>> 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, monotonicity, boundedness, maximum and minimum values, continuity, range of functions. 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 a favorable moment has come 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 the indicated 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 move this branch along the x axis to the right and left by 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 about finding the main period, we get
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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satisfying the system of inequalities:

b) Consider the set of numbers on the number axis that satisfy the system of inequalities:

Find the sum of the lengths of the segments that make up this set.

§ 7. The simplest formulas

In § 3 we established the following formula for acute angles α:

			sin2α + cos2α = 1.
				The same formula			when,
				when α is any			de-
				le, let M be a point on the trigonometry
				calic circle corresponding to
				number α (Fig. 7.1). Then		M has co-
				ordinates x = cos α, y
				However, every point (x; y) lying on
				circles of unit radius with center
				trom at the origin, satisfying
				solves the equation x2 + y2		1, from where
				cos2 α + sin2 α = 1, as required.

So, the formula cos2 α + sin2 α = 1 follows from the circle equation. It may seem that in this way we have given a new proof of this formula for acute angles (compared to that indicated in § 3, where we used the Pythagorean theorem). The difference, however, is purely external: when deriving the circle equation x2 + y2 = 1, the same Pythagorean theorem is used.

For acute angles, we also obtained other formulas, for example

symbol, the right side is always non-negative, while the left side may well be negative. For the formula to be true for all α, it must be squared. We get the equality: cos2 α = 1/(1 + tg2 α). Let us prove that this formula is true for all α:1

1/(1 + tg2			sin2α			cos2α	Cos2α.
			cos2α			sin2α + cos2α

Problem 7.1. Derive all the formulas below from the definitions and the formula sin2 α + cos2 α = 1 (we have already proved some of them):

sin2α + cos2α = 1;

tg2α =

tg2α

sin2α =

tg α ctg α = 1;

cos2α

1 + tg2α

ctg2α

Ctg2

cos2α =

1 + ctg2α

sin2

These formulas allow, knowing the value of one of the trigonometric functions of a given number, to almost find all the rest

nye. Let, for example, we know that sin x = 1/2. Then cos2 x =

1−sin2 x = 3/4, so cos x is either 3/2 or − 3/2. To find out which of these two numbers cos x is equal to, additional information is needed.

Problem 7.2. Show with examples that both of the above cases are possible.

Problem 7.3. a) Let tgx = −1. Find sinx. How many answers does this problem have?

b) Let, in addition to the conditions of point a), we know that sin x< 0. Сколько теперь ответов у задачи?

1 For which tg α is defined, i.e. cos α 6= 0.

Problem 7.4. Let sin x = 3/5, x [π/2; 3π/2]. Find tgx.

Problem 7.5. Let tg x = 3, cos x > sin x. Find cos x, sin x.

Problem 7.6. Let tgx = 3/5. Find sin x + 2 cos x . cos x − 3 sin x

Problem 7.7. Prove the identities:

tgα − sinα

c) sin α + cos α ctg α + sin α tg α + cos α =

Problem 7.8. Simplify expressions:

a) (sin α + cos α)2 + (sin α − cos α)2 ; b) (tg α + ctg α)2 + (tg α − ctg α)2 ;

c) sin α(2 + ctg α)(2 ctg α + 1) − 5 cos α.

§ 8. Periods of trigonometric functions

The numbers x, x+2π, x−2π correspond to the same point on the trigonometric circle (if you pass an extra circle along the trigonometric circle, you will end up where you were). This implies the following identities, which were already discussed in § 5:

sin(x + 2π) = sin(x − 2π) = sin x; cos(x + 2π) = cos(x − 2π) = cos x.

In connection with these identities, we have already used the term "period". We now give exact definitions.

Definition. The number T 6= 0 is called the period of the function f if the equalities f(x − T) = f(x + T) = f(x) are true for all x (it is assumed that x + T and x − T are included in the domain of the function if x is included in it). A function is called periodic if it has a period (at least one).

Periodic functions naturally arise in the description of oscillatory processes. One of these processes has already been discussed in § 5. Here are more examples:

1) Let ϕ = ϕ(t) be the angle of deviation of the swinging pendulum of the clock from the vertical at the moment t. Then ϕ is a periodic function of t.

2) The voltage (“potential difference,” as a physicist would say) between two sockets in an AC outlet, es-

whether to consider it as a function of time is a periodic function1.

3) Let us hear the musical sound. Then the air pressure at a given point is a periodic function of time.

If a function has a period T , then the periods of this function will also be the numbers −T , 2T , −2T . . . - in a word, all numbers nT , where n is an integer not equal to zero. Indeed, let's check, for example, that f(x + 2T) = f(x):

f(x + 2T) = f((x + T) + T) = f(x + T) = f(x).

Definition. The smallest positive period of the function f is - in accordance with the literal meaning of the words - a positive number T such that T is the period of f and no positive number less than T is the period of f.

A periodic function is not required to have the smallest positive period (for example, a function that is constant has a period of any number in general and, therefore, it does not have the smallest positive period). Examples can also be given of non-constant periodic functions that do not have the smallest positive period. Nevertheless, in most interesting cases, periodic functions have the smallest positive period.

1 When they say “voltage in the network is 220 volts”, they mean its “rms value”, which we will talk about in § 21. The voltage itself changes all the time.

Rice. 8.1. The period of tangent and cotangent.

In particular, the smallest positive period of both sine and cosine is 2π. Let us prove this, for example, for the function y = sin x. Let, contrary to what we say, the sine has a period T such that 0< T < 2π. При x = π/2 имеем sin x = = 1. Будем теперь увеличивать x. В точке x + T значение синуса должно быть также равно 1. Но в следующий раз синус будет равен 1 только при x = (π/2) + 2π. Поэтому период синуса быть меньше 2π не может. Доказательство для косинуса аналогично.

The smallest positive period of the function describing the oscillations (as in our examples 1-3) is simply called the period of these oscillations.

Since the number 2π is the period of sine and cosine, it will also be the period of tangent and cotangent. However, for these functions, 2π is not the smallest period: the smallest positive period of the tangent and cotangent is π. Indeed, the points corresponding to the numbers x and x + π on the trigonometric circle are diametrically opposed: from the point x to the point x + 2π one must go the distance π, which is exactly equal to half the circle. Now, if we use the definition of tangent and cotangent using the axes of tangents and cotangents, the equalities tg (x + π) = tg x and ctg (x + π) = ctg x become obvious (Fig. 8.1). It is easy to check (we will propose to do this in problems) that π is indeed the smallest positive period of the tangent and cotangent.

One note about terminology. Often the words "period of a function" are used in the sense of "the smallest positive period." So if you are asked on the exam: “Is 100π the period of the sine function?”, Take your time with the answer, but clarify whether you mean the smallest positive period or just one of the periods.

Trigonometric functions are a typical example of periodic functions: any "not very bad" periodic function can be expressed in some sense in terms of trigonometric functions.

Problem 8.1. Find the smallest positive periods of the functions:

				c) y = cos πx;

d) y = cosx + cos(1.01x).

Problem 8.2. The dependence of the voltage in the AC network on time is given by the formula U = U0 sin ωt (here t is time, U is voltage, U0 and ω are constants). The frequency of the alternating current is 50 Hertz (this means that the voltage makes 50 oscillations per second).

a) Find ω, assuming that t is measured in seconds;

b) Find the (smallest positive) period U as a function of t.

Problem 8.3. a) Prove that the smallest positive period of the cosine is 2π;

b) Prove that the smallest positive period of the tangent is π.

Problem 8.4. Let the least positive period of the function f be equal to T . Prove that all other periods are of the form nT for some integers n.

Problem 8.5. Prove that the following functions are not periodic.

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 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

1). 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/

1. Mordkovich A.G. Algebra and the beginning of analysis with in-depth study.
2. Mathematics. Preparation for the exam. Ed. Lysenko F.F., Kulabukhova S.Yu.
3. Sheremetyeva T.G. , Tarasova E.A. Algebra and beginning analysis for grades 10-11.

Argument x, then it is called periodic if there is 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 non-zero period of the 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.

With regard to simple, basic functions, the only way to establish their periodicity or non-periodicity is through calculations. But for complex functions, there are already some 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 angle of the tangent of the graph of its antiderivative at this point to the abscissa axis, and since the antiderivative repeats periodically, the derivative must also repeat. For example, the derivative of the function sin(x) is 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 non-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. Visually, this can be represented as follows: by multiplying x by some number, you seem to compress the graph of the function 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 dividing T1/T2 is a 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 later (or rather, 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 irrational, 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 ratio of periods is equal to π - an irrational number. Therefore, the function sin(x) + x mod 2 is not periodic.

Trigonometric functions periodic, that is, repeated after a certain period. As a result, it is enough to study the function on this interval and extend the discovered properties to all other periods.

Instruction

1. If you are given a primitive expression in which there is only one trigonometric function (sin, cos, tg, ctg, sec, cosec), and the angle inside the function is not multiplied by any number, and it itself is not raised to any power - use the definition. For expressions containing sin, cos, sec, cosec, boldly set the period to 2P, and if there is tg, ctg in the equation, then P. Say, for the function y \u003d 2 sinx + 5, the period will be 2P.

2. If the angle x under the sign of a trigonometric function is multiplied by some number, then in order to find the period of this function, divide the typical period by this number. Let's say you are given a function y = sin 5x. The typical period for a sine is 2P, dividing it by 5, you get 2P / 5 - this is the desired period of this expression.

3. To find the period of a trigonometric function raised to a power, evaluate the evenness of the power. For an even degree, halve the sample period. Say, if you are given a function y \u003d 3 cos ^ 2x, then the typical period 2P will decrease by 2 times, so the period will be equal to P. Please note that the functions tg, ctg are periodic to any extent P.

4. If you are given an equation containing the product or quotient of 2 trigonometric functions, first find the period for all of them separately. After that, find the minimum number that would fit the whole number of both periods. Let's say the function y=tgx*cos5x is given. For the tangent, the period is P, for the cosine 5x, the period is 2P/5. The minimum number that is allowed to fit both of these periods is 2P, so the desired period is 2P.

5. If you find it difficult to do the proposed way or doubt the result, try to do by definition. Take T as the period of the function, it is larger than zero. Substitute the expression (x + T) in the equation instead of 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 smallest period. Let's say, as a result of facilitating, you get the identity sin (T / 2) \u003d 0. The minimum value of T at which it is performed is 2P, and this will be the result of the task.

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 argument of the function does not change the value of the function.

You will need

• Knowledge of elementary mathematics and the beginnings of the survey.

Instruction

1. Let us denote the period of the function f(x) by the number K. Our task is to find this value of K. To do this, imagine that the function f(x), using the definition of a periodic function, equate f(x+K)=f(x).

2. We solve the resulting equation for the unknown K, as if x is a constant. Depending on the value of K, there will be several options.

3. If K>0, then this is the period of your function. If K=0, then the function f(x) is not periodic. If the solution of the equation f(x+K)=f(x) does not exist for any K not equal to zero, then such a function is called aperiodic and it also has no period.

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Note!
All trigonometric functions are periodic, and all polynomial functions with degree greater than 2 are aperiodic.

Helpful advice
The period of a function consisting of 2 periodic functions is the least common multiple of the periods of these functions.

Trigonometric equations are equations that contain trigonometric functions of an unknown argument (for example: 5sinx-3cosx =7). In order to learn how to solve them, you need to know some methods for this.

Instruction

1. The solution of such equations consists of 2 stages. The first is the reformation of the equation to acquire its simplest form. The simplest trigonometric equations are called the following: Sinx=a; cosx=a etc.

2. The second is the solution of the obtained simplest trigonometric equation. There are basic ways to solve equations of this kind: Solving in an algebraic way. This method is famously famous from school, from the course of algebra. It is otherwise called the method of replacing a variable and substituting. Applying the reduction formulas, we transform, make a replacement, after which we find the roots.

3. Decomposition of the equation into factors. First, we transfer all terms to the left and decompose into factors.

4. Bringing the equation to a homogeneous one. Equations are called homogeneous equations if all the terms are of the same degree and the sine, cosine of the same angle. In order to solve it, you should: first transfer all its members from the right side to the left side; move all common factors out of brackets; equate factors and brackets to zero; equated brackets give a homogeneous equation of a lesser degree, which should be divided by cos (or sin) to a higher degree; solve the resulting algebraic equation for tan.

5. The next way is to go to the half corner. Say, solve the equation: 3 sin x - 5 cos x \u003d 7. Let's move on to the half angle: 6 sin (x / 2) cos (x / 2) - 5 cos ? (x / 2) + 5 sin? (x / 2) = 7sin? (x / 2) + 7 cos? (x/ 2) , after which we reduce all the terms to one part (otherwise to the right) and solve the equation.

6. Auxiliary corner entry. When we replace the integer value cos(a) or sin(a). The sign "a" is an auxiliary angle.

7. A way to reformat a product into a sum. Here you need to apply the appropriate formulas. Let's say given: 2 sin x sin 3x = cos 4x. Let's solve it by converting the left side into a sum, that is: cos 4x - cos 8x = cos 4x, cos 8x = 0, 8x = p / 2 + pk, x = p / 16 + pk / 8.

8. The final way, called multifunction substitution. We transform the expression and make a substitution, say Cos(x/2)=u, after which we solve the equation with the parameter u. When acquiring the total, we translate the value into the opposite.

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If we consider points on a circle, then the points x, x + 2π, x + 4π, etc. match with each other. So the trigonometric functions on a straight line periodically repeat their meaning. If the period is famous functions, it is allowed to build a function on this period and repeat it on others.

Instruction

1. The period is a number T such that f(x) = f(x+T). In order to find the period, solve the corresponding equation, substituting x and x + T as an argument. In this case, the well-known periods for functions are used. For the sine and cosine functions, the period is 2π, and for the tangent and cotangent, it is π.

2. Let the function f(x) = sin^2(10x) be given. Consider the expression sin^2(10x) = sin^2(10(x+T)). Use the formula to reduce the degree: sin^2(x) = (1 - cos 2x)/2. Then get 1 - cos 20x = 1 - cos 20(x+T) or cos 20x = cos (20x+20T). Knowing that the period of the cosine is 2π, 20T = 2π. Hence, T = π/10. T is the minimum correct period, and the function will be repeated after 2T, and after 3T, and in the other direction along the axis: -T, -2T, etc.

Helpful advice
Use formulas to lower the degree of a function. If you are more familiar with the periods of some functions, try to reduce the existing function to the known ones.

Finding a function for even and odd helps to build a graph of the function and comprehend the nature of its behavior. For this research, you need to compare the given function written for the “x” argument and for the “-x” argument.

Instruction

1. Write the function you want to explore as y=y(x).

2. Replace the function argument with "-x". Substitute this argument into a functional expression.

3. Simplify the expression.

4. Thus, you got the same function written for the arguments "x" and "-x". Look at these two entries. If y(-x)=y(x), then this is an even function. If y(-x)=-y(x), then this is an odd function. That is, it is neither even nor odd.

5. Write down your results. Now you can use them in plotting a function graph or in a future analytical search for the properties of a function.

6. It is also possible to talk about even and odd functions in the case when the graph of the function is more closely defined. Let's say the graph was the result of a physical experiment. If the function graph is symmetrical about the y-axis, then y(x) is an even function. If the function graph is symmetrical about the x-axis, then x(y) is an even function. x(y) is the inverse function of y(x). If the graph of the function is symmetric about the origin (0,0), then y(x) is an odd function. The inverse function x(y) will also be odd.

7. It is significant to remember that the concept of even and odd functions has a direct relationship with the domain of the function. If, say, an even or odd function does not exist for x=5, then it does not exist for x=-5, which is impossible to say about a function of a general form. When establishing even and odd, pay attention to the domain of the function.

8. Searching for even and odd functions correlates with finding the set of function values. To find the set of values ​​of an even function, it is enough to see half of the function, to the right or to the left of zero. If for x>0 an even function y(x) takes values ​​from A to B, then it will take the same values ​​for x<0.Для нахождения множества значений, принимаемых нечетной функцией, тоже довольно разглядеть только одну часть функции. Если при x>0 odd function y(x) takes a range of values ​​from A to B, then for x<0 она будет принимать симметричный диапазон значений от (-В) до (-А).

"Trigonometric" once began to be called functions that are determined by the dependence of acute angles in a right triangle on the lengths of its sides. These functions include, first of all, sine and cosine, secondly, the secant and cosecant inverse to these functions, the tangent and cotangent derivatives of them, as well as the inverse functions arcsine, arccosine, etc. It is more positive to speak not about the “solution” of such functions, but about their “calculation”, that is, about finding a numerical value.

Instruction

1. If the argument of the trigonometric function is unknown, then it is allowed to calculate its value by an indirect method based on the definitions of these functions. To do this, you need to know the lengths of the sides of the triangle, the trigonometric function for one of the angles of which you want to calculate. Say, by definition, the sine of an acute angle in a right triangle is the ratio of the length of the leg opposite this angle to the length of the hypotenuse. It follows from this that to find the sine of an angle, it is enough to know the lengths of these 2 sides. A similar definition says that the sine of an acute angle is the ratio of the length of the leg adjacent to this angle to the length of the hypotenuse. The tangent of an acute angle can be calculated by dividing the length of the opposite leg by the length of the adjacent one, and the cotangent requires dividing the length of the adjacent leg by the length of the opposite one. To calculate the secant of an acute angle, you need to find the ratio of the length of the hypotenuse to the length of the leg adjacent to the required angle, and the cosecant is determined by the ratio of the length of the hypotenuse to the length of the opposite leg.

2. If the argument of the trigonometric function is carried out, then it is not required to know the lengths of the sides of the triangle - it is allowed to use tables of values ​​​​or calculators of trigonometric functions. Such a calculator is among the standard programs of the Windows operating system. To run it, you can press the Win + R key combination, enter the calc command and click the OK button. In the program interface, open the "View" section and select the "Engineering" or "Scientist" item. Later, it is allowed to introduce the argument of the trigonometric function. To calculate the functions sine, cosine and tangent, rather after entering the value, click on the corresponding interface button (sin, cos, tg), and to find their reciprocals of the arcsine, arccosine and arctangent, check the Inv checkbox in advance.

3. There are also alternative methods. One of them is to go to the site of the Nigma or Google search engine and enter the desired function and its argument (say, sin 0.47) as a search query. These search engines have built-in calculators, therefore, after sending such a request, you will receive the value of the trigonometric function you entered.

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Tip 7: How to detect the value of trigonometric functions

Trigonometric functions first appeared as tools for abstract mathematical calculations of the dependences of the magnitudes of acute angles in a right triangle on the lengths of its sides. Now they are widely used in both scientific and technical fields of human activity. For utilitarian calculations of trigonometric functions from given arguments, it is allowed to use various tools - a few of the most accessible of them are described below.

Instruction

1. Use, say, a calculator program installed by default with the operating system. It opens by selecting the "Calculator" item in the "Utilities" folder from the "Typical" subsection located in the "All Programs" section. This section can be found by opening the main menu of the operating system by clicking on the "Start" button. If you are using the Windows 7 version, then you can primitively enter the word "Calculator" in the "Detect programs and files" field of the main menu, and then click on the appropriate link in the search results.

2. Enter the value of the angle for which you want to calculate the trigonometric function, and then click on the button corresponding to this function - sin, cos or tan. If you are concerned about inverse trigonometric functions (arcsine, arccosine or arctangent), then first click the button labeled Inv - it reverses the functions assigned to the calculator's control buttons.

3. In earlier versions of the OS (say, Windows XP), to access trigonometric functions, you need to open the “View” section in the calculator menu and prefer the “Engineering” line. In addition, instead of the Inv button in the interface of the old versions of the program, there is a checkbox with the same inscription.

4. You can do without a calculator if you have Internet access. There are many services on the web that offer differently organized trigonometric function calculators. One particularly handy option is built into the Nigma search engine. Having gone to its main page, primitively enter the value that excites you in the search query field - say, “arc tangent of 30 degrees”. After pressing the "Discover!" the search engine will calculate and show the result of the calculation - 0.482347907101025.

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Trigonometry is a branch of mathematics for comprehending functions that express different dependences of the sides of a right triangle on the magnitudes of acute angles at the hypotenuse. Such functions are called trigonometric, and to facilitate working with them, trigonometric functions were derived. identities .

Performance identities in mathematics denotes an equality that is satisfied for any values ​​of the arguments of the functions included in it. Trigonometric identities- these are equalities of trigonometric functions, confirmed and accepted to simplify work with trigonometric formulas. A trigonometric function is an elementary function of the dependence of one of the legs of a right triangle on the magnitude of an acute angle at the hypotenuse. More often than not, six basic trigonometric functions are used: sin (sine), cos (cosine), tg (tangent), ctg (cotangent), sec (secant) and cosec (cosecant). These functions are called direct, there are also inverse functions, say, sine - arcsine, cosine - arccosine, etc. Initially, trigonometric functions were reflected in geometry, after which they spread to other areas of science: physics, chemistry, geography, optics, probability theory, as well as acoustics, music theory, phonetics, computer graphics and many others. Now it is more difficult to imagine mathematical calculations without these functions, although in the distant past they were used only in astronomy and architecture. Trigonometric identities are used to simplify work with long trigonometric formulas and bring them to a digestible form. There are six basic trigonometric identities, they are associated with direct trigonometric functions: tg ? = sin?/cos?; sin^2? + cos^2? = 1; 1 + tg^2? = 1/cos^2?; 1 + 1/tg^2? = 1/sin^2?; sin (? / 2 -?) \u003d cos ?; cos (? / 2 -?) \u003d sin?. These identities easy to confirm from the properties of the ratio of sides and angles in a right triangle: sin ? = BC/AC = b/c; cos? = AB/AC = a/c; tg? = b/a. First identity tg ? = sin?/cos? follows from the ratio of the sides in the triangle and the exclusion of the side c (hypotenuse) when dividing sin by cos. In the same way, the identity ctg is defined? = cos ?/sin ?, because ctg ? = 1/tg ?. By the Pythagorean theorem, a^2 + b^2 = c^2. Divide this equality by c^2, we get the second identity: a^2/c^2 + b^2/c^2 = 1 => sin^2 ? + cos^2 ? = 1.Third and fourth identities gets by dividing by b^2 and a^2, respectively: a^2/b^2 + 1 = c^2/b^2 => tg^2 ? + 1 = 1/cos^2 ?;1 + b^2/a^2 = c^2/a^2 => 1 + 1/tg^2 ? = 1/sin^ ? or 1 + ctg^2 ? \u003d 1 / sin ^ 2?. The fifth and sixth main identities are proved by determining the sum of the acute angles of a right triangle, which is equal to 90 ° or? / 2. More difficult trigonometric identities: formulas for adding arguments, double and triple angles, lowering the degree, reforming the sum or product of functions, as well as trigonometric substitution formulas, namely the expressions of the main trigonometric functions in terms of tg half angle: sin ?= (2*tg ?/2)/(1 + tg^2 ?/2); = (1 – tg^2 ?/2)/(1 = tg^2 ?/2);tg ? = (2*tg ?/2)/(1 – tg^2 ?/2).

The need to find the minimum meaning mathematical functions is of actual interest in solving applied problems, say, in economics. Huge meaning for entrepreneurial activity has minimization of losses.

Instruction

1. In order to find the minimum meaning functions, it is necessary to determine at what value of the argument x0 the inequality y(x0) will be satisfied? y(x), where x ? x0. As usual, this problem is solved at a certain interval or in each range of values functions, if one is not set. One aspect of the solution is finding fixed points.

2. The stationary point is called meaning the argument that the derivative functions goes to zero. According to Fermat's theorem, if a differentiable function takes an extremal meaning at some point (in this case, a local minimum), then this point is stationary.

3. Minimum meaning the function often takes exactly at this point, however, it can be determined not invariably. Moreover, it is not always possible to say exactly what the minimum is functions or he accepts an infinitely small meaning. Then, as usual, they find the limit to which it gravitates when decreasing.

4. In order to determine the minimum meaning functions, it is necessary to perform a sequence of actions consisting of four stages: finding the domain of definition functions, acquisition of fixed points, overview of values functions at these points and at the ends of the gap, the detection of a minimum.

5. It turns out that let some function y(x) be given on an interval with boundaries at points A and B. Find its domain of definition and find out if the interval is its subset.

6. Calculate Derivative functions. Equate the resulting expression to zero and find the roots of the equation. Check if these stationary points fall within the interval. If not, then at the next stage they are not taken into account.

7. Look at the gap for the type of boundaries: open, closed, compound, or dimensionless. It depends on how you find the minimum meaning. Let's say the segment [A, B] is a closed interval. Substitute them into the function and calculate the values. Do the same with the stationary point. Choose the smallest total.

8. With open and boundless intervals, the situation is somewhat more difficult. Here we have to look for one-sided limits, which do not invariably give an unambiguous result. Say, for an interval with one closed and one punctured boundary [A, B), one should find a function at x = A and a one-sided limit lim y at x? B-0.