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 = sin x
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 = sin x 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 = sin x 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 = sin x determine: a) sin 2; b) sin 4; c) sin (-3).
2.Schedule function y = sin x
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 = sin x
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").
>>Mathematics: Functions y \u003d sin x, y \u003d cos x, their properties and graphs
Functions y \u003d sin x, y \u003d cos x, their properties and graphs
In this section we discuss some properties of the functions y = sin x,y= cos x and plot their graphs.
1. Function y \u003d 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. We note some of its properties.
Properties of the function u = sint.
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 proved in § 19, for any t the equality
This means that the graph of the function u \u003d sin t, like the graph of any odd function, is symmetrical with respect to the origin at rectangular system coordinates tOi.
The function u = sin t increases on the interval 
This follows from the fact that when the point moves along the first quarter of the numerical circle, the ordinate gradually increases (from 0 to 1 - see Fig. 115), and when the point moves along the second quarter of the numerical circle, the ordinate gradually decreases (from 1 to 0 - see Fig. 115). Fig. 116).
The function u = sin t is bounded both from below and from above. This follows from the fact that, as we saw in § 19, for any t the inequality
(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 construct a graph of the function of interest to us. But (attention!) instead of u - sin t, we will write y \u003d sin x (after all, we are more accustomed to writing y \u003d f (x), and not u \u003d f (t)). This means that we will build a graph in the usual coordinate system хОу (and not tOy).
Let's make a table of function values \u200b\u200by - sin x:
Comment.
Here is one of the versions of the origin of the term "sine". In Latin, sinus means bend (bowstring).
The constructed graph to some extent justifies this terminology. 
The line that serves as a graph of the function y \u003d sin x is called a sinusoid. That part of the sinusoid, which is shown in Fig. 118 or 119, is called a sinusoid wave, and that part of the sinusoid, which is shown in fig. 117 is called a half-wave or arch of a sine wave.
2. Function y = cos x.
The study of the function y \u003d cos x could be carried out approximately according to the same scheme that was used above for the function y \u003d 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 so far have only an auxiliary value for our purposes.
For any value of t, the equalities
Proof. Let the number t correspond to the point M of the numerical n circle, and the number * + - to the point P (Fig. 124; for the sake of simplicity, we took the point M in the first quarter). The arcs AM and BP are equal, respectively, and right-angled triangles OKM and OBP are also equal. Hence, O K = Ob, MK = Pb. From these equalities and from the location of the triangles OKM and OLR in the coordinate system, we draw two conclusions:
1) the ordinate of the point P both in absolute value and in sign coincides with the abscissa of the point M; it means that
2) the abscissa of the point P is equal in absolute value to the ordinate of the point M, but differs from it in sign; it means that
Approximately the same reasoning is carried out in cases where the point M does not belong to the first quarter.
Let's use the formula 
(this is the formula proved above, only 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, so their graphs are the same.
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). Associate the function y \u003d sin x to new system coordinates - 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 \u003d sin x, is called a sinusoid (which is quite natural).
Properties of the function y = cos x.
y = cos x is an even function.
The stages of construction are shown in fig. 126:
1) we build a graph of the function y \u003d cos x (more precisely, one half-wave);
2) by stretching the constructed graph from the x-axis with a coefficient of 0.5, we get one half-wave of the required graph;
3) using the resulting half-wave, we build the entire graph of the function y \u003d 0.5 cos x.
|BD|- the length of the arc of a circle centered at a point A.
α
is an angle expressed in radians.
sine ( sinα) - This 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 π.
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.
