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Iron rusts, not finding a use for itself,
standing water rots or freezes in the cold,
and the human mind, not finding a use for itself, languishes.
Leonardo da Vinci
Used technologies: problem-based learning, critical thinking, communicative communication.
Goals:
- Development of cognitive interest in learning.
- Studying the properties of the function y \u003d sin x.
- Formation of practical skills for constructing a graph of the function y \u003d sin x based on the studied theoretical material.
Tasks:
1. Use the existing potential of knowledge about the properties of the function y \u003d sin x in specific situations.
2. Apply the conscious establishment of links between the analytical and geometric models of the function y \u003d sin x.
Develop initiative, a certain readiness and interest in finding a solution; the ability to make decisions, not to stop there, to defend one's point of view.
To educate students in cognitive activity, a sense of responsibility, respect for each other, mutual understanding, mutual support, self-confidence; culture of communication.
During the classes
Stage 1. Actualization of basic knowledge, motivation for learning new material
"Lesson Entry"
There are 3 statements written on the board:
- The trigonometric equation sin t = a always has solutions.
- An odd function can be graphed using a symmetry transformation about the y-axis.
- Schedule trigonometric function can be built using one main half-wave.
Students discuss in pairs: Are the statements true? (1 minute). The results of the initial discussion (yes, no) are then entered into the table in the "Before" column.
The teacher sets the goals and objectives of the lesson.
2. Updating knowledge (frontally on the trigonometric circle model).
We have already met with the function s = sin t.
1) What values can the variable t take. What is the scope of this function?
2) In what interval are the values of the expression sin t. Find the largest and smallest values of the function s = sin t.
3) Solve the equation sin t = 0.
4) What happens to the ordinate of the point as it moves along the first quarter? (the ordinate increases). What happens to the ordinate of a point as it moves along the second quarter? (the ordinate gradually decreases). How does this relate to the monotonicity of the function? (the function s = sin t increases on the segment and decreases on the segment ).
5) Let's write the function s = sin t in the usual form for us y = sin x (we will build in the usual xOy coordinate system) and compile a table of values for this function.
| X | 0 | ||||||
| at | 0 | 1 | 0 |
Stage 2. Perception, comprehension, primary consolidation, involuntary memorization
Stage 4. Primary systematization of knowledge and methods of activity, their transfer and application in new situations
6. No. 10.18 (b, c)
Stage 5 Final control, correction, assessment and self-assessment
7. We return to the statements (the beginning of the lesson), discuss using the properties of the trigonometric function y \u003d sin x, and fill in the "After" column in the table.
8. D / z: item 10, Nos. 10.7(a), 10.8(b), 10.11(b), 10.16(a)
In this lesson, we will consider in detail the function y \u003d sin x, its main properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y \u003d sin t on the coordinate circle and consider the graph of the function on the circle and the line. Let's show the periodicity of this function on the graph and consider the main properties of the function. At the end of the lesson, we will solve some simple problems using the graph of the function and its properties.
Topic: Trigonometric functions
Lesson: Function y=sinx, its main properties and graph
When considering a function, it is important to associate a single value of the function with each value of the argument. This law of correspondence and is called a function.
Let us define the correspondence law for .
Any real number corresponds to a single point on the unit circle. The point has a single ordinate, which is called the sine of the number (Fig. 1).
Each argument value is assigned a single function value.
Obvious properties follow from the definition of the sine.
The figure shows that 
because is the ordinate of a point on the unit circle.
Consider the function graph. Let us recall the geometric interpretation of the argument. The argument is the central angle measured in radians. On the axis, we will plot real numbers or angles in radians, along the axis, the corresponding function values.
For example, the angle on the unit circle corresponds to a point on the graph (Fig. 2)
We got the graph of the function on the site. But knowing the period of the sine, we can depict the graph of the function on the entire domain of definition (Fig. 3).
The main period of the function is This means that the graph can be obtained on a segment and then continue to the entire domain of definition.
Consider the properties of the function:
1) Domain of definition:
2) Range of values: 
3) Function odd:
4) The smallest positive period:
5) Coordinates of the points of intersection of the graph with the x-axis: 
6) Coordinates of the point of intersection of the graph with the y-axis:
7) Intervals on which the function takes positive values:
8) Intervals at which the function takes negative values:
9) Increasing intervals:
10) Descending intervals:
11) Low points: 
12) Minimum features:
13) High points: 
14) Maximum features:
We have considered the properties of a function and its graph. Properties will be repeatedly used in solving problems.
Bibliography
1. Algebra and the beginning of analysis, grade 10 (in two parts). Textbook for educational institutions ( profile level) ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and the beginning of analysis, grade 10 (in two parts). Task book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
3. Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I. Algebra and mathematical analysis for grade 10 ( tutorial for students of schools and classes with in-depth study of mathematics).-M .: Education, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. Deep Learning algebra and mathematical analysis.-M.: Education, 1997.
5. Collection of problems in mathematics for applicants to technical universities (under the editorship of M.I.Skanavi).-M.: Higher school, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic trainer.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Tasks in Algebra and the Beginnings of Analysis (a manual for students in grades 10-11 of general educational institutions).-M .: Education, 2003.
8. Karp A.P. Collection of problems in algebra and the beginnings of analysis: textbook. allowance for 10-11 cells. with a deep study mathematics.-M.: Education, 2006.
Homework
Algebra and the Beginnings of Analysis, Grade 10 (in two parts). Task book for educational institutions (profile level), ed.
A. G. Mordkovich. -M.: Mnemosyne, 2007.
№№ 16.4, 16.5, 16.8.
Additional web resources
3. Educational portal to prepare for exams ().
We found out that the behavior of trigonometric functions, and the functions y = sin x in particular, on the entire number line (or for all values of the argument X) is completely determined by its behavior in the interval 0 < X < π / 2 .
Therefore, first of all, we will plot the function y = sin x exactly in this interval.
Let's make the following table of values of our function;
By marking the corresponding points on the coordinate plane and connecting them with a smooth line, we get the curve shown in the figure
The resulting curve could also be constructed geometrically without compiling a table of function values y = sin x .
1. The first quarter of a circle of radius 1 is divided into 8 equal parts. The ordinates of the division points of the circle are the sines of the corresponding angles.
2. The first quarter of the circle corresponds to angles from 0 to π / 2 . Therefore, on the axis X Take a segment and divide it into 8 equal parts.
3.Let's draw straight lines parallel to the axis X, and from the division points we restore the perpendiculars to the intersection with the horizontal lines.
4. Connect the intersection points with a smooth line.
Now let's look at the interval π /
2
<
X <
π
.
Each argument value X from this interval can be represented as
x = π / 2 + φ
Where 0 < φ < π / 2 . According to the reduction formulas
sin( π / 2 + φ ) = cos φ = sin ( π / 2 - φ ).
Axis points X with abscissa π / 2 + φ And π / 2 - φ symmetrical to each other about the axis point X with abscissa π / 2 , and the sines at these points are the same. This allows you to get a graph of the function y = sin x in the interval [ π / 2 , π ] by simply symmetrically displaying the graph of this function in the interval relative to the straight line X = π / 2 .
Now using the property odd function y \u003d sin x,
sin(- X) = -sin X,
it is easy to plot this function in the interval [- π , 0].
The function y \u003d sin x is periodic with a period of 2π ;. Therefore, to build the entire graph of this function, it is enough to continue the curve shown in the figure to the left and right periodically with a period 2π .
The resulting curve is called sinusoid . It is the graph of the function y = sin x.
The figure well illustrates all those properties of the function y = sin x , which were previously proven by us. Recall these properties.
1) Function y = sin x defined for all values X , so that its domain is the set of all real numbers.
2) Function y = sin x limited. All values it takes are between -1 and 1, including those two numbers. Therefore, the range of this function is determined by the inequality -1 < at < 1. When X = π / 2 + 2k π function takes highest values, equal to 1, and at x = - π / 2 + 2k π - smallest values, equal to - 1.
3) Function y = sin x is odd (the sinusoid is symmetrical with respect to the origin).
4) Function y = sin x periodic with period 2 π .
5) In intervals 2n π < x < π + 2n π (n is any integer) it is positive, and in intervals π + 2k π < X < 2π + 2k π (k is any integer) it is negative. For x = k π the function goes to zero. Therefore, these values of the argument x (0; ± π ; ±2 π ; ...) are called zeros of the function y = sinx
6) In intervals - π / 2 + 2n π < X < π / 2 + 2n π function y = sin x increases monotonically, and in intervals π / 2 + 2k π < X < 3π / 2 + 2k π it monotonically decreases.
Pay special attention to the behavior of the function y = sinx near the point X = 0 .
For example, sin 0.012 ≈ 0.012; sin(-0.05) ≈ -0,05;
sin2° = sin π 2 / 180=sin π / 90 ≈ 0,03 ≈ 0,03.
However, it should be noted that for any values of x
| sin x| < | x | . (1)
Indeed, let the radius of the circle shown in the figure be equal to 1,
a /
AOB = X.
Then sin x= AC. But AU< АВ, а АВ, в свою очередь, меньше длины дуги АВ, на которую опирается угол X. The length of this arc is obviously equal to X, since the radius of the circle is 1. So, for 0< X < π / 2
sin x< х.
Hence, due to the oddness of the function y = sinx it is easy to show that when - π / 2 < X < 0
| sin x| < | x | .
Finally, at x = 0
| sin x | = | x |.
Thus, for | X | < π / 2 inequality (1) is proved. In fact, this inequality is also true for | x | > π / 2 due to the fact that | | sin X | < 1, a π / 2 > 1
Exercises
1.According to the function schedule y = sinx determine: a) sin 2; b) sin 4; c) sin (-3).
2.Schedule function y = sinx
determine which number from the interval
[ - π /
2 ,
π /
2
] has a sine equal to: a) 0.6; b) -0.8.
3. Scheduled function y = sinx
determine which numbers have a sine,
equal to 1 / 2 .
4. Find approximately (without using tables): a) sin 1°; b) sin 0.03;
c) sin (-0.015); d) sin (-2°30").
In this lesson, we will consider in detail the function y \u003d sin x, its main properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y \u003d sin t on the coordinate circle and consider the graph of the function on the circle and the line. Let's show the periodicity of this function on the graph and consider the main properties of the function. At the end of the lesson, we will solve some simple problems using the graph of the function and its properties.
Topic: Trigonometric functions
Lesson: Function y=sinx, its main properties and graph
When considering a function, it is important to associate a single value of the function with each value of the argument. This law of correspondence and is called a function.
Let us define the correspondence law for .
Any real number corresponds to a single point on the unit circle. The point has a single ordinate, which is called the sine of the number (Fig. 1).
Each argument value is assigned a single function value.
Obvious properties follow from the definition of the sine.
The figure shows that because is the ordinate of a point on the unit circle.
Consider the function graph. Let us recall the geometric interpretation of the argument. The argument is the central angle measured in radians. On the axis, we will plot real numbers or angles in radians, along the axis, the corresponding function values.
For example, the angle on the unit circle corresponds to a point on the graph (Fig. 2)
We got the graph of the function on the site. But knowing the period of the sine, we can depict the graph of the function on the entire domain of definition (Fig. 3).
The main period of the function is This means that the graph can be obtained on a segment and then continue to the entire domain of definition.
Consider the properties of the function:
1) Domain of definition:
2) Range of values:
3) Function odd:
4) The smallest positive period:
5) Coordinates of the points of intersection of the graph with the x-axis:
6) Coordinates of the point of intersection of the graph with the y-axis:
7) Intervals on which the function takes positive values:
8) Intervals at which the function takes negative values:
9) Increasing intervals:
10) Descending intervals:
11) Low points:
12) Minimum features:
13) High points:
14) Maximum features:
We have considered the properties of a function and its graph. Properties will be repeatedly used in solving problems.
Bibliography
1. Algebra and the beginning of analysis, grade 10 (in two parts). Textbook for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and the beginning of analysis, grade 10 (in two parts). Task book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
3. Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I. Algebra and mathematical analysis for grade 10 (textbook for students of schools and classes with in-depth study of mathematics). - M .: Education, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. An in-depth study of algebra and mathematical analysis.-M .: Education, 1997.
5. Collection of problems in mathematics for applicants to technical universities (under the editorship of M.I.Skanavi).-M.: Higher school, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic trainer.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Tasks in Algebra and the Beginnings of Analysis (a manual for students in grades 10-11 of general educational institutions).-M .: Education, 2003.
8. Karp A.P. Collection of problems in algebra and the beginnings of analysis: textbook. allowance for 10-11 cells. with a deep study mathematics.-M.: Education, 2006.
Homework
Algebra and the Beginnings of Analysis, Grade 10 (in two parts). Task book for educational institutions (profile level), ed.
A. G. Mordkovich. -M.: Mnemosyne, 2007.
№№ 16.4, 16.5, 16.8.
Additional web resources
3. Educational portal for exam preparation ().
|BD|- the length of the arc of a circle centered at a point A.
α
is an angle expressed in radians.
sine ( sinα) is a trigonometric function depending on the angle α between the hypotenuse and the leg right triangle, equal to the ratio the length of the opposite leg |BC| to the length of the hypotenuse |AC|.
cosine ( cosα) 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.
