Sin properties and graph. Sine (sin x) and cosine (cos x) – properties, graphs, formulas. Expressions through complex variables
FUNCTION GRAPHICS
Sine function
- a bunch of 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- a bunch of 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 all |  |
| cos x< 0 for all |  |
| 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- a bunch of 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- a bunch of 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.
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 ().
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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.
During the classes
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)
Geometric definition of sine and cosine
\(\sin \alpha = \dfrac(|BC|)(|AB|) \), \(\cos \alpha = \dfrac(|AC|)(|AB|) \)
α - angle expressed in radians.
Sine (sin α) is a trigonometric function of 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 |AB|.
Cosine (cos α) is a trigonometric function of the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AC| to the length of the hypotenuse |AB|.
Trigonometric definition
Using the formulas above, you can find the sine and cosine of an acute angle. But you need to learn how to calculate the sine and cosine of an angle of arbitrary size. A right triangle does not provide such an opportunity (it cannot have an obtuse angle, for example); Therefore, we need a more general definition of sine and cosine, containing these formulas as a special case.
The trigonometric circle comes to the rescue. Let some angle be given; it corresponds to the point of the same name on the trigonometric circle.
Rice. 2. Trigonometric definition of sine and cosine
The cosine of an angle is the abscissa of a point. The sine of an angle is the ordinate of a point.
In Fig. 2, the angle is taken to be acute, and it is easy to understand that this definition coincides with the general geometric definition. In fact, we see a right triangle with a unit hypotenuse O and an acute angle. The adjacent leg of this triangle is cos (compare with Fig. 1) and at the same time the abscissa of the point; the opposite side is sin (as in Fig. 1) and at the same time the ordinate of the point.
But now we are no longer constrained by the first quarter and have the opportunity to extend this definition to any angle. In Fig. Figure 3 shows what the sine and cosine of an angle are in the second, third and fourth quarters.
Rice. 3. Sine and cosine in the II, III and IV quarters
Table values of sine and cosine
Zero angle \(\LARGE 0^(\circ ) \)
The abscissa of point 0 is equal to 1, the ordinate of point 0 is equal to 0. Hence,
cos 0 = 1 sin 0 = 0
Fig 4. Zero angle
Angle \(\LARGE \frac(\pi)(6) = 30^(\circ )\)
We see a right triangle with a unit hypotenuse and an acute angle of 30°. As you know, the leg lying opposite the angle 30° is equal to half the hypotenuse 1; in other words, the vertical leg is equal to 1/2 and, therefore,
\[ \sin \frac(\pi)(6) =\frac(1)(2) \]
We find the horizontal leg using the Pythagorean theorem (or, which is the same, we find the cosine using the basic trigonometric identity):
\[ \cos \frac(\pi)(6) = \sqrt(1 - \left(\frac(1)(2) \right)^(2) ) =\frac(\sqrt(3) )(2 ) \]
1 Why does this happen? Cut an equilateral triangle with side 2 along its height! It will split into two right triangles with a hypotenuse of 2, an acute angle of 30° and a shorter leg of 1.
Fig 5. Angle π/6
Angle \(\LARGE \frac(\pi)(4) = 45^(\circ )\)
In this case, the right triangle is isosceles; The sine and cosine of an angle of 45° are equal to each other. Let us denote them by x for now. We have:
\[ x^(2) + x^(2) = 1 \]
whence \(x=\frac(\sqrt(2) )(2) \). Hence,
\[ \cos \frac(\pi)(4) = \sin \frac(\pi)(4) =\frac(\sqrt(2) )(2) \]
Fig 5. Angle π/4
Properties of sine and cosine
Accepted notations
\(\sin^2 x \equiv (\sin x)^2; \)\(\quad \sin^3 x \equiv (\sin x)^3; \)\(\quad \sin^n x \equiv (\sin x)^n \)\(\sin^(-1) x \equiv \arcsin x \)\((\sin x)^(-1) \equiv \dfrac1(\sin x) \equiv \cosec x \).
\(\cos^2 x \equiv (\cos x)^2; \)\(\quad \cos^3 x \equiv (\cos x)^3; \)\(\quad \cos^n x \equiv (\cos x)^n \)\(\cos^(-1) x \equiv \arccos x \)\((\cos x)^(-1) \equiv \dfrac1(\cos x) \equiv \sec x \).
Periodicity
The functions y = sin x and y = cos x are periodic with a period of 2π.
\(\sin(x + 2\pi) = \sin x; \quad \)\(\cos(x + 2\pi) = \cos x \)
Parity
The sine function is odd. The cosine function is even.
\(\sin(-x) = - \sin x; \quad \)\(\cos(-x) = \cos x \)
Areas of definition and values, extrema, increase, decrease
The basic properties of sine and cosine are presented in the table ( n- whole).
| \(\small< x < \) | \(\small -\pi + 2\pi n \) \(\small< x < \) \(\small 2\pi n \) | |
| Descending | \(\small \dfrac(\pi)2 + 2\pi n \)\(\small< x < \) \(\small \dfrac(3\pi)2 + 2\pi n \) | \(\small 2\pi n \) \(\small< x < \) \(\pi + \small 2\pi n \) |
| Maxima, \(\small x = \) \(\small \dfrac(\pi)2 + 2\pi n \) | \(\small x = 2\pi n\) | |
| Minima, \(\small x = \) \(\small -\dfrac(\pi)2 + 2\pi n \) | \(\small x = \) \(\small \pi + 2\pi n \) | |
| Zeros, \(\small x = \pi n\) | \(\small x = \dfrac(\pi)2 + \pi n \) | |
| Y-axis intersection points, x = 0 | y = 0 | y = 1 |
Basic formulas containing sine and cosine
Sum of squares
\(\sin^2 x + \cos^2 x = 1\)
Sine and cosine formulas for sum and difference
\(\sin(x + y) = \sin x \cos y + \cos x \sin y \)
\(\sin(x - y) = \sin x \cos y - \cos x \sin y \)
\(\cos(x + y) = \cos x \cos y - \sin x \sin y \)
\(\cos(x - y) = \cos x \cos y + \sin x \sin y \)
\(\sin(2x) = 2 \sin x \cos x \)
\(\cos(2x) = \cos^2 x - \sin^2 x = \)\(2 \cos^2 x - 1 = 1 - 2 \sin^2 x \)
\(\cos\left(\dfrac(\pi)2 - x \right) = \sin x \) ; \(\sin\left(\dfrac(\pi)2 - x \right) = \cos x \)
\(\cos(x + \pi) = - \cos x \) ; \(\sin(x + \pi) = - \sin x \)
Formulas for the product of sines and cosines
\(\sin x \cos y = \) \(\dfrac12 (\Large [) \sin(x - y) + \sin(x + y) (\Large ]) \)
\(\sin x \sin y = \) \(\dfrac12 (\Large [) \cos(x - y) - \cos(x + y) (\Large ]) \)
\(\cos x \cos y = \) \(\dfrac12 (\Large [) \cos(x - y) + \cos(x + y) (\Large ]) \)
\(\sin x \cos y = \dfrac12 \sin 2x \)
\(\sin^2 x = \dfrac12 (\Large [) 1 - \cos 2x (\Large ]) \)
\(\cos^2 x = \dfrac12 (\Large [) 1 + \cos 2x (\Large ]) \)
Sum and difference formulas
\(\sin x + \sin y = 2 \, \sin \dfrac(x+y)2 \, \cos \dfrac(x-y)2 \)
\(\sin x - \sin y = 2 \, \sin \dfrac(x-y)2 \, \cos \dfrac(x+y)2 \)
\(\cos x + \cos y = 2 \, \cos \dfrac(x+y)2 \, \cos \dfrac(x-y)2 \)
\(\cos x - \cos y = 2 \, \sin \dfrac(x+y)2 \, \sin \dfrac(y-x)2 \)
Expressing sine through cosine
\(\sin x = \cos\left(\dfrac(\pi)2 - x \right) = \)\(\cos\left(x - \dfrac(\pi)2 \right) = - \cos\left(x + \dfrac(\pi)2 \right) \)\(\sin^2 x = 1 - \cos^2 x \) \(\sin x = \sqrt(1 - \cos^2 x) \) \(\( 2 \pi n \leqslant x \leqslant \pi + 2 \pi n \) \)\(\sin x = - \sqrt(1 - \cos^2 x) \) \(\( -\pi + 2 \pi n \leqslant x \leqslant 2 \pi n \) \).
Expressing cosine through sine
\(\cos x = \sin\left(\dfrac(\pi)2 - x \right) = \)\(- \sin\left(x - \dfrac(\pi)2 \right) = \sin\left(x + \dfrac(\pi)2 \right) \)\(\cos^2 x = 1 - \sin^2 x \) \(\cos x = \sqrt(1 - \sin^2 x) \) \(\( -\pi/2 + 2 \pi n \leqslant x \leqslant \pi/2 + 2 \pi n \) \)\(\cos x = - \sqrt(1 - \sin^2 x) \) \(\( \pi/2 + 2 \pi n \leqslant x \leqslant 3\pi/2 + 2 \pi n \) \).
Expression through tangent
\(\sin^2 x = \dfrac(\tg^2 x)(1+\tg^2 x) \)\(\cos^2 x = \dfrac1(1+\tg^2 x) \).
At \(- \dfrac(\pi)2 + 2 \pi n< x < \dfrac{\pi}2 + 2 \pi n \) \(\sin x = \dfrac(\tg x)( \sqrt(1+\tg^2 x) ) \)\(\cos x = \dfrac1( \sqrt(1+\tg^2 x) ) \).
At \(\dfrac(\pi)2 + 2 \pi n< x < \dfrac{3\pi}2 + 2 \pi n \)
:
\(\sin x = - \dfrac(\tg x)( \sqrt(1+\tg^2 x) ) \)\(\cos x = - \dfrac1( \sqrt(1+\tg^2 x) ) \).
Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for certain values of the argument.
[ img style="max-width:500px;max-height:1080px;" src="tablitsa.png" alt="Table of sines and cosines" title="Table of sines and cosines" ]!}
Expressions through complex variables
\(i^2 = -1\)
\(\sin z = \dfrac(e^(iz) - e^(-iz))(2i) \)\(\cos z = \dfrac(e^(iz) + e^(-iz))(2) \)
Euler's formula
\(e^(iz) = \cos z + i \sin z \)
Expressions through hyperbolic functions
\(\sin iz = i \sh z \) \(\cos iz = \ch z \)
\(\sh iz = i \sin z \) \(\ch iz = \cos z \)
Derivatives
\((\sin x)" = \cos x \) \((\cos x)" = - \sin x \) . Deriving formulas > > >
Derivatives of nth order:
\(\left(\sin x \right)^((n)) = \sin\left(x + n\dfrac(\pi)2 \right) \)\(\left(\cos x \right)^((n)) = \cos\left(x + n\dfrac(\pi)2 \right) \).
Integrals
\(\int \sin x \, dx = - \cos x + C \)\(\int \cos x \, dx = \sin x + C \)
See also section Table of indefinite integrals >>>
Series expansions
\(\sin x = \sum_(n=0)^(\infty) \dfrac( (-1)^n x^(2n+1) )( (2n+1)! ) = \)\(x - \dfrac(x^3)(3 + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + ... \)
!} \(\(- \infty< x < \infty \} \)
\(\cos x = \sum_(n=0)^(\infty) \dfrac( (-1)^n x^(2n) )( (2n)! ) = \)\(1 - \dfrac(x^2)(2 + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + ... \)
!} \(\( - \infty< x < \infty \} \)
Secant, cosecant
\(\sec x = \dfrac1( \cos x ) ; \) \(\cosec x = \dfrac1( \sin x ) \)
Inverse functions
The inverse functions of sine and cosine are arcsine and arccosine, respectively.
Arcsine, arcsin
\(y = \arcsin x\) \(\left\( -1 \leqslant x \leqslant 1; \; - \dfrac(\pi)2 \leqslant y \leqslant \dfrac(\pi)2 \right\) \)
\(\sin(\arcsin x) = x\)
\(\arcsin(\sin x) = x\) \(\left\( - \dfrac(\pi)2 \leqslant x \leqslant \dfrac(\pi)2 \right\) \)
Arccosine, arccos
\(y = \arccos x\) \(\left\( -1 \leqslant x \leqslant 1; \; 0 \leqslant y \leqslant \pi \right\) \)
\(\cos(\arccos x) = x \) \(\( -1 \leqslant x \leqslant 1 \) \)
\(\arccos(\cos x) = x\) \(\( 0 \leqslant x \leqslant \pi \) \)
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
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>>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 both in magnitude and sign coincides 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, 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, 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.
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