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

Trigonometry, as a science, originated in the Ancient East. First trigonometric ratios were developed by astronomers to create an accurate calendar and navigate by the stars. These calculations related to spherical trigonometry, while in school course study the ratios of sides and angles of a plane triangle.

Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationship between the sides and angles of triangles.

During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazwi introduced functions such as tangent and cotangent, and compiled the first tables of values ​​for sines, tangents and cotangents. The concepts of sine and cosine were introduced by Indian scientists. Trigonometry received a lot of attention in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

Basic trigonometric functions numeric argument– these are sine, cosine, tangent and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.

The formulas for calculating the values ​​of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants are equal in all directions,” since the proof is given using the example of an isosceles right triangle.

Sine, cosine and other dependencies establish the relationship between sharp corners and sides of any right triangle. Let us present formulas for calculating these quantities for angle A and trace the relationships between trigonometric functions:

As you can see, tg and ctg are inverse functions. If we imagine leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, we get following formulas for tangent and cotangent:

Trigonometric circle

Graphically, the relationship between the mentioned quantities can be represented as follows:

The circle, in this case, represents all possible values ​​of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will have a “+” sign if α belongs to the 1st and 2nd quarters of the circle, that is, it is in the range from 0° to 180°. For α from 180° to 360° (III and IV quarters), sin α can only be a negative value.

Let's try to build trigonometric tables for specific angles and find out the value of the quantities.

Values ​​of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values ​​of trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen at random. The designation π in the tables is for radians. Rad is the angle at which the length of a circle's arc corresponds to its radius. This value was introduced in order to establish a universal dependence; when calculating in radians, the actual length of the radius in cm does not matter.

Angles in tables for trigonometric functions correspond to radian values:

So, it is not difficult to guess that 2π is a complete circle or 360°.

Properties of trigonometric functions: sine and cosine

In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.

Consider the comparative table of properties for sine and cosine:

Sine wave	Cosine
y = sin x	y = cos x
ODZ [-1; 1]	ODZ [-1; 1]
sin x = 0, for x = πk, where k ϵ Z	cos x = 0, for x = π/2 + πk, where k ϵ Z
sin x = 1, for x = π/2 + 2πk, where k ϵ Z	cos x = 1, at x = 2πk, where k ϵ Z
sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z	cos x = - 1, for x = π + 2πk, where k ϵ Z
sin (-x) = - sin x, i.e. the function is odd	cos (-x) = cos x, i.e. the function is even
	the function is periodic, shortest period- 2π
sin x › 0, with x belonging to the 1st and 2nd quarters or from 0° to 180° (2πk, π + 2πk)	cos x › 0, with x belonging to the I and IV quarters or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk)
sin x ‹ 0, with x belonging to the third and fourth quarters or from 180° to 360° (π + 2πk, 2π + 2πk)	cos x ‹ 0, with x belonging to the 2nd and 3rd quarters or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk)
increases in the interval [- π/2 + 2πk, π/2 + 2πk]	increases on the interval [-π + 2πk, 2πk]
decreases on intervals [π/2 + 2πk, 3π/2 + 2πk]	decreases on intervals
derivative (sin x)’ = cos x	derivative (cos x)’ = - sin x

Determining whether a function is even or not is very simple. Enough to imagine trigonometric circle with the signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs coincide, the function is even, otherwise it is odd.

The introduction of radians and the listing of the basic properties of sine and cosine waves allow us to present the following pattern:

It is very easy to verify that the formula is correct. For example, for x = π/2, the sine is 1, as is the cosine of x = 0. The check can be done by consulting tables or by tracing function curves for given values.

Properties of tangentsoids and cotangentsoids

The graphs of the tangent and cotangent functions differ significantly from the sine and cosine functions. The values ​​tg and ctg are reciprocals of each other.

1. Y = tan x.
2. The tangent tends to the values ​​of y at x = π/2 + πk, but never reaches them.
3. The smallest positive period of the tangentoid is π.
4. Tg (- x) = - tg x, i.e. the function is odd.
5. Tg x = 0, for x = πk.
6. The function is increasing.
7. Tg x › 0, for x ϵ (πk, π/2 + πk).
8. Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
9. Derivative (tg x)’ = 1/cos 2 ⁡x.

Let's consider graphic image cotangentoids below in the text.

Main properties of cotangentoids:

1. Y = cot x.
2. Unlike the sine and cosine functions, in the tangentoid Y can take on the values ​​of the set of all real numbers.
3. The cotangentoid tends to the values ​​of y at x = πk, but never reaches them.
4. The smallest positive period of a cotangentoid is π.
5. Ctg (- x) = - ctg x, i.e. the function is odd.
6. Ctg x = 0, for x = π/2 + πk.
7. The function is decreasing.
8. Ctg x › 0, for x ϵ (πk, π/2 + πk).
9. Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
10. Derivative (ctg x)’ = - 1/sin 2 ⁡x Correct

If you are already familiar with trigonometric circle , and you just want to refresh your memory of certain elements, or you are completely impatient, then here it is:

Here we will analyze everything in detail step by step.

The trigonometric circle is not a luxury, but a necessity

Trigonometry Many people associate it with an impenetrable thicket. Suddenly, so many values ​​of trigonometric functions, so many formulas pile up... But it’s like, it didn’t work out at the beginning, and... off we go... complete misunderstanding...

It is very important not to give up values ​​of trigonometric functions, - they say, you can always look at the spur with a table of values.

If you constantly look at a table with values trigonometric formulas, let's get rid of this habit!

He will help us out! You will work with it several times, and then it will pop up in your head. How is it better than a table? Yes, in the table you will find a limited number of values, but on the circle - EVERYTHING!

For example, say while looking at standard table of values ​​of trigonometric formulas , what is the sine equal to, say, 300 degrees, or -45.

No way?.. you can, of course, connect reduction formulas... And looking at the trigonometric circle, you can easily answer such questions. And you will soon know how!

And when solving trigonometric equations and inequalities without a trigonometric circle, it’s absolutely nowhere.

Introduction to the trigonometric circle

Let's go in order.

First, let's write out this series of numbers:

And now this:

And finally this one:

Of course, it is clear that, in fact, in first place is , in second place is , and in last place is . That is, we will be more interested in the chain.

But how beautiful it turned out! If something happens, we will restore this “miracle ladder.”

And why do we need it?

This chain is the main values ​​of sine and cosine in the first quarter.

Let us draw a circle of unit radius in a rectangular coordinate system (that is, we take any radius in length, and declare its length to be unit).

From the “0-Start” beam we lay the corners in the direction of the arrow (see figure).

We get the corresponding points on the circle. So, if we project the points onto each of the axes, then we will get exactly the values ​​​​from the above chain.

Why is this, you ask?

Let's not analyze everything. Let's consider principle, which will allow you to cope with other, similar situations.

Triangle AOB is rectangular and contains . And we know that opposite the angle b lies a leg half the size of the hypotenuse (we have the hypotenuse = the radius of the circle, that is, 1).

This means AB= (and therefore OM=). And according to the Pythagorean theorem

I hope something is already becoming clear?

So point B will correspond to the value, and point M will correspond to the value

Same with the other values ​​of the first quarter.

As you understand, the familiar axis (ox) will be cosine axis, and the axis (oy) – axis of sines . Later.

To the left of zero along the cosine axis (below zero along the sine axis) there will, of course, be negative values.

So, here it is, the ALMIGHTY, without whom there is nowhere in trigonometry.

But we’ll talk about how to use the trigonometric circle in.

It characterizes the maximum angle at which a car wheel will turn when the steering wheel is fully turned. And the smaller this angle, the greater the accuracy and smoothness of control. After all, to turn even a small angle, only a small movement of the steering wheel is required.

But do not forget that the smaller the maximum turning angle, the smaller the turning radius of the car. Those. It will be very difficult to turn around in a confined space. So manufacturers have to look for some kind of “golden mean”, maneuvering between a large turning radius and control accuracy.

Changing wheel alignment angles and adjusting them

The Piri Reis map has been compared with a modern map projection. Thus, he came to the conclusion that a mysterious map was taking over the world, as seen from a satellite hovering high above Cairo. In other words, above the Great Pyramid. It is surprising that Egyptologists continually defend these spaces, although one recently discovered corridor has been reviewed and has yet to yield any breakthroughs.

It is also worth noting that unusual psychotronic effects have been found in the pyramid, which, among other things, can affect human health. It's about about spatial psychotronics, creating both energy and geomagnetic " anomalous zones”, which are further explored.

Rolling shoulder - shortest distance between the middle of the tire and the steering axis of the wheel. If the axis of rotation of the wheel and the middle of the wheel coincide, then the value is considered zero. At negative value- the axis of rotation will shift outward of the wheel, and with a positive value - inward.

When the wheel turns, the tire is deformed under the influence of lateral forces. And in order to maintain the maximum contact patch with the road, the car’s wheel also tilts in the direction of the turn. But everywhere you need to know when to stop, because with a very large caster, the car’s wheel will tilt strongly and then lose traction.

Responsible for weight stabilization of the steered wheels. The point is that the moment the wheel deviates from neutral, the front end begins to rise. And since it weighs a lot, when the steering wheel is released under the influence of gravity, the system tends to take its initial position, corresponding to movement in a straight line. True, for this stabilization to work, it is necessary to maintain the (albeit small, but undesirable) positive roll-in shoulder.

Initially, the transverse angle of the steering axis was used by engineers to eliminate the shortcomings of the car's suspension. It got rid of such “illnesses” of the car as positive camber and positive rolling shoulder.

During archaeological excavations Strange funeral offerings in the form of birds with outstretched wings have also been found. Later aerodynamic studies of these subjects revealed that they were most likely ancient glider models. One of them was discovered with the inscription "gift of Amon." The god Amun in Egypt was worshiped as the god of the wind so associated with flight is obvious.

But as members of this ancient civilization came to this knowledge without a preliminary development stage? The answer in this case is only. This knowledge came from the governments of those times, which the Egyptians called their gods. It is quite possible for members technologically advanced civilization which dates back more than 000 years ago, disappeared without a trace.

Many cars use MacPherson type suspension. It makes it possible to obtain a negative or zero rolling leverage. After all, the steering axis of the wheel consists of the support of one single lever, which can easily be placed inside the wheel. But this suspension is not perfect either, because due to its design, it is almost impossible to make the angle of inclination of the turning axis small. When turning, it tilts the outer wheel at an unfavorable angle (like positive camber), while the inner wheel simultaneously leans in the opposite direction.

But such facilities are still in short supply. They fall apart, they can be destroyed, but it can also be well hidden in temples, pyramids and other iconic buildings that can lie motionless, properly secured against "treasure hunters".

The Great Pyramid's size and design precision have never been equaled. The pyramid weighs approximately six million tons. In its position as the Eiffel Tower, the Great Pyramid was the tallest building in the world. More than two million stones were used for its construction. Not a single stone weighs less than a ton.

As a result, the contact patch of the outer wheel is greatly reduced. And since the outer wheel bears the main load when turning, the entire axle loses a lot of traction. This, of course, can be partially compensated for by caster and camber. Then the grip of the outer wheel will be good, but that of the inner wheel will practically disappear.

Car wheel alignment

There are two types of car alignment: positive and negative. Determining the type of alignment is very simple: you need to draw two straight lines along the wheels of the car. If these lines intersect at the front of the car, then the toe is positive, and if at the rear, it is negative. If there is positive toe-in of the front wheels, the car will make it easier to turn and will also gain additional steering ability.

On the rear axle, with positive toe-in, the car will be more stable when moving in a straight line, but if there is negative toe-in, the car will behave inappropriately and yaw from side to side.

And some of over seventy tons. Inside, the cells are connected by corridors. Today, it is a rough stone pyramid, but once it was processed to the mirror shine of the masonry. The peak of the Great Pyramid is believed to have been decorated with pure gold. The sun's rays blinded hundreds of kilometers. For centuries, experts have speculated about the purpose of the pyramids. Traditional theory states that the pyramids were a symbolic gateway to afterworld. Others believe that the pyramid was an astronomical observatory. Some say that the help is in the geographical dimension.

But it should be remembered that excessive deviation of the car’s toe from zero will increase rolling resistance during straight-line movement; in corners this will be less noticeable.

Wheel camber

Wheel camber, like toe-in, can be either negative or positive.

If you look from the front of the car, and the wheels tilt inward, then this is negative camber, and if they lean outward of the car, then this is positive camber. Wheel camber is necessary to maintain traction between the wheel and the road surface.

One fanciful theory claims that the Great Pyramid was on granaries. However, experts today generally agree that the pyramids were much more than just a giant tomb. Scientists argue that the massive pyramid technology could not have been available to people at this point in human history when these buildings were built. For example, the height of the pyramid corresponds to the distance from the Earth to the Sun. The pyramid was precisely oriented to the four worlds with a precision that has never been achieved.

And surprisingly, the Great Pyramid lies in the exact center of the earth. Whoever built the Great Pyramid could accurately determine latitude and longitude. This is surprising because the technology for determining longitude was discovered in modern times in the sixteenth century. The pyramids were built at the exact center of the Earth. Also, the height of the pyramid can be seen from a great height, can be seen from the Moon. Moreover, the pyramid shape is one of the best for reflecting radars. These reasons lead some researchers to believe that Egyptian pyramids were built outside of their other purposes and for navigation by potential foreign explorers.

Changing the camber angle affects the behavior of the car on a straight line, because the wheels are not perpendicular to the road, which means they do not have maximum grip. But this only affects rear-wheel drive cars when starting from a stop with slipping.

› All about wheel alignment angles part 1.

For those who want to understand what Wheel Alignment Angles (Camber/Toe) mean and thoroughly understand the issue, this article has answers to all questions.

The Pyramid of Cheops is located just over eight kilometers west of Cairo. It is built on an artificially created flat with an area of ​​1.6 square kilometers. Its base extends to 900 square meters and almost a millimeter in a horizontal position. Two and three quarters of a million stone blocks were used for the construction, with the heaviest weighing up to 70 tons. They fit in such a way that this fact is a mystery. However, the technical side of creating the pyramid remains a mystery, as it would be a major challenge for today's advanced technology.

An excursion into history shows that sophisticated wheel installations were used on various vehicles long before the advent of the automobile. Here are a few more or less well-known examples.
It is no secret that the wheels of some carriages and other horse-drawn carriages, intended for “dynamic” driving, were installed with a large, clearly visible positive camber. This was done so that the dirt flying from the wheels did not fall into the carriage and important riders, but was scattered to the sides. For utilitarian carts for leisurely movement, everything was exactly the opposite. Thus, pre-revolutionary manuals on how to build a good cart recommended installing wheels with negative camber. In this case, if the dowel stopping the wheel was lost, it did not immediately jump off the axle. The driver had time to notice the damage to the chassis, which was fraught with especially big troubles if there were several dozen pounds of flour in the cart and there was no jack. In the design of gun carriages (again, vice versa), positive camber was sometimes used. It is clear that it was not intended to protect the gun from dirt. This made it convenient for the servant to roll the gun by the wheels with his hands from the side, without fear of crushing his legs. And here is her cart huge wheels, which helped to easily cross the ditches, were tilted in the other direction - towards the cart. The resulting increase in track helped to increase the stability of the Central Asian “mobile,” which was distinguished by a high center of gravity. What do these historical facts have to do with the installation of wheels on modern cars? Yes, in general, none. However, they provide a useful insight. It can be seen that the installation of wheels (in particular, their camber) is not subject to any single pattern.

Therefore, there is no hypothesis that magical powers were used in the construction of the pyramid - magical formulas written on papyrus made it possible to move heavy pieces of stone and place them on top of each other with amazing precision. Edgar Cayce said that these pyramids were built ten thousand years ago, while others believe that the pyramids were built by Atlanteans who, before the cataclysm that destroyed their continent, mainly sought refuge in Egypt. He creates scientific centers, they also created a pyramid-shaped hideout where big secrets could be hidden.

When choosing this parameter, the “manufacturer” in each specific case was guided by different considerations, which he considered priority. So, what do car suspension designers strive for when choosing a suspension system? Of course, towards the ideal. The ideal for a car that moves in a straight line is considered to be the position of the wheels when the planes of their rotation (rolling planes) are perpendicular to the road surface, parallel to each other, the axis of symmetry of the body and coincide with the trajectory of movement. In this case, power loss due to friction and wear of the tire tread is minimal, and the traction of the wheels with the road, on the contrary, is maximum. Naturally, the question arises: what makes you deliberately deviate from the ideal? Looking ahead, several considerations can be given. First, we judge wheel alignment based on a static picture when the car is stationary. Who said that when driving, accelerating, braking and maneuvering a car, it does not change? Secondly, reducing losses and extending tire life is not always a priority. Before talking about what factors suspension developers take into account, let’s agree that large number parameters describing the geometry of the car suspension, we will limit ourselves to only those included in the group of primary (primary) or basic ones. They are called so because they determine the settings and properties of the suspension, are always monitored during its diagnosis and are adjusted, if such a possibility is provided. These are the well-known toe, camber and steering angles of the steering wheels. When considering these most important parameters, we will have to remember other characteristics of the suspension.

The pyramid consists of 203 layers of stone blocks weighing from 2.5 to 15 tons. Some blocks at the bottom of the pyramid at the base weigh up to 50 tons. Originally, the entire pyramid was covered with a fine white and polished limestone shell, but the stone was used for construction, especially after frequent earthquakes in the area.

The weight of the pyramid is proportional to the weight of the Earth 1:10. The pyramid is a maximum of 280 Egyptian cubits, and the base area is 440 Egyptian cubits. If the basic pattern is divided by double the height of the pyramid, we get the Ludolf number - 3. The deviation from the Ludolf figure is only 0.05%. The base of the base is equal to the circumference of a circle with a radius equal to the height of the pyramid.

Toe-in (TOE) characterizes the orientation of the wheels relative to the longitudinal axis of the vehicle. The position of each wheel can be determined separately from the others, and then they speak of individual toe. It represents the angle between the plane of rotation of the wheel and the axis of the car when viewed from above. Total toe-in (or simply toe-in) of wheels on one axle. as the name suggests, it is the sum of individual angles. If the planes of rotation of the wheels intersect in front of the car, the toe-in is positive (toe-in), if at the rear it is negative (toe-out). In the latter case, we can talk about wheel misalignment.
In adjustment data, convergence is sometimes given not only as an angular, but also as a linear value. This is related to that. that the toe-in of the wheels is also judged by the difference in the distances between the flanges of the rims, measured at the level of their centers behind and in front of the axle.

Whatever the truth, perhaps archaeologists will, of course, recognize the skill of ancient builders, for example. Flinders Petrie concluded that the errors in the measurements were so small that he pinched his finger. The walls connecting the corridors, falling 107 m into the center of the pyramid, showed a deviation of only 0.5 cm from ideal accuracy. Can we explain the mystery of the Pharaoh's pyramid to the pedantry of the architects and builders, or to the unknown magic of Egypt, or to the simple necessity of keeping the dimensions as close as possible to achieve the maximum benefit of the pyramid?

Various sources, including serious technical literature, often give the version that wheel alignment is necessary to compensate for the side effects of camber. They say that due to the deformation of the tire in the contact patch, the “collapsed” wheel can be imagined as the base of a cone. If the wheels are installed with a positive camber angle (why is not important yet), they tend to “roll” in different directions. To counteract this, the planes of rotation of the wheels are brought together (Fig. 20)

Is it just a coincidence that this number expresses the distance from the Sun, which is reported in millions of miles? An Egyptian cubit is exactly one ten-millimeter radius of the earth. The Great Pyramid expresses the 2p relationship between the circumference and radius of the Earth. Circle The square area of ​​a circle is 023 feet.

He also discusses the similarities between the figures in Nazca, the Great Pyramid, and Egyptian hieroglyphic texts. Bowles notes that the Great Pyramid and the Nazca Plateau will be on the equator when North Pole will be located in southeast Alaska. Using coordinates and spherical trigonometry, the book demonstrates the remarkable connections between three ancient sites.

The version, it must be said, is not without grace, but does not stand up to criticism. If only because it assumes an unambiguous relationship between camber and toe. Following the proposed logic, wheels with a negative camber angle must necessarily be installed with divergence, and if the camber angle is zero, then there should be no toe-in. In reality this is not the case at all.

Of course, this connection also exists between the Great Pyramid, the Nazca Plate and the "ancient lineage" axis, regardless of where the North Pole is located. This relationship can be used to determine the distances between three points and a plane. In the royal chamber the diagonal is 309 from the eastern wall, the distance from the chamber is 412, the middle diagonal is 515.

The distances between Ollantaytambo, the Great Pyramid and the Axis Point on the Ancient Line express the same geometric relationship. 3-4 The distance of the Great Pyramid from Ollantaytambo is exactly 30% of the periphery of the Earth. Distance from Great Pyramid to Machu Picchu and the Axis Point in Alaska is 25% of the earth's perimeter. Stretching this isosceles triangle in height, we get two right triangle with sides from 15% to 20% - 25%.

Reality, as usual, is subject to more complex and ambiguous laws. When an inclined wheel rolls, a lateral force is actually present in the contact patch, which is often called camber thrust. It occurs as a result of elastic deformation of the tire in the transverse direction and acts in the direction of inclination. The greater the angle of inclination of the wheel, the greater the camber thrust. This is what drivers of two-wheeled vehicles - motorcycles and bicycles - use when cornering. They only need to tilt their horse to force it to “prescribe” a curved trajectory, which can only be corrected by steering. Camber thrust also plays an important role when maneuvering cars, which will be discussed below. So it’s unlikely that it should be intentionally compensated for by toe-in. And the message itself is that due to a positive camber angle, the wheels tend to turn outward, i.e. towards divergence, incorrect. On the contrary, the design of the steering wheel suspension in most cases is such that with positive camber its thrust tends to increase toe-in. So “compensation for the side effects of camber” has nothing to do with it. There are several known factors that determine the need for wheel alignment. The first is that the previously set toe-in compensates for the influence of longitudinal forces acting on the wheel when the car is moving. The nature and depth (and therefore the result) of the influence depend on many circumstances: the drive wheel is either free-rolling, controlled, or not, and finally, on the kinematics and elasticity of the suspension. Thus, a rolling resistance force acts on a freely rolling car wheel in the longitudinal direction. It creates a bending moment that tends to rotate the wheel relative to the suspension mounting points in the direction of divergence. If the car's suspension is rigid (for example, not a split or torsion beam), then the effect will not be very significant. Nevertheless, it will definitely happen, since “absolute rigidity” is a purely theoretical term and phenomenon. In addition, the movement of the wheel is determined not only by the elastic deformation of the suspension elements, but also by the compensation of structural gaps in their connections, wheel bearings, etc.
In the case of a suspension with high compliance (which is typical, for example, for lever structures with elastic bushings), the result will increase many times over. If the wheel is not only free-rolling, but also steerable, the situation becomes more complicated. Due to the appearance of an additional degree of freedom at the wheel, the same resistance force has a double effect. The moment that bends the front suspension is complemented by a moment that tends to turn the wheel around the turning axis. The turning moment, the magnitude of which depends on the location of the steering axis, affects the parts of the steering mechanism and, due to their compliance, also makes a significant contribution to the change in wheel toe in motion. Depending on the running arm, the contribution of the turning moment can be with a “plus” or “minus” sign. That is, it can either increase wheel divergence or counteract it. If you do not take all this into account and initially install wheels with zero toe, they will take a divergent position when moving. From this will follow the consequences characteristic of cases of violation of toe adjustment: increased fuel consumption, saw-tooth tread wear and problems with handling, which will be discussed below.
The force of resistance to movement depends on the speed of the car. Therefore, the ideal solution would be variable toe, providing the same ideal wheel position at any speed. Since this is difficult to do, the wheel is pre-adjusted so as to achieve minimal tire wear at cruising speed. The wheel located on the drive axle is exposed to traction force most of the time. It exceeds the forces of resistance to movement, so the resultant forces will be directed in the direction of movement. Applying the same logic, we find that in this case the static wheels need to be installed with a discrepancy. A similar conclusion can be drawn regarding the steered drive wheels.
The best criterion of truth is practice. If, with this in mind, you look at the adjustment data for modern cars, you may be disappointed not to find much difference in the toe-in of the steering wheels of rear- and front-wheel drive models. In most cases, for both of them this parameter will be positive. Except that among front-wheel drive cars, cases of “neutral” toe adjustment are more common. The reason is not that the above logic is not correct. It’s just that when choosing the amount of toe-in, along with compensation of longitudinal forces, other considerations are taken into account that make adjustments to the final result. One of the most important is ensuring optimal vehicle handling. With increasing speeds and the dynamism of vehicles, this factor is becoming increasingly important.
Handling is a multifaceted concept, so it is worth clarifying that wheel toe most significantly affects the stabilization of the straight trajectory of the car and its behavior when entering a turn. This influence can be clearly illustrated using the example of steered wheels.

Suppose, while moving in a straight line, one of them is subject to a random disturbance from the unevenness of the road. The increased resistance force turns the wheel in the direction of decreasing toe. Through the steering mechanism, the impact is transmitted to the second wheel, the toe of which, on the contrary, increases. If the wheels initially have positive toe-in, the drag force on the first one decreases, and on the second one it increases, which counteracts the disturbance. When the convergence is zero, there is no counteracting effect, and when it is negative, a destabilizing moment appears, contributing to the development of disturbance. A car with such a toe adjustment will wander along the road and will have to be constantly caught by steering, which is unacceptable for an ordinary road car.
This “coin” also has a reverse, positive side - negative toe-in allows you to achieve the fastest response from the steering. The slightest action by the driver immediately provokes sudden change trajectories - the car willingly maneuvers, easily “agrees” to turn. This type of toe adjustment is often used in motorsports.

Those who watch TV shows about the WRC championship have probably noticed how actively Loeb or Grönholm have to work at the wheel, even on relatively straight sections of the track. The toe-in of the rear axle wheels has a similar effect on the behavior of the car - reducing the toe-in down to a slight divergence increases the “mobility” of the axle. This effect is often used to compensate for the understeer of cars, for example, front-wheel drive models with an overloaded front axle.
Thus, the static toe-in parameters, which are given in the adjustment data, represent a kind of superposition, and sometimes a compromise, between the desire to save on fuel and tires and achieve optimal handling characteristics for the car. Moreover, it is noticeable that in recent years the latter has prevailed.

Camber is a parameter that is responsible for the orientation of the wheel relative to the road surface. We remember that ideally they should be perpendicular to each other, i.e. there shouldn't be any collapse. However, most road cars have one. What's the trick?

Reference.
Camber reflects the orientation of the wheel relative to the vertical and is defined as the angle between the vertical and the plane of rotation of the wheel. If the wheel is actually “broken”, i.e. its top is inclined outward, the camber is considered positive. If the wheel is tilted towards the body, the camber is negative.

Until recently, there was a tendency for wheels to fall apart, i.e. give camber angles positive values. Many people probably remember textbooks on automobile theory, in which the installation of cambered wheels was explained by the desire to redistribute the load between the outer and inner wheel bearings. They say that with a positive camber angle, most of it falls on the internal bearing, which is easier to make more massive and durable. As a result, the durability of the bearing assembly benefits. The thesis is not very convincing, if only because if it is true, it is only for an ideal situation - straight-line movement of a car on an absolutely flat road. It is known that when maneuvering and driving over irregularities, even the most minor ones, the bearing assembly experiences dynamic loads that are an order of magnitude greater than static forces. And they are not distributed exactly as the positive camber “dictates”.

Sometimes they try to interpret positive camber as an additional measure aimed at reducing the running-in shoulder. When we get to the point of getting to know this important parameter of the steering wheel suspension, it will become clear that this method of influence is far from the most successful. It is associated with a simultaneous change in the track width and the included angle of inclination of the wheel steering axis, which is fraught with undesirable consequences. There are more direct and less painful options for changing the break-in shoulder. In addition, its minimization is not always the goal of suspension developers.

A more convincing version is that positive camber compensates for the wheel displacement that occurs when the axle load increases (as a result of an increase in vehicle load or dynamic redistribution of its mass during acceleration and braking). The elasto-kinematic properties of most types of modern suspensions are such that as the weight on the wheel increases, the camber angle decreases. In order to ensure maximum traction of the wheels with the road, it is logical to first “break them apart” a little. Moreover, in moderate doses, camber does not significantly affect rolling resistance and tire wear.

It is reliably known that the choice of camber value is also influenced by the generally accepted profiling of the roadway. In civilized countries, where there are roads and not directions, their cross-section has a convex profile. In order for the wheel to remain perpendicular to the supporting surface in this case, it needs to be given a small positive camber angle.
Looking through the specifications on the UUK, you will notice that in last years the opposite “collapse trend” prevails. The wheels of most production cars are statically installed with negative camber. The fact is that, as already mentioned, the task of ensuring their best stability and controllability comes to the fore. Camber is a parameter that has a decisive influence on the so-called lateral reaction of the wheels. It is this that counteracts the centrifugal forces acting on the car when turning and helps keep it on a curved path. From general considerations it follows that the adhesion of the wheel to the road (lateral reaction) will be maximum with the largest contact patch area, i.e. with the wheel in a vertical position. In fact, for a standard wheel design it reaches a peak at small negative lean angles, which is due to the contribution of the mentioned camber thrust. This means that in order to make the car’s wheels extremely grippy when turning, you don’t need to break them apart, but, on the contrary, “dump them.” This effect has been known for a long time and has been used in motorsports for just as long. If you take a closer look at the “formula” car, you can clearly see that its front wheels are installed with a large negative camber.

What is good for racing cars is not entirely suitable for production cars. Excessive negative camber causes increased wear on the inner tread area. As the wheel inclination increases, the contact patch area decreases. Wheel traction during straight-line motion decreases, which in turn reduces the efficiency of acceleration and braking. Excessive negative camber affects the car’s ability to maintain a straight trajectory in the same way as insufficient toe-in; the car becomes overly nervous. The same camber thrust is to blame for this. In an ideal situation, the lateral forces caused by the camber act on both wheels of the axle and balance each other. But as soon as one of the wheels loses traction, the camber thrust of the other turns out to be uncompensated and causes the car to deviate from a straight trajectory. By the way, if you remember that the amount of traction depends on the inclination of the wheel, it is not difficult to explain the lateral pull of the car at unequal camber angles of the right and left wheels. In a word, when choosing the camber value, you also have to look for the “golden mean”.

To ensure good stability of the car, it is not enough to make the camber angles negative in static conditions. Suspension designers must ensure that the wheels maintain optimal (or close to it) orientation in all driving modes. This is not easy to do, since during maneuvers any changes in the position of the body, accompanied by displacement of the suspension elements (dive, side rolls, etc.), lead to a significant change in the camber of the wheels. Oddly enough, this problem is solved more easily on sports cars with their “brutal” suspensions, characterized by high angular rigidity and short strokes. Here, the static values ​​of camber (and toe) differ least from how they look in dynamics.

The greater the range of suspension travel, the greater the change in camber while driving. Therefore, the hardest thing is for developers of conventional road cars with maximally elastic (for the best comfort) suspensions. They have to rack their brains over how to “combine the incompatible” - comfort and stability. Usually a compromise can be found by “conjuring” the suspension kinematics.

There are solutions to minimize changes in camber angles and give these changes the desired “trend”. For example, it is desirable that when turning, the most loaded outer wheel would remain in that very optimal position - with a slight negative camber. To do this, when the body rolls, the wheel must “fall” onto it even more, which is achieved by optimizing the geometry of the suspension guide elements. In addition, they try to reduce the body roll itself by using anti-roll bars.
To be fair, it should be said that suspension elasticity is not always the enemy of stability and handling. In “good hands,” elasticity, on the contrary, contributes to them. For example, with skillful use of the “self-steering” effect of the rear axle wheels. Returning to the topic of conversation, we can summarize that the camber angles, which are indicated in the specifications for passenger cars, will differ significantly from what they will be in a turn.

Concluding the “disassembly” with alignment and camber, we can mention one more interesting aspect, which has practical significance. The regulatory data on the control unit does not provide absolute values ​​of camber and toe angles, but ranges of permissible values. The tolerances for toe-in are tighter and usually do not exceed ±10", while for camber they are several times looser (on average ±30"). This means that the master performing the adjustment of the control unit can adjust the suspension without going beyond the factory specifications. It would seem that several tens of arc minutes are nonsense. I entered the parameters into the “green corridor” - and order was achieved. But let's see what the result could be. For example, the specifications for the BMW 5 Series in the E39 body indicate: toe 0°5"±10", camber -0°13"±30". This means that, while remaining in the “green corridor”, the toe can take a value from –0°5" to 5", and the camber from –43" to 7". That is, both toe and camber can be negative, neutral or positive. Having an idea of ​​the influence of toe-in and camber on the behavior of the car, you can deliberately “tamper” these parameters so as to obtain the desired result. The effect will not be dramatic, but it will definitely be there.

The camber and toe we considered are parameters that are determined for all four wheels of the car. Next, we will talk about angular characteristics that relate only to the steered wheels and determine the spatial orientation of their rotation axis.

It is known that the position of the steering axis of a car's steering wheel is determined by two angles: longitudinal and transverse. Why not make the rotation axis strictly vertical? Unlike cases with camber and alignment, the answer to this question is more unambiguous. There is almost unanimous agreement here, at least with regard to the longitudinal angle of inclination - caster.

It is rightly noted that the main function of caster is high-speed (or dynamic) stabilization of the car’s steered wheels. Stabilization in this case is the ability of the steered wheels to resist deviation from the neutral (corresponding to straight-line motion) position and automatically return to it after the action ceases external forces that caused the deviation. A moving car wheel is constantly subject to disturbing forces that tend to push it out of its neutral position. They may be a result of driving over uneven roads, unbalanced wheels, etc. Since the magnitude and direction of disturbances are constantly changing, their impact is randomly oscillatory. Without a stabilization mechanism, the driver would have to counter the vibrations, which would make driving a pain and would certainly increase tire wear. With proper stabilization, the car moves steadily in a straight line with minimal driver intervention and even with the steering wheel released.

Deflection of the steered wheels can be caused by intentional actions of the driver associated with changing the direction of movement. In this case, the stabilizing effect assists the driver when exiting a corner by automatically returning the wheels to neutral. But at the entrance to the turn and at its apex, the “driver,” on the contrary, has to overcome the “resistance” of the wheels, applying a certain force to the steering wheel. The reaction force generated at the steering wheel creates what is called steering feel or steering feel, which is something that has received a lot of attention from both car designers and automotive journalists.

In the last lesson, we successfully mastered (or repeated, depending on who) the key concepts of all trigonometry. This trigonometric circle , angle on a circle , sine and cosine of this angle , and also mastered signs of trigonometric functions by quarters . We mastered it in detail. On the fingers, one might say.

But this is not enough yet. For successful practical application all these simple concepts we need another useful skill. Namely - correct working with corners in trigonometry. Without this skill in trigonometry, there is no way. Even in the most primitive examples. Why? Yes, because the angle is the key operating figure in all trigonometry! No, not trigonometric functions, not sine and cosine, not tangent and cotangent, namely the corner itself. No angle means no trigonometric functions, yes...

How to work with angles on a circle? To do this, we need to firmly grasp two points.

1) How Are angles measured on a circle?

2) What are they counted (measured)?

The answer to the first question is the topic of today's lesson. We will deal with the first question in detail right here and now. I will not give the answer to the second question here. Because it is quite developed. Just like the second question itself is very slippery, yes.) I won’t go into details yet. This is the topic of the next separate lesson.

Shall we get started?

How are angles measured on a circle? Positive and negative angles.

Those who read the title of the paragraph may already have their hair standing on end. How so?! Negative angles? Is this even possible?

To negative numbers We've already gotten used to it. We can depict them on the number axis: to the right of zero are positive, to the left of zero are negative. Yes, and we periodically look at the thermometer outside the window. Especially in winter, in the cold.) And the money on the phone is in the minus (i.e. duty) sometimes they leave. This is all familiar.

What about the corners? It turns out that negative angles in mathematics there are too! It all depends on how to measure this very angle... no, not on the number line, but on number circle! That is, on a circle. The circle - here it is, an analogue of the number line in trigonometry!

So, How are angles measured on a circle? There’s nothing we can do, we’ll have to draw this very circle first.

I'll draw this beautiful picture:

It is very similar to the pictures from the last lesson. There are axes, there is a circle, there is an angle. But there is also new information.

I also added 0°, 90°, 180°, 270° and 360° numbers on the axes. Now this is more interesting.) What kind of numbers are these? Right! These are the angle values ​​measured from our fixed side that fall to the coordinate axes. We remember that the fixed side of the angle is always tightly tied to the positive semi-axis OX. And any angle in trigonometry is measured precisely from this semi-axis. This basic starting point for angles must be kept firmly in mind. And the axes – they intersect at right angles, right? So we add 90° in each quarter.

And more added red arrow. With a plus. Red is on purpose so that it catches the eye. And it is well etched in my memory. Because this must be remembered reliably.) What does this arrow mean?

So it turns out that if we twist our corner along the arrow with a plus(counterclockwise, according to the numbering of quarters), then the angle will be considered positive! As an example, the figure shows an angle of +45°. By the way, please note that the axial angles 0°, 90°, 180°, 270° and 360° are also rewound in the positive direction! Follow the red arrow.

Now let's look at another picture:

Almost everything is the same here. Only the angles on the axes are numbered reversed. Clockwise. And they have a minus sign.) Still drawn blue arrow. Also with a minus. This arrow is the direction of the negative angles on the circle. She shows us that if we put off our corner clockwise, That the angle will be considered negative. For example, I showed an angle of -45°.

By the way, please note that the numbering of quarters never changes! It doesn’t matter whether we move the angles to plus or minus. Always strictly counterclockwise.)

Remember:

1. The starting point for angles is from the positive semi-axis OX. By the clock – “minus”, against the clock – “plus”.

2. The numbering of quarters is always counterclockwise, regardless of the direction in which the angles are calculated.

By the way, labeling angles on the axes 0°, 90°, 180°, 270°, 360°, each time drawing a circle, is not at all mandatory. This is done purely for the sake of understanding the point. But these numbers must be present in your head when solving any trigonometry problem. Why? Yes, because this basic knowledge provides answers to so many other questions in all of trigonometry! Most main question – Which quarter does the angle we are interested in fall into? Believe it or not, answering this question correctly solves the lion's share of all other trigonometry problems. We will deal with this important task (distributing angles into quarters) in the same lesson, but a little later.

The values ​​of the angles lying on the coordinate axes (0°, 90°, 180°, 270° and 360°) must be remembered! Remember it firmly, until it becomes automatic. And both a plus and a minus.

But from this moment the first surprises begin. And along with them, tricky questions addressed to me, yes...) What happens if there is a negative angle on a circle coincides with the positive? It turns out that the same point on a circle can be denoted by both a positive and a negative angle???

Absolutely right! This is true.) For example, a positive angle of +270° occupies a circle same situation , the same as a negative angle of -90°. Or, for example, a positive angle of +45° on a circle will take same situation , the same as the negative angle -315°.

We look at the next drawing and see everything:

In the same way, a positive angle of +150° will fall in the same place as a negative angle of -210°, a positive angle of +230° will fall in the same place as a negative angle of -130°. And so on…

And now what i can do? How exactly to count angles, if you can do it this way and that? Which is correct?

Answer: in every way correct! Mathematics does not prohibit either of the two directions for counting angles. And the choice of a specific direction depends solely on the task. If the assignment does not say anything in plain text about the sign of the angle (such as "define the largest negative corner" etc.), then we work with the angles that are most convenient for us.

Of course, for example, in such cool topics as trigonometric equations and inequalities, the direction of calculating angles can have a huge impact on the answer. And in the relevant topics we will consider these pitfalls.

Remember:

Any point on a circle can be designated by either a positive or a negative angle. Anyone! Whatever we want.

Now let's think about this. We found out that an angle of 45° is exactly the same as an angle of -315°? How did I find out about these same 315° ? Can't you guess? Yes! Through a full rotation.) In 360°. We have an angle of 45°. How long does it take to complete a full revolution? Subtract 45° from 360° - so we get 315° . Let's go to negative side– and we get an angle of -315°. Still not clear? Then look at the picture above again.

And this should always be done when converting positive angles to negative (and vice versa) - draw a circle, mark approximately a given angle, we calculate how many degrees are missing to complete a full revolution, and move the resulting difference in the opposite direction. That's all.)

What else is interesting about angles that occupy the same position on a circle, do you think? And the fact that at such corners exactly the same sine, cosine, tangent and cotangent! Always!

For example:

Sin45° = sin(-315°)

Cos120° = cos(-240°)

Tg249° = tg(-111°)

Ctg333° = ctg(-27°)

But this is extremely important! For what? Yes, all for the same thing!) To simplify expressions. Because simplifying expressions is a key procedure successful solution any math assignments. And in trigonometry as well.

So, with general rule We figured out how to count angles on a circle. Well, if we started talking about full turns, about quarter turns, then it’s time to twist and draw these very corners. Shall we draw?)

Let's start with positive corners They will be easier to draw.

We draw angles within one revolution (between 0° and 360°).

Let's draw, for example, an angle of 60°. Everything is simple here, no hassles. We draw coordinate axes and a circle. You can do it directly by hand, without any compass or ruler. Let's draw schematically: We are not drawing with you. You don’t need to comply with any GOSTs, you won’t be punished.)

You can (for yourself) mark the angle values ​​on the axes and point the arrow in the direction against the clock. After all, we are going to save as a plus?) You don’t have to do this, but you need to keep everything in your head.

And now we draw the second (moving) side of the corner. In what quarter? In the first, of course! Because 60 degrees is strictly between 0° and 90°. So we draw in the first quarter. At an angle approximately 60 degrees to the fixed side. How to count approximately 60 degrees without a protractor? Easily! 60° is two thirds of right angle! We mentally divide the first devil of the circle into three parts, taking two thirds for ourselves. And we draw... How much we actually get there (if you attach a protractor and measure) - 55 degrees or 64 - it doesn’t matter! It’s important that it’s still somewhere about 60°.

We get the picture:

That's all. And no tools were needed. Let's develop our eye! It will come in handy in geometry problems.) This unsightly drawing is indispensable when you need to quickly scribble a circle and an angle, without really thinking about beauty. But at the same time scribble Right, without errors, with all necessary information. For example, like aid when solving trigonometric equations and inequalities.

Let's now draw an angle, for example, 265°. Let's figure out where it might be located? Well, it’s clear that not in the first quarter and not even in the second: they end at 90 and 180 degrees. You can figure out that 265° is 180° plus another 85°. That is, to the negative semi-axis OX (where 180°) you need to add approximately 85°. Or, even simpler, guess that 265° does not reach the negative semi-axis OY (where 270° is) some unfortunate 5°. In short, in the third quarter there will be this angle. Very close to the negative semi-axis OY, to 270 degrees, but still in the third!

Let's draw:

Again, absolute precision is not required here. Let in reality this angle turn out to be, say, 263 degrees. But to the most important question (what quarter?) we answered correctly. Why is this the most important question? Yes, because any work with an angle in trigonometry (it doesn’t matter whether we draw this angle or not) begins with the answer to exactly this question! Always. If you ignore this question or try to answer it mentally, then mistakes are almost inevitable, yes... Do you need it?

Remember:

Any work with an angle (including drawing this very angle on a circle) always begins with determining the quarter in which this angle falls.

Now, I hope you can accurately depict angles, for example, 182°, 88°, 280°. IN correct quarters. In the third, first and fourth, if that...)

The fourth quarter ends with an angle of 360°. This is one full revolution. It is clear that this angle occupies the same position on the circle as 0° (i.e., the origin). But the angles don't end there, yeah...

What to do with angles greater than 360°?

“Are there really such things?”- you ask. They do happen! There is, for example, an angle of 444°. And sometimes, say, an angle of 1000°. There are all kinds of angles.) It’s just that visually such exotic angles are perceived a little more difficult than the angles we are used to within one revolution. But you also need to be able to draw and calculate such angles, yes.

To correctly draw such angles on a circle, you need to do the same thing - find out Which quarter does the angle we are interested in fall into? Here, the ability to accurately determine the quarter is much more important than for angles from 0° to 360°! The procedure for determining the quarter itself is complicated by just one step. You'll see what it is soon.

So, for example, we need to figure out which quadrant the 444° angle falls into. Let's start spinning. Where? A plus, of course! They gave us a positive angle! +444°. We twist, we twist... We twisted it one turn - we reached 360°.

How long is there left until 444°?We count the remaining tail:

444°-360° = 84°.

So, 444° is one full rotation (360°) plus another 84°. Obviously this is the first quarter. So, the angle 444° falls in the first quarter. Half the battle is done.

Now all that remains is to depict this angle. How? Very simple! We make one full turn along the red (plus) arrow and add another 84°.

Like this:

Here I didn’t bother cluttering the drawing - labeling the quarters, drawing angles on the axes. All this good stuff should have been in my head for a long time.)

But I used a “snail” or a spiral to show exactly how an angle of 444° is formed from angles of 360° and 84°. The dotted red line is one full revolution. To which 84° (solid line) are additionally screwed. By the way, please note that if this full revolution is discarded, this will not affect the position of our angle in any way!

But this is important! Angle position 444° completely coincides with an angle position of 84°. There are no miracles, that’s just how it turns out.)

Is it possible to discard not one full revolution, but two or more?

Why not? If the angle is hefty, then it’s not only possible, but even necessary! The angle won't change! More precisely, the angle itself will, of course, change in magnitude. But his position on the circle is absolutely not!) That’s why they full revolutions, that no matter how many copies you add, no matter how many you subtract, you will still end up at the same point. Nice, isn't it?

Remember:

If you add (subtract) any angle to an angle whole the number of full revolutions, the position of the original angle on the circle will NOT change!

For example:

Which quarter does the 1000° angle fall into?

No problem! We count how many full revolutions sit in a thousand degrees. One revolution is 360°, another is already 720°, the third is 1080°... Stop! Too much! This means that it sits at an angle of 1000° two full turns. We throw them out of 1000° and calculate the remainder:

1000° - 2 360° = 280°

So, the position of the angle is 1000° on the circle the same, as at an angle of 280°. Which is much more pleasant to work with.) And where does this corner fall? It falls into the fourth quarter: 270° (negative semi-axis OY) plus another ten.

Let's draw:

Here I no longer drew two full turns with a dotted spiral: it turns out to be too long. I just drew the remaining tail from zero, discarding All extra turns. It’s as if they didn’t exist at all.)

Once again. In a good way, the angles 444° and 84°, as well as 1000° and 280°, are different. But for sine, cosine, tangent and cotangent these angles are - the same!

As you can see, in order to work with angles greater than 360°, you need to determine how many full revolutions sits in a given large angle. This is the very additional step that must be done first when working with such angles. Nothing complicated, right?

Rejecting full revolutions is, of course, a pleasant experience.) But in practice, when working with absolutely terrible angles, difficulties arise.

For example:

Which quarter does the angle 31240° fall into?

So what, are we going to add 360 degrees many, many times? It's possible, if it doesn't burn too much. But we can not only add.) We can also divide!

So let’s divide our huge angle into 360 degrees!

With this action we will find out exactly how many full revolutions are hidden in our 31240 degrees. You can divide it into a corner, you can (whisper in your ear:)) on a calculator.)

We get 31240:360 = 86.777777….

The fact that the number turned out to be fractional is not scary. Only us whole I'm interested in the revs! Therefore, there is no need to divide completely.)

So, in our shaggy coal sits as many as 86 full revolutions. Horror…

It will be in degrees86·360° = 30960°

Like this. This is exactly how many degrees can be painlessly thrown out of a given angle of 31240°. Remains:

31240° - 30960° = 280°

All! The position of the angle 31240° is fully identified! Same place as 280°. Those. fourth quarter.) I think we've already depicted this angle before? When was the 1000° angle drawn?) There we also went 280 degrees. Coincidence.)

So, the moral of this story is:

If we are given a scary hefty angle, then:

1. Determine how many full revolutions sit in this corner. To do this, divide the original angle by 360 and discard the fractional part.

2. We count how many degrees there are in the resulting number of revolutions. To do this, multiply the number of revolutions by 360.

3. We subtract these revolutions from the original angle and work with the usual angle ranging from 0° to 360°.

How to work with negative angles?

No problem! Exactly the same as with positive ones, only with one single difference. Which one? Yes! You need to turn the corners reverse side, minus! Going clockwise.)

Let's draw, for example, an angle of -200°. First, everything is as usual for positive angles - axes, circle. Let's also draw a blue arrow with a minus and sign the angles on the axes differently. Naturally, they will also have to be counted in a negative direction. These will be the same angles, stepping through 90°, but counted in the opposite direction, to the minus: 0°, -90°, -180°, -270°, -360°.

The picture will look like this:

When working with negative angles, there is often a feeling of slight bewilderment. How so?! It turns out that the same axis is, say, +90° and -270° at the same time? No, something is fishy here...

Yes, everything is clean and transparent! We already know that any point on a circle can be called either a positive or a negative angle! Absolutely any. Including on some of the coordinate axes. In our case we need negative angle calculus. So we snap all the corners to minus.)

Now drawing the angle -200° correctly is not difficult at all. This is -180° and minus another 20°. We begin to swing from zero to minus: we fly through the fourth quarter, we also miss the third, we reach -180°. Where should I spend the remaining twenty? Yes, everything is there! By the hour.) Total angle -200° falls within second quarter.

Now do you understand how important it is to firmly remember the angles on the coordinate axes?

The angles on the coordinate axes (0°, 90°, 180°, 270°, 360°) must be remembered precisely in order to accurately determine the quarter where the angle falls!

What if the angle is large, with several full turns? It's OK! What difference does it make whether these full revolutions are turned to positive or negative? A point on a circle will not change its position!

For example:

Which quarter does the -2000° angle fall into?

All the same! First, we count how many full revolutions sit in this evil corner. In order not to mess up the signs, let’s leave the minus alone for now and simply divide 2000 by 360. We’ll get 5 with a tail. We don’t care about the tail for now, we’ll count it a little later when we draw the corner. We count five full revolutions in degrees:

5 360° = 1800°

Wow. This is exactly how many extra degrees we can safely throw out of our corner without harming our health.

We count the remaining tail:

2000° – 1800° = 200°

But now we can remember about the minus.) Where will we wind the 200° tail? Minus, of course! We are given a negative angle.)

2000° = -1800° - 200°

So we draw an angle of -200°, only without any extra revolutions. I just drew it, but so be it, I’ll draw it one more time. By hand.

It is clear that the given angle -2000°, as well as -200°, falls within second quarter.

So, let’s go crazy... sorry... on our head:

If a very large negative angle is given, then the first part of working with it (finding the number of full revolutions and discarding them) is the same as when working with a positive angle. The minus sign does not play any role at this stage of the solution. The sign is taken into account only at the very end, when working with the angle remaining after removing full revolutions.

As you can see, drawing negative angles on a circle is no more difficult than positive ones.

Everything is the same, only in the other direction! By the hour!

Now comes the most interesting part! We looked at positive angles, negative angles, large angles, small angles - the full range. We also found out that any point on a circle can be called a positive and negative angle, we discarded full revolutions... Any thoughts? It must be postponed...

Yes! Whatever point on the circle you take, it will correspond to infinite set corners! Big ones and not so big ones, positive ones and negative ones - all kinds! And the difference between these angles will be whole number of full revolutions. Always! That’s how the trigonometric circle works, yes...) That’s why reverse the task is to find the angle using the known sine/cosine/tangent/cotangent - solvable ambiguous. And much more difficult. In contrast to the direct problem - given an angle, find the entire set of its trigonometric functions. And in more serious topics of trigonometry ( arches, trigonometric equations And inequalities ) we will encounter this trick all the time. We're getting used to it.)

1. Which quarter does the -345° angle fall into?

2. Which quarter does the angle 666° fall into?

3. Which quarter does the angle 5555° fall into?

4. Which quarter does the -3700° angle fall into?

5. What sign doescos999°?

6. What sign doesctg999°?

And did it work? Wonderful! There is a problem? Then you.

Answers:

1. 1

2. 4

3. 2

4. 3

5. "+"

6. "-"

This time the answers are given in order, breaking with tradition. For there are only four quarters, and there are only two signs. You won’t really run away...)

In the next lesson we will talk about radians, about mysterious number"pi", let's learn how to easily and simply convert radians to degrees and vice versa. And we will be surprised to discover that even this simple knowledge and skills will be quite enough for us to successfully solve many non-trivial trigonometry problems!