Lesson "Periodicity of functions y=sinx, y=cosx". Investigation of a function for periodicity

>> 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

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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

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Formulas for the product of sines and cosines

Sum and difference formulas

Expression of sine through cosine

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Expression of cosine through sine

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Expression in terms of tangent

; .

For , we have:
; .

At :
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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

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Euler formula

Expressions in terms of hyperbolic functions

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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.

A 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 values ​​of the argument, and therefore any number can be considered its period.

Usually interested in the smallest zero function period. 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- not the only periodic ones.

Regarding the simple basic functions the only way to establish their periodicity or non-periodicity is by 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 the point x is equal to the tangent of the tangent of the graph of its antiderivative at this point to the x-axis, and since it repeats periodically, it must repeat. For example, the derivative of sin functions(x) is equal to cos(x), and it is periodic. Taking the derivative of cos(x) gives you -sin(x). Periodicity remains unchanged.

However, the reverse 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 - also a periodic function, and its period is equal to T/k. For example sin(2x) is a periodic function and its period is π. Visually, this can be represented as follows: by multiplying x by some number, you kind of compress the functions horizontally exactly as 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 LCM (12, 15) = 60.

Visually, this can be 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 , then the total function will not be periodic at all. For example, let F1(x) = x mod 2 (the remainder of x divided by 2) and F2(x) = sin(x). T1 here will be equal to 2, and T2 is equal to 2π. The period ratio is π - irrational number. Therefore, the function sin(x) + x mod 2 is not periodic.

Sources:

• Function Theory

Many mathematical functions have one feature that facilitates their construction - it is periodicity, that is, the repeatability of the graph on the coordinate grid at regular intervals.

Instruction

The best known periodic functions of mathematics are the sinusoid and the cosine wave. These functions have a wavelike and basic 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 zero and satisfies the rule f(x)=f(x+N), thus ensuring repeatability. The period of the function is smallest number N, but not zero. That is, for example, the sin x function is equal to the sin (x + 2PN) function, where N \u003d ± 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