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) On 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:
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 ().
How to graph the function y=sin x? First, let's look at the sine graph on the interval.
We take a single segment 2 cells long in the notebook. On the Oy axis we mark one.
For convenience, we round the number π/2 to 1.5 (and not to 1.6, as required by the rounding rules). In this case, a segment of length π/2 corresponds to 3 cells.
On the Ox axis we mark not single segments, but segments of length π/2 (every 3 cells). Accordingly, a segment of length π corresponds to 6 cells, and a segment of length π/6 corresponds to 1 cell.
With this choice of a unit segment, the graph depicted on a sheet of notebook in a box corresponds as much as possible to the graph of the function y=sin x.
Let's make a table of sine values on the interval:
We mark the resulting points on the coordinate plane:
Since y=sin x is an odd function, the sine graph is symmetrical with respect to the origin - point O(0;0). Taking this fact into account, let’s continue plotting the graph to the left, then the points -π:
The function y=sin x is periodic with period T=2π. Therefore, the graph of a function taken on the interval [-π;π] is repeated an infinite number of times to the right and to the left.
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 graph of the 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. 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").
Lesson and presentation on the topic: "Function y=sin(x). Definitions and properties"
Additional materials
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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]
