What is sine alpha. What is sine and cosine. Sine in geometry

This article has collected tables of sines, cosines, tangents and cotangents. First, we give a table of basic values ​​​​of trigonometric functions, that is, a table of sines, cosines, tangents and cotangents of angles 0, 30, 45, 60, 90, ..., 360 degrees ( 0, π/6, π/4, π/3, π/2, …, 2π radian). After that, we will give a table of sines and cosines, as well as a table of tangents and cotangents by V. M. Bradis, and show how to use these tables when finding the values ​​of trigonometric functions.

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Table of sines, cosines, tangents and cotangents for angles 0, 30, 45, 60, 90, ... degrees

Bibliography.

• Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
• Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
• Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
• Bradis V. M. Four-digit mathematical tables: For general education. textbook establishments. - 2nd ed. - M.: Bustard, 1999.- 96 p.: ill. ISBN 5-7107-2667-2

|BD|- the length of the arc of a circle centered at a point A.
α is an angle expressed in radians.

sine ( sinα) 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 ( 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

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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 pi.

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

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Formulas for the product of sines and cosines

Sum and difference formulas

Expression of sine through cosine

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Expression of cosine through sine

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Expression in terms of tangent

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For , we have:
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At :
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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

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Euler formula

Expressions in terms of hyperbolic functions

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Derivatives

; . Derivation of formulas > > >

Derivatives of the nth order:
{ -∞ < x < +∞ }

Secant, cosecant

Inverse functions

The inverse functions to 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 Students of Higher Educational Institutions, Lan, 2009.

See also:

In this article, we will show how definitions of sine, cosine, tangent and cotangent of angle and number in trigonometry. Here we will talk about notation, give examples of records, give graphic illustrations. In conclusion, we draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.

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Definition of sine, cosine, tangent and cotangent

Let's follow how the concept of sine, cosine, tangent and cotangent is formed in the school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in a right triangle is given. And later trigonometry is studied, which refers to the sine, cosine, tangent and cotangent of the angle of rotation and the number. We give all these definitions, give examples and give the necessary comments.

Acute angle in a right triangle

From the course of geometry, the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle are known. They are given as the ratio of the sides of a right triangle. We present their formulations.

Definition.

Sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse.

Definition.

Cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse.

Definition.

Tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent leg.

Definition.

Cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite leg.

The notation of sine, cosine, tangent and cotangent is also introduced there - sin, cos, tg and ctg, respectively.

For example, if ABC is a right triangle with a right angle C, then the sine of the acute angle A is equal to the ratio of the opposite leg BC to the hypotenuse AB, that is, sin∠A=BC/AB.

These definitions allow you to calculate the values ​​of the sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from the known values ​​of the sine, cosine, tangent, cotangent and the length of one of the sides, find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is 3 and the hypotenuse AB is 7 , then we could calculate the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7 .

Angle of rotation

In trigonometry, they begin to look at the angle more widely - they introduce the concept of angle of rotation. The angle of rotation, unlike an acute angle, is not limited by frames from 0 to 90 degrees, the angle of rotation in degrees (and in radians) can be expressed by any real number from −∞ to +∞.

In this light, the definitions of the sine, cosine, tangent and cotangent are no longer an acute angle, but an angle of arbitrary magnitude - the angle of rotation. They are given through the x and y coordinates of the point A 1 , into which the so-called initial point A(1, 0) passes after it rotates through an angle α around the point O - the beginning of a rectangular Cartesian coordinate system and the center of the unit circle.

Definition.

Sine of rotation angleα is the ordinate of the point A 1 , that is, sinα=y .

Definition.

cosine of the angle of rotationα is called the abscissa of the point A 1 , that is, cosα=x .

Definition.

Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tgα=y/x .

Definition.

The cotangent of the angle of rotationα is the ratio of the abscissa of the point A 1 to its ordinate, that is, ctgα=x/y .

The sine and cosine are defined for any angle α , since we can always determine the abscissa and ordinate of a point, which is obtained by rotating the starting point through the angle α . And tangent and cotangent are not defined for any angle. The tangent is not defined for such angles α at which the initial point goes to a point with zero abscissa (0, 1) or (0, −1) , and this takes place at angles 90°+180° k , k∈Z (π /2+π k rad). Indeed, at such angles of rotation, the expression tgα=y/x does not make sense, since it contains division by zero. As for the cotangent, it is not defined for such angles α at which the starting point goes to to a point with zero ordinate (1, 0) or (−1, 0) , and this is the case for angles 180° k , k ∈Z (π k rad).

So, the sine and cosine are defined for any rotation angles, the tangent is defined for all angles except 90°+180° k , k∈Z (π/2+π k rad), and the cotangent is for all angles except 180° ·k , k∈Z (π·k rad).

The notations already known to us appear in the definitions sin, cos, tg and ctg, they are also used to denote the sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the notation tan and cotcorresponding to tangent and cotangent). So the sine of the rotation angle of 30 degrees can be written as sin30°, the records tg(−24°17′) and ctgα correspond to the tangent of the rotation angle −24 degrees 17 minutes and the cotangent of the rotation angle α . Recall that when writing the radian measure of an angle, the notation "rad" is often omitted. For example, the cosine of a rotation angle of three pi rads is usually denoted cos3 π .

In conclusion of this paragraph, it is worth noting that in talking about the sine, cosine, tangent and cotangent of the angle of rotation, the phrase “angle of rotation” or the word “rotation” is often omitted. That is, instead of the phrase "sine of the angle of rotation alpha", the phrase "sine of the angle of alpha" is usually used, or even shorter - "sine of alpha". The same applies to cosine, and tangent, and cotangent.

Let's also say that the definitions of the sine, cosine, tangent, and cotangent of an acute angle in a right triangle are consistent with the definitions just given for the sine, cosine, tangent, and cotangent of a rotation angle ranging from 0 to 90 degrees. We will substantiate this.

Numbers

Definition.

Sine, cosine, tangent and cotangent of a number t is a number equal to the sine, cosine, tangent and cotangent of the angle of rotation in t radians, respectively.

For example, the cosine of 8 π is, by definition, a number equal to the cosine of an angle of 8 π rad. And the cosine of the angle in 8 π rad is equal to one, therefore, the cosine of the number 8 π is equal to 1.

There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is assigned a point of the unit circle centered at the origin of the rectangular coordinate system, and the sine, cosine, tangent and cotangent are determined in terms of the coordinates of this point. Let's dwell on this in more detail.

Let us show how the correspondence between real numbers and points of the circle is established:

• the number 0 is assigned the starting point A(1, 0) ;
• a positive number t is associated with a point on the unit circle, which we will get to if we move around the circle from the starting point in a counterclockwise direction and go through a path of length t;
• a negative number t is associated with a point on the unit circle, which we will get to if we move around the circle from the starting point in a clockwise direction and go through a path of length |t| .

Now let's move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point of the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1) ).

Definition.

The sine of a number t is the ordinate of the unit circle point corresponding to the number t , that is, sint=y .

Definition.

The cosine of a number t is called the abscissa of the point of the unit circle corresponding to the number t , that is, cost=x .

Definition.

Tangent of a number t is the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of the number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost .

Definition.

Cotangent of a number t is the ratio of the abscissa to the ordinate of the point of the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is as follows: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t : ctgt=cost/sint .

Here we note that the definitions just given agree with the definition given at the beginning of this subsection. Indeed, the point of the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point through an angle of t radians.

It is also worth clarifying this point. Let's say we have a sin3 entry. How to understand whether the sine of the number 3 or the sine of the rotation angle of 3 radians is in question? This is usually clear from the context, otherwise it probably doesn't matter.

Trigonometric functions of angular and numerical argument

According to the definitions given in the previous paragraph, each rotation angle α corresponds to a well-defined value sin α , as well as the value cos α . In addition, all rotation angles other than 90°+180° k , k∈Z (π/2+π k rad) correspond to the values ​​tgα , and other than 180° k , k∈Z (π k rad ) are the values ​​of ctgα . Therefore sinα, cosα, tgα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.

Similarly, we can talk about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a well-defined value of sint , as well as cost . In addition, all numbers other than π/2+π·k , k∈Z correspond to the values ​​tgt , and the numbers π·k , k∈Z correspond to the values ​​ctgt .

The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.

It is usually clear from the context that we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can consider the independent variable as both a measure of the angle (the angle argument) and a numeric argument.

However, the school mainly studies numeric functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking about functions, then it is advisable to consider trigonometric functions as functions of numerical arguments.

Connection of definitions from geometry and trigonometry

If we consider the angle of rotation α from 0 to 90 degrees, then the data in the context of trigonometry of the definition of the sine, cosine, tangent and cotangent of the angle of rotation are fully consistent with the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's substantiate this.

Draw a unit circle in the rectangular Cartesian coordinate system Oxy. Note the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get the point A 1 (x, y) . Let's drop the perpendicular A 1 H from the point A 1 to the Ox axis.

It is easy to see that in a right triangle the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of the point A 1, that is, |OH|=x, the length of the leg opposite to the angle A 1 H is equal to the ordinate of the point A 1 , that is, |A 1 H|=y , and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of an acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y . And by definition from trigonometry, the sine of the angle of rotation α is equal to the ordinate of the point A 1, that is, sinα=y. This shows that the definition of the sine of an acute angle in a right triangle is equivalent to the definition of the sine of the angle of rotation α for α from 0 to 90 degrees.

Similarly, it can be shown that the definitions of the cosine, tangent, and cotangent of an acute angle α are consistent with the definitions of the cosine, tangent, and cotangent of the angle of rotation α.

Bibliography.

1. Geometry. 7-9 grades: studies. for general education institutions / [L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev and others]. - 20th ed. M.: Education, 2010. - 384 p.: ill. - ISBN 978-5-09-023915-8.
2. Pogorelov A.V. Geometry: Proc. for 7-9 cells. general education institutions / A. V. Pogorelov. - 2nd ed. - M.: Enlightenment, 2001. - 224 p.: ill. - ISBN 5-09-010803-X.
3. Algebra and elementary functions: Textbook for students of grade 9 of secondary school / E. S. Kochetkov, E. S. Kochetkova; Edited by Doctor of Physical and Mathematical Sciences O. N. Golovin. - 4th ed. Moscow: Education, 1969.
4. Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
5. Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
6. Mordkovich A. G. Algebra and the beginnings of analysis. Grade 10. At 2 h. Part 1: a textbook for educational institutions ( profile level)/ A. G. Mordkovich, P. V. Semenov. - 4th ed., add. - M.: Mnemosyne, 2007. - 424 p.: ill. ISBN 978-5-346-00792-0.
7. Algebra and the beginning of mathematical analysis. Grade 10: textbook. for general education institutions: basic and profile. levels /[Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; ed. A. B. Zhizhchenko. - 3rd ed. - I .: Education, 2010. - 368 p.: Ill. - ISBN 978-5-09-022771-1.
8. Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
9. Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Lecture: Sine, cosine, tangent, cotangent of an arbitrary angle

Sine, cosine of an arbitrary angle

To understand what trigonometric functions are, let's turn to a circle with a unit radius. This circle is centered at the origin on the coordinate plane. To determine the given functions, we will use the radius vector OR, which starts at the center of the circle, and the point R is a point on the circle. This radius vector forms an angle alpha with the axis OH. Since the circle has a radius equal to one, then OR = R = 1.

If from the point R drop a perpendicular on the axis OH, then we get a right triangle with hypotenuse equal to one.

If the radius vector moves clockwise, then this direction is called negative, but if it moves counter-clockwise - positive.

The sine of an angle OR, is the ordinate of the point R vectors on a circle.

That is, to obtain the value of the sine of a given angle alpha, it is necessary to determine the coordinate At on surface.

How was this value obtained? Since we know that the sine of an arbitrary angle in a right triangle is the ratio of the opposite leg to the hypotenuse, we get that

And since R=1, then sin(α) = y 0 .

In the unit circle, the ordinate value cannot be less than -1 and greater than 1, which means that

The sine is positive in the first and second quarters of the unit circle, and negative in the third and fourth.

Cosine of an angle given circle formed by the radius vector OR, is the abscissa of the point R vectors on a circle.

That is, to obtain the value of the cosine of a given angle alpha, it is necessary to determine the coordinate X on surface.

The cosine of an arbitrary angle in a right triangle is the ratio of the adjacent leg to the hypotenuse, we get that

And since R=1, then cos(α) = x 0 .

In the unit circle, the value of the abscissa cannot be less than -1 and greater than 1, which means that

The cosine is positive in the first and fourth quadrants of the unit circle, and negative in the second and third.

tangentarbitrary angle the ratio of sine to cosine is calculated.

If we consider a right triangle, then this is the ratio of the opposite leg to the adjacent one. If we are talking about a unit circle, then this is the ratio of the ordinate to the abscissa.

Judging by these relationships, it can be understood that the tangent cannot exist if the value of the abscissa is zero, that is, at an angle of 90 degrees. The tangent can take all other values.

The tangent is positive in the first and third quarters of the unit circle, and negative in the second and fourth.