Power functions y sin. Function y=sinx, these are the main powers and graph. Lesson and presentation on the topic: "Function y=sin(x). Significance and power"

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A leak of rust, not knowing that it has stagnated,
standing water rots or freezes in the cold,
and the mind of the people, not knowing themselves, is stagnant, withering away.
Leonardo Da Vinci

Vickory technologies: problematic thinking, critical thinking, communicative thinking

Goals:

• Development of cognitive interest before beginning.
• Vychennya power functions y = sin x.
• Formation of practical skills based on the graph of the function y = sin x based on the learned theoretical material.

Zavdannya:

1. Vikoristovat the obvious potential of knowing the power of the function y = sin x in specific situations.

2. Be aware of the establishment of connections between analytical and geometric models of the function y = sin x.

Develop initiative, willingness and interest until a solution is found; Make a decision, don’t rest on what you have achieved, stick to your thoughts.

Involve yourself in educational cognitive activity, according to the sense of consistency, one-to-one relationships, mutual understanding, mutual encouragement, and self-confidence; spilkuvannya culture.

Lesson progress

Stage 1 Updating of basic knowledge, motivation for learning new material

"Enter before class."

There are 3 affirmations written on the doshta:

1. Trigonometric equation sin t = a solution.
2. The graph of an unpaired function can be rearranged by reversing the symmetry along the Oy axis.
3. The graph of a trigonometric function can be graphed in a simple manner.

Learn to discuss in pairs: what are your true principles? (1 hvilina). Then the results of the initial discussion (so, no) are entered into the table in the “Before” section.

The teacher sets goals and objectives for the lesson.

2. Updating knowledge (frontally on a trigonometric stake model).

We learned about the function s = sin t.

1) What values ​​can be changed t. What is the area of ​​significance of this function?

2) Which gap has the value of sin t. Find the highest and lowest values ​​of the function s = sin t.

3) Unleash jealousy sin t = 0.

4) What does the ordinate of the point under the hour of the first quarter mean? (The ordinate gets bigger). What does the ordinate of the point under the hour and the direction of the other quarter mean? (The ordinate changes step by step). How is this related to the monotony of the function? (the function s = sin t increases by a subcut and changes by a subcut).

5) We write the function s = sin t in the basic form y = sin x (we will have the basic coordinate system xОу) and create a table of the values ​​of this function.

Stage 2 Acceptance, understanding, primary consolidation, fleeting memorization

Stage 4 The primary systematization of knowledge and methods of action, their transfer and stagnation in new situations

6. No. 10.18 (b, c)

Stage 5 Sub-bag control, correction, assessment and self-esteem

7. We turn to solidity (the beginning of the lesson), we discuss the power of the trigonometric function y = sin x, which is remembered in the table in the section “Pislya”.

8. D/z: clause 10, No. 10.7(a), 10.8(b), 10.11(b), 10.16(a)

Lesson and presentation on the topic: "Function y=sin(x). Significance and power"

Additional materials
Shanny koristuvachs, do not forget to deprive your comments, vogues, tributes! All materials have been verified by anti-virus software.

Resource books and simulators in the online store "Integral" for 10th grade under 1C
There are problems with geometry. Interactive daily assignments for grades 7-10
Software middleware "1C: Mathematical Constructor 6.1"

What we need to know:

• Power of the function Y = sin (X).
• Function graph.
• How will the schedule be and what scale will it be?
• apply it.

Power to the sinus. Y=sin(X)

Children, we have already become familiar with trigonometric functions of a numerical argument. Do you remember them?

Let's take a closer look at the function Y=sin(X)

Let's write down the powers of this function:
1) The area of ​​significance is the impersonality of active numbers.
2) The function is unpaired. The meaning of the unpaired function is guessable. The function is called unpaired because the equation is equal: y(-x)=-y(x). We remember the following formulas: sin(-x)=-sin(x). The value was determined, so Y = sin (X) is an unpaired function.
3) The function Y=sin(X) increases by a section and changes to a section [π/2; π]. When we collapse along the first quarter (against the year arrow), the ordinate increases, and when the collapse occurs along the other quarter, it changes.

4) The function Y=sin(X) is bounded at the bottom. This power flows from the fact that
-1 ≤ sin(X) ≤ 1
5) The smallest value of the function is -1 (at x = - π/2+ πk). The highest value of the function is equal to 1 (at x = π/2+ πk).

Let's take a quick look at authorities 1-5 and figure out the graph of the function Y = sin (X). Our schedule will be consistent, stagnant with our authorities. There will soon be a schedule for breaks.

I will especially appreciate the zoom in on the scale. On the ordinate axis, it is better to take a single section equal to two cells, and on the abscis axis, a single section (two cells) is taken to be equal to π/3 (marvel the little ones).

Pobudova graph of the sine function x, y=sin(x)

Let's look at the meaning of the functions in our section:

Let's create a graph following our points, with the third component in place.

Rework table for ghost formulas

It would be quick for another authority to say that our function is unpaired, which means that it can be represented symmetrically around the coordinates:

We know that sin(x+2π) = sin(x). This means that the cut [- π; π] the graph looks just like the section [π; 3π] or or [-3π; - π] and so on. We are deprived of carefully drawing up the graph on the front page for the entire abscise.

The graph of the function Y=sin(X) is called a sine curve.

Let's write a few more authorities today with the required schedule:
6) The function Y=sin(X) grows in any form: [- π/2+ 2πk; π/2+ 2πk], k is an integer number and changes to any form: [π/2+ 2πk; 3π/2+ 2πk], k – whole number.
7) Function Y=sin(X) is a non-interruptible function. Looking at the graph of the function and changing it over, our function does not have any disruptions, which means there is no interruption.
8) Value area: increments [-1; 1]. This is also clearly visible from the graph of the function.
9) Function Y = sin (X) is a periodic function. Looking at the graph again, it is important that the function accumulates the same values, through intervals.

Apply command iz sine

1. Unravel the equation sin(x)= x-π

Solution: We will create 2 graphs of the function: y=sin(x) and y=x-π (div. figure).
Our graphs change at one point A(π;0), and this is the answer: x = π

2. Create a graph of the function y=sin(π/6+x)-1

Solution: The search for a graph results in a way to move the graph of the function y=sin(x) by π/6 units to the left and 1 unit down.

Solution: Let’s look at the graph of the function and look at our section [π/2; 5π/4].
The graph of the function shows that the highest and lowest values ​​are reached at the ends of the section, at points π/2 and 5π/4 in a row.
Example: sin(π/2) = 1 – the largest value, sin(5π/4) = the smallest value.

Prerequisite for sine for independent virtuous

• Unravel the equation: sin(x)= x+3π, sin(x)= x-5π
• Graph the function y=sin(π/3+x)-2
• Graph the function y=sin(-2π/3+x)+1
• Find the highest and lowest value of the function y=sin(x) per section
• Find the largest and smallest value of the function y=sin(x) per section [- π/3; 5π/6]

Functiony = sinx

The graph of the function is a sinusoid.

The repeated part of a sinusoid is called a sinusoid.

Half of the sinusoid is called the superior sinusoid (or arch).

Authority functionsy = sinx:

For daily schedule functions y= sin x It’s easy to create the following scale:

On arkush in a clique, for one cut we will accept two clints.

On the axis x We are celebrating the end of the day. With a handiness of 3.14, we can imagine 3 - without a fraction. Todi on arkush in a klitin π warehouse 6 klіtin (three 2 klіtin). And the skin tissue takes on its natural name (from the first to the third): π/6, π/3, π/2, 2π/3, 5π/6, π. This is significant x.

On the y-axis the value is 1, which includes two cells.

We create a table of function values, summarizing our values x:

The following is a flexible schedule. You will see the top point, which is (π/2; 1). This function graph y= sin x for a break. I'll add a symmetrical graphic to the bottom of the page (symmetrical to the beginning of the coordinates, then for the cut -?). The ridge is upright - under the weight x with coordinates (-1; -1). As a result, you will see a hive. This function graph y= sin x per cut [-π; π].

You can continue to live by staying for a break [π; 3π], [π; 5π], [π; 7π] etc. In all these sections of the graph, the function looks the same as in the section [-π; π]. There is an uninterrupted wriggling line with new wiggles.

Functiony = cosx.

The graph of the function is a sine wave (sometimes called a cosine wave).

Authority functionsy = cosx:

Functiony = mf(x).

Let's take the forward function y=cos x. As you already know, this graph is a sinusoid. If we multiply the cosine of this function by the number m, let it stretch along the axis x(or shrink, depending on the size of m).
This will be the graph of the function y = mf(x), where m is an active number.

Thus, the function y = mf(x) is the most important function for us, the function y = f(x), multiplied by m.

Yakshchom< 1, то синусоида сжимается к оси x for the coefficientm. Yakshchom > 1, then the sinusoid expands along the axisx for the coefficientm.

By either stretching or squeezing, you can initially get just one sinusoidal impulse, and then complete the entire schedule.

Functiony = f(kx).

What is the function y =mf(x) cause the sinusoid to stretch along the axis x or squeeze to the axle x, then the function y = f(kx) is brought to extension along the axis y or squeeze to the axle y.

Moreover, k is a real number.

At 0< k< 1 синусоида растягивается от оси y for the coefficientk. Yakshchok > 1, then the sinusoid is compressed to the axisy for the coefficientk.

By adding a graph of this function, you can initially generate one waveform of the sinusoid, and then use it to obtain the entire graph.

Functiony = tgx.

Function graph y= tg xє tangent.

It is enough to spend part of the graph on the interval from 0 to π/2, and then you can symmetrically continue on the interval from 0 to 3π/2.

Authority functionsy = tgx:

Functiony = ctgx

Function graph y=ctg x also a tangentoid (sometimes called a cotangentoid).

Authority functionsy = ctgx:

Geometrically different values ​​of sine and cosine

\(\sin \alpha = \dfrac(|BC|)(|AB|) \), \(\cos \alpha = \dfrac(|AC|)(|AB|) \)

α - Kut, expressions in radians.

Sine (sin α)– this is a trigonometric function of the cut between the hypotenuse and the leg of the rectum tricumus, equal to the length of the protilage leg |BC| before the hypotenuse | AB |.

Cosine (cos α)– this is a trigonometric function of the cut between the hypotenuse and the leg of the rectum tricumus, equal to the length of the adjacent leg |AC| before the hypotenuse | AB |.

Trigonometric value

Using additional formulas and meanings, you can find out the sine and cosine of the highest value. You just need to learn how to calculate the sine and cosine of a significant amount. The straight-cut tricut does not provide such capacity (a blunt cut, for example, cannot be found in anyone else); However, it is necessary to use more formal meanings of sine and cosine in order to place the meaning of the formula as a derivative.

Trigonometric colo come to help. Let it be given to you; This is illustrated by the same point on the trigonometric ring.

Small 2. Trigonometric value of sine and cosine

The cosine of the cut is the abscis of the point. The sine of the cut is the ce ordinate of the point.

In Fig. 2 rounds of takings are welcome, and it is easy to understand that this value is avoided from the hidden geometric meanings. In fact, it is a very straight-cut tricutnik with a single hypotenuse O and a gostrykut. Adjacent cathedral tricuputum є cos (align with Fig. 1) і at the same time abscis point; the proximal leg is sin (as in Fig. 1) and the hour ordinate of the point.

But now we are no longer limited by the first quarter and the possibility of expanding this value at any level is eliminated. In Fig. Figure 3 shows that the same sine and cosine are in the other, third and fourth quarters.

Small 3. Sine and cosine in the II, III and IV quarters

Table values ​​of sine and cosine

Null cut \(\LARGE 0^(\circ ) \)

The abscis of point 0 is equal to 1, the ordinate of point 0 is equal to 0. Also,

cos 0 = 1 sin 0 = 0

Fig 4. Null cut

Kut \(\LARGE \frac(\pi)(6) = 30^(\circ ) \)

Mi bachimo straight-cut tricut with a single hypotenuse and gostream cut 30°. Apparently, the leg lies opposite the 30° half of the hypotenuse 1; Otherwise, it seems that the vertical leg is older than 1/2 and, therefore,

\[ \sin \frac(\pi)(6) =\frac(1)(2) \]

The horizontal leg is known from the Pythagorean theorem (or, at the same time, the cosine is known from the basic trigonometric identity):

\[ \cos \frac(\pi)(6) = \sqrt(1 - \left(\frac(1)(2) \right)^(2) ) =\frac(\sqrt(3) )(2 ) \]

1 Why act like this? Cut the even-sided tricutule with side 2 of its height! It is divided into two rectilinear tricutudes with hypotenuse 2, a side of 30° and a smaller leg of 1.

Fig 5. Cut π/6

Kut \(\LARGE \frac(\pi)(4) = 45^(\circ ) \)

If the tricut is straight, it is isosceles; sine and cosine of 45 ° equal one to one. їх is significant in terms of x. Maemo:

\[ x^(2) + x^(2) = 1 \]

stars \(x=\frac(\sqrt(2) )(2) \). Otje,

\[ \cos \frac(\pi)(4) = \sin \frac(\pi)(4) =\frac(\sqrt(2) )(2) \]

Fig 5. Cut π/4

Power of sine and cosine

Accepted appointments

\(\sin^2 x \equiv (\sin x)^2; \)\(\quad \sin^3 x \equiv (\sin x)^3; \)\(\quad \sin^n x \equiv (\sin x)^n \)\(\sin^(-1) x \equiv \arcsin x \)\((\sin x)^(-1) \equiv \dfrac1(\sin x) \equiv \cosec x \).

\(\cos^2 x \equiv (\cos x)^2; \)\(\quad \cos^3 x \equiv (\cos x)^3; \)\(\quad \cos^n x \equiv (\cos x)^n \)\(\cos^(-1) x \equiv \arccos x \)\((\cos x)^(-1) \equiv \dfrac1(\cos x) \equiv \sec x \).

Frequency

The functions y = sin x and y = cos x are periodic with a period of 2π.

\(\sin(x + 2\pi) = \sin x; \quad \)\(\cos(x + 2\pi) = \cos x \)

Parity

The sine function is unpaired. The cosine function is parna.

\(\sin(-x) = - \sin x; \quad \)\(\cos(-x) = \cos x \)

Areas of significance, extremes, growth, decline

The main powers of sine and cosine are presented in the table ( n- whole).

Basic formulas for sine and cosine

Sum of squares

\(\sin^2 x + \cos^2 x = 1\)

Formulas for sine and cosine sum and difference

\(\sin(x + y) = \sin x \cos y + \cos x \sin y \)
\(\sin(x - y) = \sin x \cos y - \cos x \sin y \)
\(\cos(x + y) = \cos x \cos y - \sin x \sin y \)
\(\cos(x - y) = \cos x \cos y + \sin x \sin y \)

\(\sin(2x) = 2 \sin x \cos x \)
\(\cos(2x) = \cos^2 x - \sin^2 x = \)\(2 \cos^2 x - 1 = 1 - 2 \sin^2 x \)
\(\cos\left(\dfrac(\pi)2 - x \right) = \sin x \) ; \(\sin\left(\dfrac(\pi)2 - x \right) = \cos x \)
\(\cos(x + \pi) = - \cos x \); \(\sin(x + \pi) = - \sin x \)

Formulas for creating sines and cosines

\(\sin x \cos y = \) \(\dfrac12 (\Large [) \sin(x - y) + \sin(x + y) (\Large ]) \)
\(\sin x \sin y = \) \(\dfrac12 (\Large [) \cos(x - y) - \cos(x + y) (\Large ]) \)
\(\cos x \cos y = \) \(\dfrac12 (\Large [) \cos(x - y) + \cos(x + y) (\Large ]) \)

\(\sin x \cos y = \dfrac12 \sin 2x \)
\(\sin^2 x = \dfrac12 (\Large [) 1 - \cos 2x (\Large )) \)
\(\cos^2 x = \dfrac12 (\Large [) 1 + \cos 2x (\Large ]) \)

Formulas sumi and rіznitsi

\(\sin x + \sin y = 2 \, \sin \dfrac(x+y)2 \, \cos \dfrac(x-y)2 \)
\(\sin x - \sin y = 2 \, \sin \dfrac(x-y)2 \, \cos \dfrac(x+y)2 \)
\(\cos x + \cos y = 2 \, \cos \dfrac(x+y)2 \, \cos \dfrac(x-y)2 \)
\(\cos x - \cos y = 2 \, \sin \dfrac(x+y)2 \, \sin \dfrac(y-x)2 \)

Viraz sine through cosine

\(\sin x = \cos\left(\dfrac(\pi)2 - x \right) = \)\(\cos\left(x - \dfrac(\pi)2 \right) = - \cos\left(x + \dfrac(\pi)2 \right) \)\(\sin^2 x = 1 - \cos^2 x\) \(\sin x = \sqrt(1 - \cos^2 x) \) \(\( 2 \pi n \leqslant x \leqslant \pi + 2 \pi n \) \)\(\sin x = - \sqrt(1 - \cos^2 x) \) \(\( -\pi + 2 \pi n \leqslant x \leqslant 2 \pi n \) \).

Viraz cosine through sine

\(\cos x = \sin\left(\dfrac(\pi)2 - x \right) = \)\(- \sin\left(x - \dfrac(\pi)2 \right) = \sin\left(x + \dfrac(\pi)2 \right) \)\(\cos^2 x = 1 - \sin^2 x \) \(\cos x = \sqrt(1 - \sin^2 x) \) \(\( -\pi/2 + 2 \pi n \leqslant x \leqslant \pi/2 + 2 \pi n \) \)\(\cos x = - \sqrt(1 - \sin^2 x) \) \(\( \pi/2 + 2 \pi n \leqslant x \leqslant 3\pi/2 + 2 \pi n \) \).

Viraz via tangent

\(\sin^2 x = \dfrac(\tg^2 x)(1+\tg^2 x) \)\(\cos^2 x = \dfrac1(1+\tg^2 x) \).

At \(- \dfrac(\pi)2 + 2 \pi n< x < \dfrac{\pi}2 + 2 \pi n \) \(\sin x = \dfrac(\tg x)( \sqrt(1+\tg^2 x) ) \)\(\cos x = \dfrac1( \sqrt(1+\tg^2 x) ) \).

At \(\dfrac(\pi)2 + 2 \pi n< x < \dfrac{3\pi}2 + 2 \pi n \) :
\(\sin x = - \dfrac(\tg x)( \sqrt(1+\tg^2 x) ) \)\(\cos x = - \dfrac1( \sqrt(1+\tg^2 x) ) \).

Table of sines and cosines, tangents and cotangents

This table presents the values ​​of sines and cosines for various values ​​of the argument.
[ img style="max-width:500px;max-height:1080px;" src="tablitsa.png" alt="Table of sines and cosines" title="Table of sines and cosines" ]!}!}

Viruses through complex changes

\(i^2 = -1\)
\(\sin z = \dfrac(e^(iz) - e^(-iz))(2i) \)\(\cos z = \dfrac(e^(iz) + e^(-iz))(2) \)

Euler's formula

\(e^(iz) = \cos z + i \sin z \)

Expressions through hyperbolic functions

\(\sin iz = i \sh z \) \(\cos iz = \ch z \)
\(\sh iz = i \sin z \) \(\ch iz = \cos z \)

Pokhіdni

\((\sin x)" = \cos x \) \((\cos x)" = - \sin x \) . Summary of formulas > > >

Proceedings of the nth order:
\(\left(\sin x \right)^((n)) = \sin\left(x + n\dfrac(\pi)2 \right) \)\(\left(\cos x \right)^((n)) = \cos\left(x + n\dfrac(\pi)2 \right) \).

Integrals

\(\int \sin x \, dx = - \cos x + C \)\(\int \cos x \, dx = \sin x + C \)
also section Table of non-valuable integrals >>>

Unpacking to the lavas

\(\sin x = \sum_(n=0)^(\infty) \dfrac( (-1)^n x^(2n+1) )( (2n+1)! ) = \)\(x - \dfrac(x^3)(3 + \dfrac(x^5)(5) - \dfrac{x^7}{7!} + ... \) !}!} \(\(- \infty< x < \infty \} \)
\(\cos x = \sum_(n=0)^(\infty) \dfrac( (-1)^n x^(2n) )( (2n)! ) = \)\(1 - \dfrac(x^2)(2 + \dfrac(x^4)(4 - \dfrac{x^6}{6!} + ... \) !}!} \(\( - \infty< x < \infty \} \)

Secant, cosecant

\(\sec x = \dfrac1( \cos x ) ; \) \(\cosec x = \dfrac1( \sin x ) \)

Gate functions

Return functions to sine and cosine, arcsine and arccosine, obviously.

Arcsinus, arcsin

\(y = \arcsin x\) \(\left\( -1 \leqslant x \leqslant 1; \; - \dfrac(\pi)2 \leqslant y \leqslant \dfrac(\pi)2 \right\) \)
\(\sin(\arcsin x) = x\)
\(\arcsin(\sin x) = x\) \(\left\( - \dfrac(\pi)2 \leqslant x \leqslant \dfrac(\pi)2 \right\) \)

Arccosine, arccos

\(y = \arccos x\) \(\left\( -1 \leqslant x \leqslant 1; \; 0 \leqslant y \leqslant \pi \right\) \)
\(\cos(\arccos x) = x \) \(\( -1 \leqslant x \leqslant 1 \) \)
\(\arccos(\cos x) = x\) \(\( 0 \leqslant x \leqslant \pi \) \)

Wikorystan literature:
I.M. Bronstein, K.A. Semendyaev, Adviser on mathematics for engineers and university students, “Lan”, 2009.

FUNCTION GRAPHICS

Sine function

- impersonal R all active numbers.

Impersonal meaning of function- Vidrazok [-1; 1], then. sine function - lined.

The function is unpaired: sin(−x)=−sin x for all x ∈ R.

The function is periodic

sin(x+2π k) = sin x, where k ∈ Z for all x ∈ R.

sin x = 0 for x = π k, k ∈ Z.

sin x > 0(positive) for all x ∈ (2π·k , π+2π·k ), k ∈ Z.

sin x< 0 (negative) for all x ∈ (π+2π·k , 2π+2π·k ), k ∈ Z.

Cosine function

Function area- impersonal R all active numbers.

Impersonal meaning of function- Vidrazok [-1; 1], that is. cosine function - lined.

Parn function: cos(−x)=cos x for all x ∈ R.

The function is periodic with the smallest positive period 2π:

cos(x+2π k) = cos x, de k ∈ Z for all x ∈ R.

cos x = 0 at
cos x > 0 for everyone
cos x< 0 for everyone
The function is growing from −1 to 1 on intervals:
Function changes from −1 to 1 on intervals:
The highest value of the function sin x = 1 at points:
The smallest value of the function sin x = −1 at points:

Tangent function

Impersonal meaning of function- Wusya is numerically straight, tobto. tangent - function unbound.

The function is unpaired: tg(−x)=−tg x
The graph of the function is symmetrical along the OY axis.

The function is periodic with the smallest positive period π, then. tg(x+π k) = tan x, k ∈ Z for everyone in the area of ​​significance.

Cotangent function

Impersonal meaning of function- Wusya is numerically straight, tobto. cotangent - function unbound.

The function is periodic with the smallest positive period π, then. cotg(x+π k)=ctg x, k ∈ Z for everyone in the area of ​​significance.

Arcsine function

Function area- Vidrazok [-1; 1]

Impersonal meaning of function- Vidrazok -π / 2 arcsin x π / 2, then. arcsine - function lined.

The function is unpaired: arcsin(−x)=−arcsin x for all x ∈ R.
The graph of the function is symmetrical around the coordinates.

Throughout the region there is a difference.

Arc cosine function

Function area- Vidrazok [-1; 1]

Impersonal meaning of function- Vidrazok 0 arccos x π, then. arc cosine - function lined.

The function is growing throughout the entire area of ​​​​significance.

Arctangent function

Function area- impersonal R all active numbers.

Impersonal meaning of function- Vidrezok 0 π, then. arctangent - function lined.

The function is unpaired: arctg(−x)=−arctg x for all x ∈ R.
The graph of the function is symmetrical around the coordinates.

The function is growing throughout the entire area of ​​​​significance.

Arc tangent function

Function area- impersonal R all active numbers.

Impersonal meaning of function- Vidrezok 0 π, then. arccotangent - function lined.

The function is neither paired nor unpaired.
The graph of the function is asymmetrical around the coordinates and not around the Oy axis.

The function is depressed throughout the entire area of ​​​​significance.