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Iron rusts without finding any use,
standing water rots or freezes in the cold,
and a person’s mind, not finding any use for itself, languishes.
Leonardo da Vinci
Technologies used: problem-based learning, critical thinking, communicative communication.
Goals:
- Development of cognitive interest in learning.
- Studying the properties of the function y = sin x.
- Formation of practical skills in constructing a graph of the function y = sin x based on the studied theoretical material.
Tasks:
1. Use the existing potential of knowledge about the properties of the function y = sin x in specific situations.
2. Apply conscious establishment of connections between analytical and geometric models of the function y = sin x.
Develop initiative, a certain willingness and interest in finding a solution; the ability to make decisions, not stop there, and defend your point of view.
To foster in students cognitive activity, a sense of responsibility, respect for each other, mutual understanding, mutual support, and self-confidence; culture of communication.
Lesson progress
Stage 1. Updating basic knowledge, motivating learning new material
"Entering the lesson."
There are 3 statements written on the board:
- The trigonometric equation sin t = a always has solutions.
- The graph of an odd function can be constructed using a symmetry transformation about the Oy axis.
- A trigonometric function can be graphed using one principal 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 a model of a trigonometric circle).
We have already become acquainted 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 contained? 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 a 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 form y = sin x that is familiar to us (we will construct it in the usual xOy coordinate system) and compile a table of the values of 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 (beginning of the lesson), discuss using the properties of the trigonometric function y = sin x, and fill in the “After” column in the table.
8. D/z: clause 10, No. 10.7(a), 10.8(b), 10.11(b), 10.16(a)
In this lesson we will take a detailed look at the function y = sin x, its basic properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y = sin t on the coordinate circle and consider the graph of the function on the circle and 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 several simple problems using the graph of a function and its properties.
Topic: Trigonometric functions
Lesson: Function y=sinx, its basic properties and graph
When considering a function, it is important to associate each argument value with a single function value. 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. A point has a single ordinate, which is called the sine of the number (Fig. 1).
Each argument value is associated with a single function value.
Obvious properties follow from the definition of sine.
The figure shows that 
because is the ordinate of a point on the unit circle.
Consider the graph of the function. Let us recall the geometric interpretation of the argument. The argument is the central angle, measured in radians. Along the axis we will plot real numbers or angles in radians, along the axis the corresponding values of the function.
For example, an angle on the unit circle corresponds to a point on the graph (Fig. 2)
We have obtained a graph of the function in the area. But knowing the period of the sine, we can depict the graph of the function over 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 continued throughout the entire domain of definition.
Consider the properties of the function:
1) Scope of definition:
2) Range of values: 
3) Odd function:
4) Smallest positive period:
5) Coordinates of the points of intersection of the graph with the abscissa axis: 
6) Coordinates of the point of intersection of the graph with the ordinate axis:
7) Intervals at which the function takes positive values:
8) Intervals at which the function takes negative values:
9) Increasing intervals:
10) Decreasing intervals:
11) Minimum points: 
12) Minimum functions:
13) Maximum points: 
14) Maximum functions:
We looked at the properties of the function and its graph. The properties will be used repeatedly when solving problems.
References
1. Algebra and beginning of analysis, grade 10 (in two parts). Textbook for general education institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and beginning of analysis, grade 10 (in two parts). Problem 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.: Prosveshchenie, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. In-depth study of algebra and mathematical analysis.-M.: Education, 1997.
5. Collection of problems in mathematics for applicants to higher educational institutions (edited by M.I. Skanavi). - M.: Higher School, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic simulator.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Problems on algebra and principles of analysis (a manual for students in grades 10-11 of general education institutions). - M.: Prosveshchenie, 2003.
8. Karp A.P. Collection of problems on algebra and principles of analysis: textbook. allowance for 10-11 grades. with depth studied Mathematics.-M.: Education, 2006.
Homework
Algebra and beginning of analysis, grade 10 (in two parts). Problem 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 preparing for exams ().
Functiony = sinx
The graph of the function is a sinusoid.
The complete non-repeating portion of a sine wave is called a sine wave.
Half a sine wave is called a half sine wave (or arc).
Function propertiesy =
sinx:
3) This is an odd function. 4) This is a continuous function.
6) On the segment [-π/2; π/2] function increases on the interval [π/2; 3π/2] – decreases. 7) At intervals the function takes positive values. 8) Intervals of increasing function: [-π/2 + 2πn; π/2 + 2πn]. 9) Minimum points of the function: -π/2 + 2πn. |
To graph a function y= sin x It is convenient to use the following scales:
On a sheet of paper with a square, we take the length of two squares as a unit of segment.
On axis x Let's measure the length π. At the same time, for convenience, we present 3.14 in the form of 3 - that is, without a fraction. Then on a sheet of paper in a cell π will be 6 cells (three times 2 cells). And each cell will receive its own natural name (from the first to the sixth): π/6, π/3, π/2, 2π/3, 5π/6, π. These are the meanings x.
On the y-axis we mark 1, which includes two cells.
Let's create a table of function values using our values x:
√3 | √3 |
Next we will create a schedule. The result is a half-wave, the highest point of which is (π/2; 1). This is the graph of the function y= sin x on the segment. Let's add a symmetrical half-wave to the constructed graph (symmetrical relative to the origin, that is, on the segment -π). The crest of this half-wave is under the x-axis with coordinates (-1; -1). The result will be a wave. This is the graph of the function y= sin x on the segment [-π; π].
You can continue the wave by constructing it on the segment [π; 3π], [π; 5π], [π; 7π], etc. On all these segments, the graph of the function will look the same as on the segment [-π; π]. You will get a continuous wavy line with identical waves.
Functiony = cosx.
The graph of a function is a sine wave (sometimes called a cosine wave).
Function propertiesy = cosx:
1) The domain of definition of a function is the set of real numbers. 2) The range of function values is the segment [–1; 1] 3) This is an even function. 4) This is a continuous function. 5) Coordinates of the intersection points of the graph: 6) On the segment the function decreases, on the segment [π; 2π] – increases. 7) On intervals [-π/2 + 2πn; π/2 + 2πn] function takes positive values. 8) Increasing intervals: [-π + 2πn; 2πn]. 9) Minimum points of the function: π + 2πn. 10) The function is limited from above and below. The smallest value of the function is –1, 11) This is a periodic function with a period of 2π (T = 2π) |
Functiony = mf(x).
Let's take the previous function y=cos x. As you already know, its graph is a sine wave. If we multiply the cosine of this function by a certain number m, then the wave will expand from the axis x(or will shrink, depending on the value of m).
This new wave will be the graph of the function y = mf(x), where m is any real number.
Thus, the function y = mf(x) is the familiar function y = f(x) multiplied by m.
Ifm< 1, то синусоида сжимается к оси x by the coefficientm. Ifm > 1, then the sinusoid is stretched from the axisx by the coefficientm.
When performing stretching or compression, you can first plot only one half-wave of a sine wave, and then complete the entire graph.
Functiony= f(kx).
If the function y=mf(x) leads to stretching of the sinusoid from the axis x or compression towards the axis x, then the function y = f(kx) leads to stretching from the axis y or compression towards the axis y.
Moreover, k is any real number.
At 0< k< 1 синусоида растягивается от оси y by the coefficientk. Ifk > 1, then the sinusoid is compressed towards the axisy by the coefficientk.
When graphing this function, you can first build one half-wave of a sine wave, and then use it to complete the entire graph.
Functiony = tgx.
Function graph y= tg x is a tangent.
It is enough to construct part of the graph in the interval from 0 to π/2, and then you can symmetrically continue it in the interval from 0 to 3π/2.
Function propertiesy = tgx:
Functiony = ctgx
Function graph y=ctg x is also a tangentoid (it is sometimes called a cotangentoid).
Function propertiesy = ctgx:
FUNCTION GRAPHICS
Sine function
- many R all real numbers.
Multiple Function Values— segment [-1; 1], i.e. sine function - limited.
Odd function: 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 Domain- many R all real numbers.
Multiple Function Values— segment [-1; 1], i.e. cosine function - limited.
Even 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, where k ∈ Z for all x ∈ R.
| cos x = 0 at |  |
| cos x > 0 for everyone |  |
| cos x< 0 for everyone |  |
| Function increases from −1 to 1 on intervals: | |
| The function is decreasing from −1 to 1 on intervals: | |
| The largest value of the function sin x = 1 at points: |  |
| The smallest value of the function sin x = −1 at points: |
Tangent function
Multiple Function Values— the entire number line, i.e. tangent - function unlimited.
Odd function: tg(−x)=−tg x
The graph of the function is symmetrical about the OY axis.
The function is periodic with the smallest positive period π, i.e. tg(x+π k) = tan x, k ∈ Z for all x from the domain of definition.
Cotangent function
Multiple Function Values— the entire number line, i.e. cotangent - function unlimited.
Odd function: ctg(−x)=−ctg x for all x from the domain of definition.The graph of the function is symmetrical about the OY axis.
The function is periodic with the smallest positive period π, i.e. cotg(x+π k)=ctg x, k ∈ Z for all x from the domain of definition.
Arcsine function
Function Domain— segment [-1; 1]
Multiple Function Values- segment -π /2 arcsin x π /2, i.e. arcsine - function limited.
Odd function: arcsin(−x)=−arcsin x for all x ∈ R.
The graph of the function is symmetrical about the origin.
Throughout the entire definition area.
Arc cosine function
Function Domain— segment [-1; 1]
Multiple Function Values— segment 0 arccos x π, i.e. arccosine - function limited.
The function is increasing over the entire definition area.
Arctangent function
Function Domain- many R all real numbers.
Multiple Function Values— segment 0 π, i.e. arctangent - function limited.
Odd function: arctg(−x)=−arctg x for all x ∈ R.
The graph of the function is symmetrical about the origin.
The function is increasing over the entire definition area.
Arc tangent function
Function Domain- many R all real numbers.
Multiple Function Values— segment 0 π, i.e. arccotangent - function limited.
The function is neither even nor odd.
The graph of the function is asymmetrical neither with respect to the origin nor with respect to the Oy axis.
The function is decreasing over the entire definition area.
>>Mathematics: Functions y = sin x, y = cos x, their properties and graphs
Functions y = sin x, y = cos x, their properties and graphs
In this section we will discuss some properties of the functions y = sin x, y = cos x and construct their graphs.
1. Function y = sin X.
Above, in § 20, we formulated a rule that allows each number t to be associated with the number cos t, i.e. characterized the function y = sin t. Let us note some of its properties.
Properties of the function u = sin t.
The domain of definition is the set K of real numbers.
This follows from the fact that any number 2 corresponds to a point M(1) on the number circle, which has a well-defined ordinate; this ordinate is cos t.
u = sin t is an odd function.
This follows from the fact that, as was proven in § 19, for any t the equality
This means that the graph of the function u = sin t, like the graph of any odd function, is symmetrical with respect to the origin in the rectangular coordinate system tOi.
The function u = sin t increases on the interval 
This follows from the fact that when a point moves along the first quarter of the number circle, the ordinate gradually increases (from 0 to 1 - see Fig. 115), and when the point moves along the second quarter of the number circle, the ordinate gradually decreases (from 1 to 0 - see Fig. 116).
The function u = sint is bounded both below and above. This follows from the fact that, as we saw in § 19, for any t the inequality holds
(the function reaches this value at any point of the form 
(the function reaches this value at any point of the form
Using the obtained properties, we will construct a graph of the function of interest to us. But (attention!) instead of u - sin t we will write y = sin x (after all, we are more accustomed to writing y = f(x), and not u = f(t)). This means that we will build a graph in the usual xOy coordinate system (and not tOy).
Let's make a table of the values of the function y - sin x:
Comment.
Let us give one of the versions of the origin of the term “sine”. In Latin, sinus means bend (bow string).
The constructed graph to some extent justifies this terminology. 
The line that serves as a graph of the function y = sin x is called a sine wave. That part of the sinusoid that is shown in Fig. 118 or 119 is called a sine wave, and that part of the sine wave that is shown in Fig. 117 is called a half-wave or arc of a sine wave.
2. Function y = cos x.
The study of the function y = cos x could be carried out approximately according to the same scheme that was used above for the function y = sin x. But we will choose the path that leads to the goal faster. First, we will prove two formulas that are important in themselves (you will see this in high school), but for now have only auxiliary significance for our purposes.
For any value of t the following equalities are valid:
Proof. Let the number t correspond to point M of the numerical circle n, and the number * + - point P (Fig. 124; for the sake of simplicity, we took point M in the first quarter). The arcs AM and BP are equal, and the right triangles OKM and OLBP are correspondingly equal. This means O K = Ob, MK = Pb. From these equalities and from the location of triangles OCM and OBP in the coordinate system, we draw two conclusions:
1) the ordinate of point P coincides in absolute value and sign with the abscissa of point M; this means that
2) the abscissa of point P is equal in absolute value to the ordinate of point M, but differs in sign from it; this means that
Approximately the same reasoning is carried out in cases where point M does not belong to the first quarter.
Let's use the formula 
(this is the formula proven above, but instead of the variable t we use the variable x). What does this formula give us? It allows us to assert that the functions
are identical, which means their graphs coincide.
Let's plot the function 
To do this, let's move on to an auxiliary coordinate system with the origin at a point (the dotted line is drawn in Fig. 125). Let's bind the function y = sin x to the new coordinate system - this will be the graph of the function 
(Fig. 125), i.e. graph of the function y - cos x. It, like the graph of the function y = sin x, is called a sine wave (which is quite natural).
Properties of the function y = cos x.
y = cos x is an even function.
The construction stages are shown in Fig. 126:
1) build a graph of the function y = cos x (more precisely, one half-wave);
2) by stretching the constructed graph from the x-axis with a factor of 0.5, we obtain one half-wave of the required graph;
3) using the resulting half-wave, we construct the entire graph of the function y = 0.5 cos x.
Graph of sin 0. Math lesson. Topic: "Function y=sin x, its properties and graph." Expressing sine through cosine
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