Lesson "Periodicity of functions y=sinx, y=cosx". Sine (sin x) and cosine (cos x) - properties, graphs, formulas

>> Periodicity of functions y = sin x, y = cos x

§ 11. Periodicity of functions y \u003d sin x, y \u003d cos x

In the previous paragraphs, we have used seven properties functions: domain, even or odd, monotonic, limited, largest and smallest value, continuity, range of the function. We used these properties either to construct a function graph (as it was, for example, in § 9), or in order to read the constructed graph (as it was, for example, in § 10). Now has come auspicious moment to introduce one more (eighth) property of functions, which is perfectly visible on the above-constructed charts functions y \u003d sin x (see Fig. 37), y \u003d cos x (see Fig. 41).

Definition. A function is called periodic if there exists a non-zero number T such that for any x from the sets, the double equality:

The number T that satisfies specified condition, is called the period of the function y \u003d f (x).
It follows that, since for any x, the equalities are true:

then the functions y \u003d sin x, y \u003d cos x are periodic and the number 2 P serves as the period of both functions.
The periodicity of a function is the promised eighth property of functions.

Now look at the graph of the function y \u003d sin x (Fig. 37). To build a sinusoid, it is enough to build 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 \u003d cos x (Fig. 41). We see that here, too, 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 make the following conclusion.

If the function y \u003d f (x) has a period T, then to plot the graph of the function, you must first plot a branch (wave, part) of the graph on any interval of length T (most often, they 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 3T is a period, and -T is a period; in general, a period is any number of the form KT, where k \u003d ± 1, ± 2, ± 3 ... Usually, if possible, they try to single out the smallest positive period, it is called the main period.
So, any number of the form 2pc, where k \u003d ± 1, ± 2, ± 3, is the period of the functions y \u003d sinn x, y \u003d cos x; 2p is the main period of both functions.

Example. Find the main period of a function:

a) Let T be the main period of the function y \u003d sin x. Let's put

For the number T to be the period of the function, the identity Ho must hold, since we are talking on finding the main period, we obtain
b) Let T be the main period of the function y = cos 0.5x. Let f(x)=cos 0.5x. Then f (x + T) \u003d cos 0.5 (x + T) \u003d cos (0.5x + 0.5 T).

For the number T to be the period of the function, the identity cos (0.5x + 0.5T) = cos 0.5x must be satisfied.

So, 0.5t = 2pp. But, since we are talking about finding the main period, we get 0.5T = 2 l, T = 4l.

The generalization of the results obtained in the example is the following statement: the main period of the function

A.G. Mordkovich Algebra Grade 10

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Centered at a point A.
α is an angle expressed in radians.

Definition
Sinus 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 leg |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 designations

;
;
.

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 a period 2 pi.

Parity

The sine function is odd. The cosine function is even.

Domain of definition and values, extrema, increase, decrease

The functions sine and cosine are continuous on their domain of definition, that is, for all x (see the 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
Ascending
Descending
Maximums, y= 1
Minima, y = - 1
Zeros, y= 0
Points of intersection with the y-axis, x = 0	y= 0	y= 1

Basic formulas

Sum of squared sine and cosine

Sine and cosine formulas for sum and difference

;
;

Formulas for the product of sines and cosines

Sum and difference formulas

Expression of sine through cosine

;
;
;
.

Expression of cosine through sine

;
;
;
.

Expression in terms of tangent

; .

For , we have:
; .

At :
; .

Table of sines and cosines, tangents and cotangents

This table shows the values of sines and cosines for some values of the argument.

Expressions through complex variables

;

Euler formula

Expressions in terms of hyperbolic functions

;
;

Derivatives

; . Derivation of formulas > > >

Derivatives of the nth order:
{ -∞ < x < +∞ }

Secant, cosecant

Inverse functions

Inverse functions to sine and cosine are the arcsine and arccosine, respectively.

Arcsine, arcsin

Arccosine, arccos

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

Instruction

To find the period of a trigonometric function raised to a power, evaluate the evenness of the power. To reduce the standard period by half. For example, if you are given a function y \u003d 3 cos ^ 2x, then the standard period 2P will decrease by 2 times, so the period will be equal to P. Note that the functions tg, ctg are periodic to any degree of P.

If you are given an equation that contains or is a quotient of two trigonometric functions, first find the period for each of them separately. Then find the minimum number that would fit an integer amount of both . For example, given the function y=tgx*cos5x. For the tangent, the period is P, for the cosine 5x, the period is 2P/5. The minimum number that can fit both of these periods is 2P, so the required period is 2P.

If you find it difficult to act in the proposed way or doubt the answer, try to act by definition. Take T as the period of the function, it is greater than zero. Substitute the expression (x + T) in the equation for x and solve the resulting equality as if T were a parameter or a number. As a result, you will find the value of the trigonometric function and be able to choose the minimum period. For example, as a result of simplification, you get the identity sin (T / 2) \u003d 0. The minimum value of T at which it is performed is 2P, this will be the task.

Sources:

sin period

A periodic function is a function that repeats its values after some non-zero period. The period of a function is a number whose addition to the function argument does not change the value of the function.

You will need

Knowledge of elementary mathematics and the beginnings of analysis.

Instruction

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Trigonometry is a branch of mathematics for studying, expressing various dependences of the sides of a right triangle on the magnitudes of acute angles at the hypotenuse. Such functions are called trigonometric, and to simplify working with them, trigonometric functions were derived. identities.

concept identities in means equality, which is satisfied for any values of the arguments of the functions included in it. Trigonometric identities- these are the equalities of trigonometric functions, proven and accepted to facilitate work with trigonometric formulas. A trigonometric function is an elementary function of the dependence of one of the legs of a right-angled triangle on the magnitude of an acute angle at the hypotenuse. There are six basic trigonometric functions most commonly used: sin (sine), cos (cosine), tg (tangent), ctg (cotangent), sec (secant), and cosec (cosecant). These functions are called direct, there are also

Purpose: to generalize and systematize students' knowledge on the topic "Periodicity of functions"; to form skills in applying the properties of a periodic function, finding the smallest positive period of a function, plotting periodic functions; promote interest in the study of mathematics; cultivate observation, accuracy.

Equipment: computer, multimedia projector, task cards, slides, clocks, ornament tables, folk craft elements

“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. Presentation of the topic and objectives of the lesson.

II. Checking homework.

We check homework according to samples, discuss the most difficult points.

III. Generalization and systematization of knowledge.

1. Oral frontal work.

Questions of theory.

1) Form the definition of the period of the function
2) What is 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) Use the circle to 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 the angle of 540º is one of the periods of the function y= cos(2x)

3. Prove that the 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 meet with the words PERIOD, PERIODICITY?

Students' answers: A period in music is a construction in which a more or less complete musical thought is stated. The geological period is part of an era and is divided into epochs with a period of 35 to 90 million years.

The half-life of a radioactive substance. Periodic fraction. Periodicals are printed publications that appear on strictly defined dates. Periodic system of Mendeleev.

6. The figures show parts of the graphs of periodic functions. Define 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 met with the construction of repeating elements?

Students answer: Elements of ornaments, folk art.

IV. Collective problem solving.

(Problem solving on slides.)

Let us consider one of the ways to study a function for periodicity.

This method bypasses the difficulties associated with proving that one or another period is the smallest, and also there is no need to touch on questions about arithmetic operations on periodic functions and about 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: Let's 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 x=-0.25 we get

(T)=0<=>T=n, n ∈ Z

We have obtained that all periods of the considered function (if they exist) are among integers. Choose among these numbers the smallest positive number. This is 1 . Let's check if it is actually 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.

Task 2. Show that the function f(x)=cos 2 (x) is periodic and find its main period.

Task 3. Find the main period of the function

f(x)=sin(1.5x)+5cos(0.75x)

Assume the T-period of the function, then for any X the ratio

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(1.5Т)+5cos(0.75Т)=5

Adding, we get:

10cos(0.75T)=10

2π n, n € Z

Let's choose from all numbers "suspicious" for the period the smallest positive one 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)

Hence, is the main period of the function f.

Task 4. Check if the function f(x)=sin(x) is periodic

Let T be the period of the function f. Then for any x

sin|x+T|=sin|x|

If x=0, then sin|T|=sin0, sin|T|=0 T=π n, n ∈ Z.

Suppose. That for some n the number π n is a period

considered function π n>0. Then sin|π n+x|=sin|x|

This implies that n must be both even and odd at the same time, which is impossible. Therefore, this function is not periodic.

Task 5. Check if the function is periodic

f(x)=

Let T be the period f, then

, hence sinT=0, T=π n, n € Z. Let us assume that for some n the number π n is indeed the period of the given function. Then the number 2π n will also be a period

Since the numerators are equal, so are their denominators, so

Hence, the function f is not periodic.

Group work.

Tasks for group 1.

Tasks for group 2.

Check if the function f is periodic and find its main period (if it exists).

f(x)=cos(2x)+2sin(2x)

Tasks for group 3.

At the end of the work, the groups present their solutions.

VI. Summing up the lesson.

Reflection.

The teacher gives students cards with drawings and offers to paint over part of the first drawing in accordance with the extent to which, as it seems to them, they have mastered the methods of studying the function for periodicity, and in part of the second drawing, in accordance with their contribution to the work in the lesson.

VII. Homework

one). Check if function f is periodic and find its main 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 the beginning of analysis with in-depth study.
Mathematics. Preparation for the exam. Ed. Lysenko F.F., Kulabukhova S.Yu.
Sheremetyeva T.G. , Tarasova E.A. Algebra and beginning analysis for grades 10-11.