Angular measure along the cosine and sine axis. Sine, cosine, tangent, cotangent of an acute angle. trigonometric functions. Transition from sum to product

Initially, sine and cosine arose due to the need to calculate quantities in right triangles. It was noticed that if the value of the degree measure of angles in right triangle do not change, then the aspect ratio, no matter how much these sides change in length, always remains the same.

This is how the concepts of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse, and the cosine is the ratio of the adjacent leg to the hypotenuse.

Theorems of cosines and sines

But cosines and sines can be used not only in right triangles. To find the value of an obtuse or acute angle, the side of any triangle, it is enough to apply the cosine and sine theorem.

The cosine theorem is quite simple: "The square of the side of a triangle is equal to the sum squares of the other two sides minus twice the product of these sides by the cosine of the angle between them.

There are two interpretations of the sine theorem: small and extended. According to the small: "In a triangle, the angles are proportional to the opposite sides." This theorem is often extended due to the property of the circumcircle of a triangle: "In a triangle, the angles are proportional to opposite sides, and their ratio is equal to the diameter of the circumscribed circle."

Derivatives

A derivative is a mathematical tool that shows how quickly a function changes with respect to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.

When solving problems, you need to know the tabular values \u200b\u200bof the derivatives of trigonometric functions: sine and cosine. The derivative of the sine is the cosine, and the derivative of the cosine is the sine, but with a minus sign.

Application in mathematics

Especially often, sines and cosines are used in solving right triangles and problems related to them.

The convenience of sines and cosines is also reflected in technology. Angles and sides were easy to evaluate using the cosine and sine theorems, breaking complex shapes and objects into "simple" triangles. Engineers and, often dealing with calculations of aspect ratio and degree measures, spent a lot of time and effort calculating cosines and sines of non-table angles.

Then Bradis tables came to the rescue, containing thousands of values of sines, cosines, tangents and cotangents of different angles. IN Soviet time some teachers forced their wards to memorize the pages of the Bradys tables.

Radian - the angular value of the arc, along the length equal to the radius or 57.295779513 ° degrees.

Degree (in geometry) - 1/360th part of a circle or 1/90th part right angle.

π = 3.141592653589793238462… (approximate value of pi).

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 school course mathematics. 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 an acute angle A is equal to the ratio opposite leg BC to 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 to 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 the point, which is obtained by rotating the starting point by 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 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", they usually use the phrase "sine of the angle of alpha" 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 is 8 π rad 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 through 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 of sinα, as well as the value of cosα. 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 specifically about functions, it is expedient 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 in a rectangular Cartesian coordinate system Oxy unit circle. 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 A 1 H opposite to the angle 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.

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.
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.
Algebra and elementary functions : Tutorial for 9th grade students high school/ E. S. Kochetkov, E. S. Kochetkova; Edited by Doctor of Physical and Mathematical Sciences O. N. Golovin. - 4th ed. Moscow: Education, 1969.
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
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.
Mordkovich A. G. Algebra and the beginnings of analysis. Grade 10. At 2 p. Ch. 1: a tutorial 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.
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.
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.
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How to find the sine?

The study of geometry helps to develop thinking. This item must be included in schooling. In life, knowledge of this subject can be useful - for example, when planning an apartment.

From the history

As part of the geometry course, trigonometry is also studied, which explores trigonometric functions. In trigonometry, we study the sines, cosines, tangents, and cotangents of an angle.

But for now, let's start with the simplest - sine. Let's take a closer look at the very first concept - the sine of an angle in geometry. What is a sine and how to find it?

The concept of "sine of the angle" and sinusoids

The sine of an angle is the ratio of the values of the opposite leg and the hypotenuse of a right triangle. This is a direct trigonometric function, which is written in writing as "sin (x)", where (x) is the angle of the triangle.

On the graph, the sine of an angle is indicated by a sinusoid with its own characteristics. A sinusoid looks like a continuous wavy line that lies within certain limits on the coordinate plane. The function is odd, therefore it is symmetric with respect to 0 on the coordinate plane (it leaves the origin of the coordinates).

The domain of this function lies in the range from -1 to +1 on the Cartesian coordinate system. The period of the sine angle function is 2 Pi. This means that every 2 Pi the pattern is repeated and the sine wave goes through a full cycle.

Sinusoid equation

sin x = a / c
where a is the leg opposite to the angle of the triangle
c - hypotenuse of a right triangle

Properties of the sine of an angle

sin(x) = - sin(x). This feature demonstrates that the function is symmetrical, and if the values x and (-x) are set aside on the coordinate system in both directions, then the ordinates of these points will be opposite. They will be on equal distance from each other.
Another feature of this function is that the graph of the function increases on the segment [- P / 2 + 2 Pn]; [P/2 + 2Pn], where n is any integer. A decrease in the graph of the sine of the angle will be observed on the segment: [P / 2 + 2 Pn]; [ 3P/2 + 2Pn].
sin (x) > 0 when x is in the range (2Pn, P + 2Pn)
(x)< 0, когда х находится в диапазоне (-П+2Пn, 2Пn)

The values of the sines of the angle are determined by special tables. Such tables have been created to facilitate the process of calculating complex formulas and equations. It is easy to use and contains values not only sin functions(x), but also the values of other functions.

Moreover, the table of standard values of these functions is included in the mandatory memory study, like the multiplication table. This is especially true for classes with a physical and mathematical bias. In the table you can see the values of the main angles used in trigonometry: 0, 15, 30, 45, 60, 75, 90, 120, 135, 150, 180, 270 and 360 degrees.

There is also a table that defines the values of the trigonometric functions of non-standard angles. Using different tables, you can easily calculate the sine, cosine, tangent and cotangent of some angles.

Equations are made with trigonometric functions. Solving these equations is easy if you know simple trigonometric identities and reductions of functions, for example, such as sin (P / 2 + x) \u003d cos (x) and others. A separate table has also been compiled for such casts.

How to find the sine of an angle

When the task is to find the sine of an angle, and by condition we have only the cosine, tangent, or cotangent of the angle, we can easily calculate what we need using trigonometric identities.

sin 2 x + cos 2 x = 1

From this equation, we can find both sine and cosine, depending on which value is unknown. We will succeed trigonometric equation with one unknown:

sin 2 x = 1 - cos 2 x
sin x = ± √ 1 - cos 2 x
ctg 2 x + 1 = 1 / sin 2 x

From this equation, you can find the value of the sine, knowing the value of the cotangent of the angle. To simplify, replace sin 2 x = y, and then you have a simple equation. For example, the value of the cotangent is 1, then:

1 + 1 = 1/y
2 = 1 / y
2y = 1
y = 1/2

Now we perform the reverse replacement of the player:

sin 2 x = ½
sin x = 1 / √2

Since we took the value of the cotangent for the standard angle (45 0), the obtained values \u200b\u200bcan be checked against the table.

If you have a tangent value, but you need to find the sine, another trigonometric identity will help:

tg x * ctg x = 1

It follows that:

ctg x = 1 / tg x

In order to find the sine of a non-standard angle, for example, 240 0, you need to use the angle reduction formulas. We know that π corresponds to 180 0 for us. Thus, we will express our equality using standard angles by expansion.

240 0 = 180 0 + 60 0

We need to find the following: sin (180 0 + 60 0). In trigonometry, there are reduction formulas that are useful in this case. This is the formula:

sin (π + x) = - sin (x)

Thus, the sine of an angle of 240 degrees is:

sin (180 0 + 60 0) = - sin (60 0) = - √3/2

In our case, x = 60, and P, respectively, 180 degrees. We found the value (-√3/2) from the table of values of the functions of standard angles.

In this way, non-standard angles can be decomposed, for example: 210 = 180 + 30.

As you can see, this circle is built in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin, the initial position of the radius vector is fixed along the positive direction of the axis (in our example, this is the radius).

Each point of the circle corresponds to two numbers: the coordinate along the axis and the coordinate along the axis. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, remember about the considered right-angled triangle. In the figure above, you can see two whole right triangles. Consider a triangle. It is rectangular because it is perpendicular to the axis.

What is equal to from a triangle? That's right. In addition, we know that is the radius of the unit circle, and therefore, . Substitute this value into our cosine formula. Here's what happens:

And what is equal to from a triangle? Well, of course, ! Substitute the value of the radius into this formula and get:

So, can you tell me what are the coordinates of a point that belongs to the circle? Well, no way? And if you realize that and are just numbers? What coordinate does it correspond to? Well, of course, the coordinate! What coordinate does it correspond to? That's right, coordinate! Thus, the point.

And what then are equal and? That's right, let's use the appropriate definitions of tangent and cotangent and get that, a.

What if the angle is larger? Here, for example, as in this picture:

What has changed in this example? Let's figure it out. To do this, we again turn to a right-angled triangle. Consider a right triangle: an angle (as adjacent to an angle). What is the value of the sine, cosine, tangent and cotangent of an angle? That's right, we adhere to the corresponding definitions of trigonometric functions:

Well, as you can see, the value of the sine of the angle still corresponds to the coordinate; the value of the cosine of the angle - the coordinate; and the values of tangent and cotangent to the corresponding ratios. Thus, these relations are applicable to any rotations of the radius vector.

It has already been mentioned that the initial position of the radius vector is along the positive direction of the axis. So far we have rotated this vector counterclockwise, but what happens if we rotate it clockwise? Nothing extraordinary, you will also get an angle of a certain size, but only it will be negative. Thus, when rotating the radius vector counterclockwise, we get positive angles , and when rotating clockwise - negative.

So, we know that a whole revolution of the radius vector around the circle is or. Is it possible to rotate the radius vector by or by? Well, of course you can! In the first case, therefore, the radius vector will make one complete revolution and stop at position or.

In the second case, that is, the radius vector will make three complete revolutions and stop at position or.

Thus, from the above examples, we can conclude that angles that differ by or (where is any integer) correspond to the same position of the radius vector.

The figure below shows an angle. The same image corresponds to the corner, and so on. This list can be continued indefinitely. All these angles can be written with the general formula or (where is any integer)

Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values \u200b\u200bare equal to:

Here's a unit circle to help you:

Any difficulties? Then let's figure it out. So we know that:

From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the corner at corresponds to a point with coordinates, therefore:

Does not exist;

Further, adhering to the same logic, we find out that the corners in correspond to points with coordinates, respectively. Knowing this, it is easy to determine the values of trigonometric functions at the corresponding points. Try it yourself first, then check the answers.

Answers:

Does not exist

Thus, we can make the following table:

There is no need to remember all these values. It is enough to remember the correspondence between the coordinates of points on the unit circle and the values of trigonometric functions:

But the values \u200b\u200bof the trigonometric functions of the angles in and, given in the table below, must be remembered:

Do not be afraid, now we will show one of the examples rather simple memorization of the corresponding values:

To use this method, it is vital to remember the values of the sine for all three measures of the angle (), as well as the value of the tangent of the angle in. Knowing these values, it is quite easy to restore the entire table - the cosine values are transferred in accordance with the arrows, that is:

Knowing this, you can restore the values for. The numerator " " will match and the denominator " " will match. Cotangent values are transferred in accordance with the arrows shown in the figure. If you understand this and remember the diagram with arrows, then it will be enough to remember the entire value from the table.

Coordinates of a point on a circle

Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation?

Well, of course you can! Let's bring out general formula for finding the coordinates of a point.

Here, for example, we have such a circle:

We are given that the point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the point by degrees.

As can be seen from the figure, the coordinate of the point corresponds to the length of the segment. The length of the segment corresponds to the coordinate of the center of the circle, that is, it is equal to. The length of a segment can be expressed using the definition of cosine:

Then we have that for the point the coordinate.

By the same logic, we find the value of the y coordinate for the point. In this way,

So in general view point coordinates are determined by the formulas:

Circle center coordinates,

circle radius,

Angle of rotation of the radius vector.

As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are zero, and the radius is equal to one:

Well, let's try these formulas for a taste, practicing finding points on a circle?

1. Find the coordinates of a point on a unit circle obtained by turning a point on.

2. Find the coordinates of a point on a unit circle obtained by rotating a point on.

3. Find the coordinates of a point on a unit circle obtained by turning a point on.

4. Point - the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.

5. Point - the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.

Having trouble finding the coordinates of a point on a circle?

Solve these five examples (or understand the solution well) and you will learn how to find them!

It can be seen that. And we know what corresponds to a full turn of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:

2. The circle is unit with a center at a point, which means that we can use simplified formulas:

It can be seen that. We know what corresponds to two complete rotations of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:

Sine and cosine are tabular values. We remember their values and get:

Thus, the desired point has coordinates.

3. The circle is unit with a center at a point, which means that we can use simplified formulas:

It can be seen that. Let's depict the considered example in the figure:

The radius makes angles with the axis equal to and. Knowing that the table values of the cosine and sine are equal, and having determined that the cosine here takes negative meaning, and the sine is positive, we have:

More similar examples understand when studying formulas for reducing trigonometric functions in the topic.

Thus, the desired point has coordinates.

Angle of rotation of the radius vector (by condition)

To determine the corresponding signs of sine and cosine, we construct a unit circle and an angle:

As you can see, the value, that is, is positive, and the value, that is, is negative. Knowing the tabular values of the corresponding trigonometric functions, we obtain that:

Let's substitute the obtained values into our formula and find the coordinates:

Thus, the desired point has coordinates.

5. To solve this problem, we use formulas in general form, where

The coordinates of the center of the circle (in our example,

Circle radius (by condition)

Angle of rotation of the radius vector (by condition).

Substitute all the values into the formula and get:

and - table values. We remember and substitute them into the formula:

Thus, the desired point has coordinates.

SUMMARY AND BASIC FORMULA

The sine of an angle is the ratio of the opposite (far) leg to the hypotenuse.

The cosine of an angle is the ratio of the adjacent (close) leg to the hypotenuse.

The tangent of an angle is the ratio of the opposite (far) leg to the adjacent (close).

The cotangent of an angle is the ratio of the adjacent (close) leg to the opposite (far).

Trigonometry - section mathematical science, which studies trigonometric functions and their use in geometry. The development of trigonometry began in the days of ancient Greece. During the Middle Ages, scientists from the Middle East and India made an important contribution to the development of this science.

This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the main trigonometric functions: sine, cosine, tangent and cotangent. Their meaning in the context of geometry is explained and illustrated.

Initially, the definitions of trigonometric functions, whose argument is an angle, were expressed through the ratio of the sides of a right triangle.

Definitions of trigonometric functions

The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.

The cosine of the angle (cos α) is the ratio of the adjacent leg to the hypotenuse.

The tangent of the angle (t g α) is the ratio of the opposite leg to the adjacent one.

The cotangent of the angle (c t g α) is the ratio of the adjacent leg to the opposite one.

These definitions are given for an acute angle of a right triangle!

Let's give an illustration.

In triangle ABC with right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.

The definitions of sine, cosine, tangent, and cotangent make it possible to calculate the values of these functions from the known lengths of the sides of a triangle.

Important to remember!

The range of sine and cosine values: from -1 to 1. In other words, sine and cosine take values from -1 to 1. The range of tangent and cotangent values is the entire number line, that is, these functions can take any value.

The definitions given above refer to acute angles. In trigonometry, the concept of the angle of rotation is introduced, the value of which, unlike an acute angle, is not limited by frames from 0 to 90 degrees. The angle of rotation in degrees or radians is expressed by any real number from - ∞ to + ∞.

In this context, one can define the sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Imagine a unit circle centered at the origin of the Cartesian coordinate system.

The starting point A with coordinates (1 , 0) rotates around the center of the unit circle by some angle α and goes to point A 1 . The definition is given through the coordinates of the point A 1 (x, y).

Sine (sin) of the rotation angle

The sine of the rotation angle α is the ordinate of the point A 1 (x, y). sinα = y

Cosine (cos) of the angle of rotation

The cosine of the angle of rotation α is the abscissa of the point A 1 (x, y). cos α = x

Tangent (tg) of rotation angle

The tangent of the angle of rotation α is the ratio of the ordinate of the point A 1 (x, y) to its abscissa. t g α = y x

Cotangent (ctg) of rotation angle

The cotangent of the angle of rotation α is the ratio of the abscissa of the point A 1 (x, y) to its ordinate. c t g α = x y

Sine and cosine are defined for any angle of rotation. This is logical, because the abscissa and ordinate of the point after the rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is not defined when the point after the rotation goes to the point with zero abscissa (0 , 1) and (0 , - 1). In such cases, the expression for the tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with the cotangent. The difference is that the cotangent is not defined in cases where the ordinate of the point vanishes.

Important to remember!

Sine and cosine are defined for any angles α.

The tangent is defined for all angles except α = 90° + 180° k , k ∈ Z (α = π 2 + π k , k ∈ Z)

The cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)

When solving practical examples, do not say "sine of the angle of rotation α". The words "angle of rotation" are simply omitted, implying that from the context it is already clear what is at stake.

Numbers

What about the definition of the sine, cosine, tangent and cotangent of a number, and not the angle of rotation?

Sine, cosine, tangent, cotangent of a number

Sine, cosine, tangent and cotangent of a number t a number is called, which is respectively equal to the sine, cosine, tangent and cotangent in t radian.

For example, the sine of 10 π is equal to the sine of the rotation angle of 10 π rad.

There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. Let's consider it in more detail.

Any real number t a point on the unit circle is put in correspondence with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are defined in terms of the coordinates of this point.

The starting point on the circle is point A with coordinates (1 , 0).

positive number t

Negative number t corresponds to the point to which the starting point will move if it moves counterclockwise around the circle and passes the path t .

Now that the connection between the number and the point on the circle has been established, we proceed to the definition of sine, cosine, tangent and cotangent.

Sine (sin) of the number t

Sine of a number t- ordinate of the point of the unit circle corresponding to the number t. sin t = y

Cosine (cos) of t

Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x

Tangent (tg) of t

Tangent of a number t- the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t. t g t = y x = sin t cos t

The latter definitions are consistent with and do not contradict the definition given at the beginning of this section. Point on a circle corresponding to a number t, coincides with the point to which the starting point passes after turning through the angle t radian.

Trigonometric functions of angular and numerical argument

Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° · k , k ∈ Z (α = π 2 + π · k , k ∈ Z) corresponds to a certain value of the tangent. The cotangent, as mentioned above, is defined for all α, except for α = 180 ° k , k ∈ Z (α = π k , k ∈ Z).

We can say that sin α , cos α , t g α , c t g α are functions of the angle alpha, or functions of the angular argument.

Similarly, one can speak of sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a specific value of the sine or cosine of a number t. All numbers other than π 2 + π · k , k ∈ Z, correspond to the value of the tangent. The cotangent is similarly defined for all numbers except π · k , k ∈ Z.

Basic functions of trigonometry

Sine, cosine, tangent and cotangent are the basic trigonometric functions.

From the context, it is usually clear with which argument of the trigonometric function ( angular argument or a numeric argument) we are dealing with.

Let's return to the data at the very beginning of the definitions and the angle alpha, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent and cotangent are fully consistent with geometric definitions, given by the ratios of the sides of a right triangle. Let's show it.

Take a unit circle centered on a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw from the resulting point A 1 (x, y) perpendicular to the x-axis. In the resulting right triangle, the angle A 1 O H is equal to the angle of rotation α, the length of the leg O H is equal to the abscissa of the point A 1 (x, y) . The length of the leg opposite the corner is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.

In accordance with the definition from geometry, the sine of the angle α is equal to the ratio of the opposite leg to the hypotenuse.

sin α \u003d A 1 H O A 1 \u003d y 1 \u003d y

This means that the definition of the sine of an acute angle in a right triangle through the aspect ratio is equivalent to the definition of the sine of the angle of rotation α, with alpha lying in the range from 0 to 90 degrees.

Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.