = 2πn (five te equals two pi en), but according to the condition you need to find the main period, which means 5T = 2π. We get T=
(the period of this function is two pi divided by five).
Answer: T=.
EXAMPLE 2. Find the main period of the function y = sin (the y is equal to the sine of the quotient of two x by seven).
Solution. Let T be the main period of the function y = sin. Let's put
f (x) = sin, then f (x + T) = sin (x + T) = sin (x + T) (ef of x plus te is equal to the sine of the product of two sevenths and the sum of x and te is equal to the sine of the sum of two sevenths x and two sevenths te).
For the number T to be the period of the function, the identity must be satisfied
sin (x + T) = sin. Hence T= 2πn (two sevenths te is equal to two pi en), but according to the condition you need to find the main period, which means T= 2π. We get T=7
(the period of this function is seven pi).
Answer: T=7.
Summarizing the results obtained in the examples, we can conclude: the main period of the functions y = sin kx or y = cos kx (y is equal to sine ka x or y is equal to cosine ka x) is equal to (two pi divided by ka).