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 obtain 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. Divide the first quarter of a circle of radius 1 into 8 equal parts. The ordinates of the dividing 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 Let's take a segment and divide it into 8 equal parts.
3. Let's draw straight lines parallel to the axes X, and from the division points we construct perpendiculars until they intersect with 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 reduction formulas
sin ( π / 2 + φ ) = cos φ = sin( π / 2 - φ ).
Axis points X with abscissas 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 us to obtain 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 parity function y = sin x,
sin(- X) = - sin X,
it is easy to plot this function in the interval [- π , 0].
The function y = sin x is periodic with a period of 2π ;. Therefore, to construct 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 represents the graph of the function y = sin x.
The figure illustrates well all the properties of the function y = sin x , which we have previously proven. Let us recall these properties.
1) Function y = sin x defined for all values X , so its domain is the set of all real numbers.
2) Function y = sin x limited. All the values it accepts are between -1 and 1, including these two numbers. Consequently, the range of variation of this function is determined by the inequality -1 < at < 1. When X = π / 2 + 2k π the function takes the largest values equal to 1, and for x = - π / 2 + 2k π - the smallest values equal to - 1.
3) Function y = sin x is odd (the sinusoid is symmetrical about the origin).
4) Function y = sin x periodic with period 2 π .
5) In 2n intervals 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. At x = k π the function goes to zero. Therefore, these values of the argument x (0; ± π ; ±2 π ; ...) are called function zeros y = sin x
6) At 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 decreases monotonically.
You should 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;
sin 2° = sin π 2 / 180 = sin π / 90 ≈ 0,03 ≈ 0,03.
At the same time, 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 AC< 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, at 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, when x = 0
| sin x | = | x |.
Thus, for | X | < π / 2 inequality (1) has been proven. 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 graph of the function y = sin x determine: a) sin 2; b) sin 4; c) sin (-3).
2.According to the function graph 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. According to the graph of the 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").
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.
Bibliography
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 exam preparation ().
Reference information on the trigonometric functions sine (sin x) and cosine (cos x). Geometric definition, properties, graphs, formulas. Table of sines and cosines, derivatives, integrals, series expansions, secant, cosecant. Expressions through complex variables. Connection with hyperbolic functions.
Geometric definition of sine and cosine
|BD|- length of the arc of a circle with center at a point A.
α
- angle expressed in radians.
Definition
Sine (sin α) 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 opposite leg |BC| to the length of the hypotenuse |AC|.
Cosine (cos α) 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 notations
;
;
.
;
;
.
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 period 2π.
Parity
The sine function is odd. The cosine function is even.
Domain of definition and values, extrema, increase, decrease
The sine and cosine functions are continuous in their domain of definition, that is, for all x (see 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 |
| Increasing | ||
| Descending | ||
| Maxima, y = 1 | ||
| Minima, y = - 1 | ||
| Zeros, y = 0 | ||
| Intercept points with the ordinate axis, x = 0 | y= 0 | y= 1 |
Basic formulas
Sum of squares of sine and cosine
Formulas for sine and cosine from sum and difference
;
;
Formulas for the product of sines and cosines
Sum and difference formulas
Expressing sine through cosine
;
;
;
.
Expressing cosine through sine
;
;
;
.
Expression through tangent
; .
When , we have:
;
.
At :
;
.
Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for certain values of the argument. 
Expressions through complex variables
;
Euler's formula
{ -∞ < x < +∞ }
Secant, cosecant
Inverse functions
The inverse functions of sine and cosine are arcsine and arccosine, respectively.
Arcsine, arcsin
Arccosine, arccos
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
Lesson and presentation on the topic: "Function y=sin(x). Definitions and properties"
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Manuals and simulators in the Integral online store for grade 10 from 1C
We solve problems in geometry. Interactive construction tasks for grades 7-10
Software environment "1C: Mathematical Constructor 6.1"
What we will study:
- Properties of the function Y=sin(X).
- Function graph.
- How to build a graph and its scale.
- Examples.
Properties of sine. Y=sin(X)
Guys, we have already become acquainted 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 some properties of this function:
1) The domain of definition is the set of real numbers.
2) The function is odd. Let's remember the definition of an odd function. A function is called odd if the equality holds: y(-x)=-y(x). As we remember from the ghost formulas: sin(-x)=-sin(x). The definition is fulfilled, which means Y=sin(X) is an odd function.
3) The function Y=sin(X) increases on the segment and decreases on the segment [π/2; π]. When we move along the first quarter (counterclockwise), the ordinate increases, and when we move through the second quarter it decreases.
4) The function Y=sin(X) is limited from below and from above. This property follows from the fact that
-1 ≤ sin(X) ≤ 1
5) The smallest value of the function is -1 (at x = - π/2+ πk). The largest value of the function is 1 (at x = π/2+ πk).
Let's use properties 1-5 to plot the function Y=sin(X). We will build our graph sequentially, applying our properties. Let's start building a graph on the segment.
Particular attention should be paid to the scale. On the ordinate axis it is more convenient to take a unit segment equal to 2 cells, and on the abscissa axis it is more convenient to take a unit segment (two cells) equal to π/3 (see figure).
_2.png)
Plotting the sine function x, y=sin(x)
Let's calculate the values of the function on our segment:
_3.png)
Let's build a graph using our points, taking into account the third property.
_4.png)
Conversion table for ghost formulas
Let's use the second property, which says that our function is odd, which means that it can be reflected symmetrically with respect to the origin:
_5.png)
We know that sin(x+ 2π) = sin(x). This means that on the interval [- π; π] the graph looks the same as on the segment [π; 3π] or or [-3π; - π] and so on. All we have to do is carefully redraw the graph in the previous figure along the entire x-axis.
_6.png)
The graph of the function Y=sin(X) is called a sinusoid.
Let's write a few more properties according to the constructed graph:
6) The function Y=sin(X) increases on any segment of the form: [- π/2+ 2πk; π/2+ 2πk], k is an integer and decreases on any segment of the form: [π/2+ 2πk; 3π/2+ 2πk], k – integer.
7) Function Y=sin(X) is a continuous function. Let's look at the graph of the function and make sure that our function has no breaks, this means continuity.
8) Range of values: segment [- 1; 1]. This is also clearly visible from the graph of the function.
9) Function Y=sin(X) - periodic function. Let's look at the graph again and see that the function takes the same values at certain intervals.
Examples of problems with sine
1. Solve the equation sin(x)= x-π
Solution: Let's build 2 graphs of the function: y=sin(x) and y=x-π (see figure).
Our graphs intersect at one point A(π;0), this is the answer: x = π
_7.png)
2. Graph the function y=sin(π/6+x)-1
Solution: The desired graph will be obtained by moving the graph of the function y=sin(x) π/6 units to the left and 1 unit down.
_8.png)
Solution: Let's plot the function and consider our segment [π/2; 5π/4].
The graph of the function shows that the largest and smallest values are achieved at the ends of the segment, at points π/2 and 5π/4, respectively.
Answer: sin(π/2) = 1 – the largest value, sin(5π/4) = the smallest value.
_9.png)
Sine problems for independent solution
- Solve 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 largest and smallest value of the function y=sin(x) on the segment
- Find the largest and smallest value of the function y=sin(x) on the interval [- π/3; 5π/6]
>>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 a cos t number, 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; it 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; it 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.