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Small angle of attack - [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics power engineering in general Synonyms low angle of attack EN negative incidencelow incidence ...

negative cutting angle- - Topics oil and gas industry EN negative cutting anglenegative cutting anglenegative rake ... Technical Translator's Guide

negative bevel angle of the upper surface of the brush- [GOST 21888 82 (IEC 276 68, IEC 560 77)] Topics of electrical rotating machines in general... Technical Translator's Guide

wing angle Encyclopedia "Aviation"

wing angle- Wing installation angle. wing installation angle angle φ0 between the central chord of the wing and the base axis of the aircraft (see figure). Depending on the aerodynamic configuration of the aircraft, this angle can be either positive or negative. Usually … Encyclopedia "Aviation"

Wing angle- angle (φ)0 between the central chord of the wing and the base axis of the aircraft. Depending on the aerodynamic configuration of the aircraft, this angle can be either positive or negative. Usually it is in the range from ―2(°) to +3(°). Angle (φ)0… … Encyclopedia of technology

DECEPTION ANGLE- (Depressed angle) the angle formed by the elevation line (cm) with the horizon when the first one passes below the horizon, i.e. a negative elevation angle. Samoilov K.I. Marine dictionary. M.L.: State Naval Publishing House of the NKVMF Union... ... Marine Dictionary

ANGLE OF OPTICAL AXES- acute angle between opt. axles in biaxial shafts. U. o. O. called positive when the acute bisector is Ng and negative when the acute bisector is Np (see Optically biaxial crystal). True U. o. O. is designated... ... Geological encyclopedia

Castor (angle)- This term has other meanings, see Castor. θ castor, red line is the steering axis of the wheel. In the figure, the castor is positive (the angle is measured clockwise, the front of the car is on the left) ... Wikipedia

Castor (Rotation angle)- θ castor, red line is the steering axis of the wheel. In the figure, the caster is positive (the angle is measured clockwise, the front of the car is on the left) Castor (English caster) is the longitudinal inclination angle of the car's wheel turning axis. Castor... ...Wikipedia

rake angle- 3.2.9 rake angle: The angle between the rake surface and the base plane (see Figure 5). 1 negative rake angle; 2 positive rake angle Figure 5 Rake angles

An important concept in trigonometry is angle of rotation. Below we will consistently give an idea of ​​the turn and introduce all the related concepts. Let's start with a general idea of ​​a turn, say a full rotation. Next, let's move on to the concept of rotation angle and consider its main characteristics, such as the direction and magnitude of rotation. Finally, we give the definition of rotation of a figure around a point. We will provide the entire theory in the text with explanatory examples and graphic illustrations.

Page navigation.

What is called the rotation of a point around a point?

Let us immediately note that, along with the phrase “rotation around a point,” we will also use the phrases “rotation about a point” and “rotation about a point,” which mean the same thing.

Let's introduce concept of turning a point around a point.

First, let's define the center of rotation.

Definition.

The point about which the rotation is made is called center of rotation.

Now let's say what happens as a result of rotating the point.

As a result of rotating a certain point A relative to the center of rotation O, a point A 1 is obtained (which, in the case of a certain number, may coincide with A), and point A 1 lies on a circle with a center at point O of radius OA. In other words, when rotated relative to point O, point A goes to point A 1 lying on a circle with a center at point O of radius OA.

It is believed that point O, when turning around itself, turns into itself. That is, as a result of rotation around the center of rotation O, point O turns into itself.

It is also worth noting that the rotation of point A around point O should be considered as a displacement as a result of the movement of point A in a circle with a center at point O of radius OA.

For clarity, we will give an illustration of the rotation of point A around point O; in the figures below, we will show the movement of point A to point A 1 using an arrow.

Full turn

It is possible to rotate point A relative to the center of rotation O, such that point A, having passed all the points of the circle, will be in the same place. In this case, they say that point A has moved around point O.

Let's give a graphic illustration of a complete revolution.

If you do not stop at one revolution, but continue to move the point around the circle, then you can perform two, three, and so on full revolutions. The drawing below shows how two full turns can be made on the right and three turns on the left.

Rotation angle concept

From the concept of rotating a point introduced in the first paragraph, it is clear that there are an infinite number of options for rotating point A around point O. Indeed, any point on a circle with a center at point O of radius OA can be considered as point A 1 obtained as a result of rotating point A. Therefore, to distinguish one turn from another, we introduce concept of rotation angle.

One of the characteristics of the rotation angle is direction of rotation. The direction of rotation determines whether the point is rotated clockwise or counterclockwise.

Another characteristic of the rotation angle is its magnitude. Rotation angles are measured in the same units as: degrees and radians are the most common. It is worth noting here that the angle of rotation can be expressed in degrees by any real number from minus infinity to plus infinity, in contrast to the angle in geometry, the value of which in degrees is positive and does not exceed 180.

Lowercase letters of the Greek alphabet are usually used to indicate rotation angles: etc. To designate a large number of rotation angles, one letter with subscripts is often used, for example, .

Now let's talk about the characteristics of the rotation angle in more detail and in order.

Turning direction

Let points A and A 1 be marked on a circle with center at point O. You can get to point A 1 from point A by turning around the center O either clockwise or counterclockwise. It is logical to consider these turns different.

Let's illustrate rotations in a positive and negative direction. The drawing below shows rotation in a positive direction on the left, and in a negative direction on the right.

Rotation angle value, angle of arbitrary value

The angle of rotation of a point other than the center of rotation is completely determined by indicating its magnitude; on the other hand, by the magnitude of the angle of rotation one can judge how this rotation was carried out.

As we mentioned above, the rotation angle in degrees is expressed as a number from −∞ to +∞. In this case, the plus sign corresponds to a clockwise rotation, and the minus sign corresponds to a counterclockwise rotation.

Now it remains to establish a correspondence between the value of the rotation angle and the rotation it corresponds to.

Let's start with a rotation angle of zero degrees. This angle of rotation corresponds to the movement of point A towards itself. In other words, when rotated 0 degrees around point O, point A remains in place.

We proceed to the rotation of point A around point O, in which the rotation occurs within half a revolution. We will assume that point A goes to point A 1. In this case, the absolute value of angle AOA 1 in degrees does not exceed 180. If the rotation occurred in a positive direction, then the value of the rotation angle is considered equal to the value of the angle AOA 1, and if the rotation occurred in a negative direction, then its value is considered equal to the value of the angle AOA 1 with a minus sign. As an example, here is a picture showing rotation angles of 30, 180 and −150 degrees.

Rotation angles greater than 180 degrees and less than −180 degrees are determined based on the following fairly obvious properties of successive turns: several successive rotations of point A around center O are equivalent to one rotation, the magnitude of which is equal to the sum of the magnitudes of these rotations.

Let us give an example illustrating this property. Let's rotate point A relative to point O by 45 degrees, and then rotate this point by 60 degrees, after which we rotate this point by −35 degrees. Let us denote the intermediate points during these turns as A 1, A 2 and A 3. We could get to the same point A 3 by performing one rotation of point A at an angle of 45+60+(−35)=70 degrees.

So, we will represent rotation angles greater than 180 degrees as several successive turns by angles, the sum of which gives the value of the original rotation angle. For example, a rotation angle of 279 degrees corresponds to successive rotations of 180 and 99 degrees, or 90, 90, 90, and 9 degrees, or 180, 180, and −81 degrees, or 279 successive rotations of 1 degree.

Rotation angles smaller than −180 degrees are determined similarly. For example, a rotation angle of −520 degrees can be interpreted as successive rotations of the point by −180, −180, and −160 degrees.

Summarize. We have determined the angle of rotation, the value of which in degrees is expressed by some real number from the interval from −∞ to +∞. In trigonometry, we will work specifically with angles of rotation, although the word “rotation” is often omitted and they simply say “angle.” Thus, in trigonometry we will work with angles of arbitrary magnitude, by which we mean rotation angles.

To conclude this point, we note that a full rotation in the positive direction corresponds to a rotation angle of 360 degrees (or 2 π radians), and in a negative direction - a rotation angle of −360 degrees (or −2 π rad). In this case, it is convenient to represent large rotation angles as a certain number of full revolutions and another rotation at an angle ranging from −180 to 180 degrees. For example, let's take a rotation angle of 1,340 degrees. It’s easy to imagine 1,340 as 360·4+(−100) . That is, the initial rotation angle corresponds to 4 full turns in the positive direction and a subsequent rotation of −100 degrees. Another example: a rotation angle of −745 degrees can be interpreted as two turns counterclockwise followed by a rotation of −25 degrees, since −745=(−360) 2+(−25) .

Rotate a shape around a point by an angle

The concept of point rotation is easily extended to rotate any shape around a point by an angle(we are talking about such a rotation that both the point about which the rotation is carried out and the figure that is being rotated lie in the same plane).

By rotating a figure we mean the rotation of all points of the figure around a given point by a given angle.

As an example, let's illustrate the following action: rotate the segment AB by an angle relative to the point O; this segment, when rotated, will turn into the segment A 1 B 1.

Bibliography.

• Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Education, 1990.- 272 p.: ill.- isbn 5-09-002727-7
• Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
• Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

A pair of different rays Oa and Ob emanating from one point O is called an angle and is denoted by the symbol (a, b). Point O is called the vertex of the angle, and the rays Oa and Ob are called the sides of the angle. If A and B are two points of the rays Oa and Ob, then (a, b) is also denoted by the symbol AOB (Fig. 1.1).

An angle (a, b) is called unfolded if the rays Oa and Ob emerging from the same point lie on the same straight line and do not coincide (i.e., opposite directions).

Fig.1.1

Two angles are considered equal if one angle can be superimposed on the other so that the sides of the angles coincide. The bisector of an angle is a ray with its origin at the vertex of the angle, dividing the angle into two equal angles.

They say that ray OS emanating from the vertex of angle AOB lies between its sides if it intersects segment AB (Fig. 1.2). Point C is said to lie between the sides of an angle if through this point it is possible to draw a ray with its origin at the vertex of the angle and lying between the sides of the angle. The set of all points of the plane lying between the sides of the angle forms the internal region of the angle (Fig. 1.3). The set of points of the plane that do not belong to the internal region and sides of the angle forms the external region of the angle.

Angle (a, b) is considered greater than angle (c, d) if angle (c, d) can be superimposed on angle (a, b) so that after combining one pair of sides, the second side of angle (c, d) will lie between sides of the angle (a, b). In Fig. 1.4 AOB is greater than AOC.

Let the ray c lie between the sides of the angle (a, b) (Fig. 1.5). Pairs of rays a, c and c, b form two angles. The angle (a, b) is said to be the sum of two angles (a, c) and (c, b), and they write: (a, b) = (a, c) + (c, b).

Fig.1.3

Usually in geometry we deal with angles smaller than the unfolded angle. However, the addition of two angles can result in an angle larger than the unfolded one. In this case, that part of the plane that is considered the inner area of ​​the angle is marked with an arc. In Fig. 1.6, the inner part of the angle AOB, obtained by adding the angles AOS and COB and the larger unfolded one, is marked with an arc.

Fig.1.5

There are also angles greater than 360°. Such angles are formed, for example, by the rotation of an airplane propeller, the rotation of a drum on which a rope is wound, etc.

In the future, when considering each angle, we will agree to consider one of the sides of this angle as its initial side, and the other as its final side.

Any angle, for example angle AOB (Fig. 1.7), can be obtained by rotating a moving beam around vertex O from the initial side of the angle (OA) to its final side (OB). We will measure this angle, taking into account the total number of revolutions made around point O, as well as the direction in which the rotation occurred.

Positive and negative angles.

Let us have an angle formed by rays OA and OB (Fig. 1.8). The moving beam, rotating around point O from its initial position (OA), can take its final position (OB) in two different directions of rotation. These directions are shown in Figure 1.8 by the corresponding arrows.

Fig.1.7

Just as on the number axis one of the two directions is considered positive and the other negative, two different directions of rotation of the moving beam are also distinguished. We agreed to consider the positive direction of rotation to be the direction opposite to the direction of clockwise rotation. The direction of rotation coinciding with the direction of rotation clockwise is considered negative.

According to these definitions, angles are also classified into positive and negative.

A positive angle is the angle formed by rotating the moving beam around the starting point in a positive direction.

Figure 1.9 shows some positive angles. (The direction of rotation of the moving beam is shown in the drawings by arrows.)

A negative angle is the angle formed by rotating the moving beam around the starting point in a negative direction.

Figure 1.10 shows some negative angles. (The direction of rotation of the moving beam is shown in the drawings by arrows.)

But two coinciding rays can also form angles +360°n and -360°n (n = 0,1,2,3,...). Let us denote by b the smallest possible non-negative angle of rotation that transfers the beam OA to position OB. If now ray OB makes an additional full revolution around point O, then we obtain a different angle value, namely: ABO = b + 360°.

Measuring angles using circular arcs. Units for arcs and angles

In some cases it turns out to be convenient to measure angles using circular arcs. The possibility of such a measurement is based on the well-known proposal of planimetry that in one circle (or in equal circles) the central angles and the corresponding arcs are in direct proportion.

Let some arc of a given circle be taken as the unit of measurement of arcs. We take the central angle corresponding to this arc as the unit of measurement for angles. Under this condition, any arc of a circle and the central angle corresponding to this arc will contain the same number of units of measurement. Therefore, by measuring the arcs of a circle, it is possible to determine the value of the central angles corresponding to these arcs.

Let's look at the two most common systems for measuring arcs and angles.

Degree measure of angles

When measuring angles by degrees, an angle of one degree (denoted 1?) is taken as the basic unit of measurement of angles (the reference angle with which various angles are compared). An angle of one degree is an angle equal to 1/180 of the reversed angle. An angle equal to 1/60th of an angle of 1° is an angle of one minute (denoted 1"). An angle equal to 1/60th of an angle of one minute is an angle of one second (denoted 1").

Radian measure of angles

Along with the degree measure of angles, geometry and trigonometry also use another measure of angles, called the radian. Let's consider a circle of radius R with center O. Let's draw two radii O A and OB so that the length of the arc AB is equal to the radius of the circle (Fig. 1.12). The resulting central angle AOB will be an angle of one radian. An angle of 1 radian is taken as the radian unit of measurement for angles. When measuring angles in radians, the rotated angle is equal to p radians.

The degree and radian units of measurement of angles are related by the equalities:

1 radian =180?/р57° 17" 45"; 1?=p/180 radians0.017453 radians;

1"=p/180*60 radian0.000291 radian;

1""=p/180*60*60 radian0.000005 radian.

The degree (or radian) measure of an angle is also called the angle magnitude. The angle AOB is sometimes denoted /

Classification of angles

An angle equal to 90°, or in radian measure p/2, is called a right angle; it is often denoted by the letter d. An angle less than 90° is called acute; An angle greater than 90° but less than 180° is called obtuse.

Two angles that have one common side and add up to 180° are called adjacent angles. Two angles that have one common side and add up to 90° are called supplementary angles.

Corner: ° π rad =

Convert to: radians degrees 0 - 360° 0 - 2π positive negative Calculate

When lines intersect, there are four different areas relative to the point of intersection.
These new areas are called corners.

The picture shows 4 different angles formed by the intersection of lines AB and CD

Angles are usually measured in degrees, which is denoted as °. When an object makes a complete circle, that is, moving from point D through B, C, A, and then back to D, then it is said to have turned 360 degrees (360°). So a degree is $\frac(1)(360)$ of a circle.

Angles greater than 360 degrees

We talked about how when an object makes a full circle around a point, it goes 360 degrees, however, when an object makes more than one circle, it makes an angle of more than 360 degrees. This is a common occurrence in everyday life. The wheel goes around many circles when the car is moving, that is, it forms an angle of more than 360°.

To find out the number of cycles (circles completed) when rotating an object, we count the number of times we need to add 360 to itself to get a number equal to or less than a given angle. In the same way, we find a number that we multiply by 360 to get a number that is smaller but closest to the given angle.

Example 2
1. Find the number of circles described by an object forming an angle
a) 380°
b) 770°
c) 1000°
Solution
a) 380 = (1 × 360) + 20
The object described one circle and 20°
Since $20^(\circ) = \frac(20)(360) = \frac(1)(18)$ circle
The object described $1\frac(1)(18)$ circles.

B) 2 × 360 = 720
770 = (2 × 360) + 50
The object described two circles and 50°
$50^(\circ) = \frac(50)(360) = \frac(5)(36)$ circle
The object described $2\frac(5)(36)$ of a circle
c)2 × 360 = 720
1000 = (2 × 360) + 280
$280^(\circ) = \frac(260)(360) = \frac(7)(9)$ circles
The object described $2\frac(7)(9)$ circles

When an object rotates clockwise, it forms a negative angle of rotation, and when it rotates counterclockwise, it forms a positive angle. Up to this point, we have only considered positive angles.

In diagram form, a negative angle can be depicted as shown below.

The figure below shows the sign of the angle, which is measured from a common straight line, the 0 axis (x-axis - x-axis)

This means that if there is a negative angle, we can get a corresponding positive angle.
For example, the bottom of a vertical line is 270°. When measured in the negative direction, we get -90°. We simply subtract 270 from 360. Given a negative angle, we add 360 to get the corresponding positive angle.
When the angle is -360°, it means the object has made more than one clockwise circle.

Example 3
1. Find the corresponding positive angle
a) -35°
b) -60°
c) -180°
d) - 670°

2. Find the corresponding negative angle of 80°, 167°, 330° and 1300°.
Solution
1. In order to find the corresponding positive angle, we add 360 to the angle value.
a) -35°= 360 + (-35) = 360 - 35 = 325°
b) -60°= 360 + (-60) = 360 - 60 = 300°
c) -180°= 360 + (-180) = 360 - 180 = 180°
d) -670°= 360 + (-670) = -310
This means one circle clockwise (360)
360 + (-310) = 50°
The angle is 360 + 50 = 410°

2. In order to get the corresponding negative angle, we subtract 360 from the angle value.
80° = 80 - 360 = - 280°
167° = 167 - 360 = -193°
330° = 330 - 360 = -30°
1300° = 1300 - 360 = 940 (one lap completed)
940 - 360 = 580 (second round completed)
580 - 360 = 220 (third round completed)
220 - 360 = -140°
The angle is -360 - 360 - 360 - 140 = -1220°
Thus 1300° = -1220°

Radian

A radian is the angle from the center of a circle that encloses an arc whose length is equal to the radius of the circle. This is a unit of measurement for angular magnitude. This angle is approximately 57.3°.
In most cases, this is denoted as glad.
Thus $1 rad \approx 57.3^(\circ)$

Radius = r = OA = OB = AB
Angle BOA is equal to one radian

Since the circumference is given as $2\pi r$, then there are $2\pi$ radii in the circle, and therefore in the whole circle there are $2\pi$ radians.

Radians are usually expressed in terms of $\pi$ to avoid decimals in calculations. In most books, the abbreviation glad does not occur, but the reader should know that when it comes to angle, it is specified in terms of $\pi$, and the units of measurement automatically become radians.

$360^(\circ) = 2\pi\rad$
$180^(\circ) = \pi\rad$,
$90^(\circ) = \frac(\pi)(2) rad$,
$30^(\circ) = \frac(30)(180)\pi = \frac(\pi)(6) rad$,
$45^(\circ) = \frac(45)(180)\pi = \frac(\pi)(4) rad$,
$60^(\circ) = \frac(60)(180)\pi = \frac(\pi)(3) rad$
$270^(\circ) = \frac(270)(180)\pi = \frac(27)(18)\pi = 1\frac(1)(2)\pi\ rad$

Example 4
1. Convert 240°, 45°, 270°, 750° and 390° to radians using $\pi$.
Solution
Let's multiply the angles by $\frac(\pi)(180)$.
$240^(\circ) = 240 \times \frac(\pi)(180) = \frac(4)(3)\pi=1\frac(1)(3)\pi$
$120^(\circ) = 120 \times \frac(\pi)(180) = \frac(2\pi)(3)$
$270^(\circ) = 270 \times \frac(1)(180)\pi = \frac(3)(2)\pi=1\frac(1)(2)\pi$
$750^(\circ) = 750 \times \frac(1)(180)\pi = \frac(25)(6)\pi=4\frac(1)(6)\pi$
$390^(\circ) = 390 \times \frac(1)(180)\pi = \frac(13)(6)\pi=2\frac(1)(6)\pi$

2. Convert the following angles to degrees.
a) $\frac(5)(4)\pi$
b) $3.12\pi$
c) 2.4 radians
Solution
$180^(\circ) = \pi$
a) $\frac(5)(4) \pi = \frac(5)(4) \times 180 = 225^(\circ)$
b) $3.12\pi = 3.12 \times 180 = 561.6^(\circ)$
c) 1 rad = 57.3°
$2.4 = \frac(2.4 \times 57.3)(1) = 137.52$

Negative angles and angles greater than $2\pi$ radians

To convert a negative angle to a positive one, we add it to $2\pi$.
To convert a positive angle to a negative one, we subtract $2\pi$ from it.

Example 5
1. Convert $-\frac(3)(4)\pi$ and $-\frac(5)(7)\pi$ to positive angles in radians.

Solution
Add $2\pi$ to the angle
$-\frac(3)(4)\pi = -\frac(3)(4)\pi + 2\pi = \frac(5)(4)\pi = 1\frac(1)(4)\ pi$

$-\frac(5)(7)\pi = -\frac(5)(7)\pi + 2\pi = \frac(9)(7)\pi = 1\frac(2)(7)\ pi$

When an object rotates by an angle greater than $2\pi$;, it makes more than one circle.
In order to determine the number of revolutions (circles or cycles) in such an angle, we find a number, multiplying it by $2\pi$, the result is equal to or less, but as close as possible to this number.

Example 6
1. Find the number of circles traversed by the object at given angles
a) $-10\pi$
b) $9\pi$
c) $\frac(7)(2)\pi$

Solution
a) $-10\pi = 5(-2\pi)$;
$-2\pi$ implies one cycle in a clockwise direction, this means that
the object made 5 clockwise cycles.

b) $9\pi = 4(2\pi) + \pi$, $\pi =$ half cycle
the object made four and a half cycles counterclockwise

c) $\frac(7)(2)\pi=3.5\pi=2\pi+1.5\pi$, $1.5\pi$ is equal to three quarters of the cycle $(\frac(1.5\pi)(2\pi)= \frac(3)(4))$
the object has gone through one and three quarters of a cycle counterclockwise