Negative camber. Camber-convergence: what does it affect in the car. The positive angle is too large. Trigonometric circle. A Comprehensive Guide (2019) Angles in Trigonometry Count

Trigonometry, as a science, originated in the Ancient East. The first trigonometric relationships were derived by astronomers to create an accurate calendar and star orientation. These calculations were related to spherical trigonometry, while in the school course the aspect and angle ratios of a flat triangle are studied.

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 of 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-Marazvi introduced functions such as tangent and cotangent, compiled the first tables of values ​​for sines, tangents and cotangents. The concept of sine and cosine was introduced by Indian scientists. A lot of attention is devoted to trigonometry in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

The basic trigonometric functions of a numeric argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sinusoid, cosine, tangent and cotangent.

The formulas for calculating the values ​​of these quantities are based on the Pythagorean theorem. Schoolchildren know it better in the wording: "Pythagorean pants, equal in all directions," since the proof is given on the example of an isosceles right-angled triangle.

Sine, cosine, and other dependencies establish a relationship between acute angles and the sides of any right triangle. Let's give formulas for calculating these values ​​for angle A and trace the relationship of trigonometric functions:

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

Trigonometric circle

Graphically, the ratio of these quantities can be represented as follows:

The circle, in this case, represents all possible values ​​of the angle α - from 0 ° to 360 °. As you can see from the figure, each function takes on a negative or positive value, depending on the value of the angle. For example, sin α will be with a "+" sign if α belongs to I and II quarters of a circle, that is, is in the range from 0 ° to 180 °. When α is from 180 ° to 360 ° (III and IV quarters), sin α can only be negative.

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

The 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 by chance. The designation π in the tables stands for radians. Rad is the angle at which the length of a circular 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.

The angles in the tables for trigonometric functions correspond to the values ​​of radians:

So, it's not hard to guess that 2π is a full 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 a comparative table of properties for a sine wave and a cosine wave:

Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with the signs of trigonometric quantities and mentally "fold" the graph about the OX axis. If the signs match, the function is even; otherwise, it is odd.

The introduction of radians and the enumeration of the main properties of the sinusoid and cosine allow us to give the following pattern:

It is very easy to verify the correctness of the formula. For example, for x = π / 2, the sine is 1, as is the cosine x = 0. The check can be carried out by referring to tables or by tracing the curves of functions for given values.

Tangentoid and cotangentoid properties

Plots of tangent and cotangent functions differ significantly from sine and cosine. The tg and ctg values ​​are inverse to each other.

1. Y = tg x.
2. The tangentoid tends to the y-values ​​at x = π / 2 + πk, but never reaches them.
3. The smallest positive period of the tangentoid is π.
4. Tg (- x) = - tg x, that is, 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.

Consider a graphical representation of a cotangentoid below in the text.

The main properties of a cotangensoid:

1. Y = ctg 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 cotangensoid tends to the values ​​of y at x = πk, but never reaches them.
4. The smallest positive period of a cotangensoid is π.
5. Ctg (- x) = - ctg x, that is, 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

Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take as an example an infinite set of natural numbers, then the considered examples can be presented in the following form:

For a visual proof of their correctness, mathematicians have come up with many different methods. Personally, I look at all these methods as dancing shamans with tambourines. Essentially, they all boil down to the fact that either some of the rooms are not occupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of the century. Of course, the time factor can be stupidly ignored, but it will already be from the category "the law is not written for fools." It all depends on what we are doing: adjusting reality to match mathematical theories or vice versa.

What is an "endless hotel"? An endless hotel is a hotel that always has any number of vacant places, no matter how many rooms are occupied. If all the rooms in the endless visitor corridor are occupied, there is another endless corridor with the guest rooms. There will be an endless number of such corridors. Moreover, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, however, are not able to distance themselves from commonplace everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. Here are mathematicians and are trying to manipulate the serial numbers of hotel rooms, convincing us that it is possible to "shove the stuff in."

I will demonstrate the logic of my reasoning to you on the example of an infinite set of natural numbers. First, you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves, in Nature there are no numbers. Yes, Nature is excellent at counting, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers there are. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. And if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:

I wrote down the actions in the algebraic notation system and in the notation system adopted in set theory, with a detailed enumeration of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one subtracts from it and adds the same unit.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:

Subscripts "one" and "two" indicate that these items belonged to different sets. Yes, if you add one to the infinite set, the result will also be an infinite set, but it will not be the same as the original set. If we add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

Lots of natural numbers are used for counting in the same way as a ruler for measurements. Now imagine adding one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - it's your own business. But if you ever run into mathematical problems, think about whether you are not following the path of false reasoning trodden by generations of mathematicians. After all, doing mathematics, first of all, form a stable stereotype of thinking in us, and only then add mental abilities to us (or, on the contrary, deprive us of free thought).

Sunday, 4 August 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical foundation of Babylonian mathematics did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it hard for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:

The rich theoretical basis of modern mathematics is not holistic and is reduced to a set of disparate sections devoid of a common system and evidence base.

I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, 3 August 2019

How do you divide a set into subsets? To do this, it is necessary to enter a new unit of measurement that is present for some of the elements of the selected set. Let's look at an example.

Let us have many A consisting of four people. This set was formed on the basis of "people" Let us denote the elements of this set by the letter a, a subscript with a digit will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sex" and denote it by the letter b... Since sexual characteristics are inherent in all people, we multiply each element of the set A by gender b... Note that now our multitude of "people" has become a multitude of "people with sex characteristics." After that, we can divide the sex characteristics into masculine bm and women bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sex characteristics, it does not matter which one is male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.

After multiplication, reduction and rearrangement, we got two subsets: the subset of men Bm and a subset of women Bw... Mathematicians think about the same when they apply set theory in practice. But they do not devote us to the details, but give a finished result - "a lot of people consist of a subset of men and a subset of women." Naturally, you may wonder how correctly the mathematics is applied in the above transformations? I dare to assure you, in fact, the transformations were done correctly, it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I'll tell you about it.

As for supersets, you can combine two sets into one superset by choosing the unit of measurement that is present for the elements of these two sets.

As you can see, units of measurement and common mathematics make set theory a thing of the past. An indication that set theory is not all right is that mathematicians have come up with their own language and notation for set theory. Mathematicians did what shamans once did. Only shamans know how to "correctly" apply their "knowledge". They teach us this "knowledge".

Finally, I want to show you how mathematicians manipulate with.

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". This is how it sounds:

Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.

This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them has become a generally accepted solution to the question ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn over the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly catch up with the turtle."

How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:

During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia Zeno tells about a flying arrow:

The flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs are needed, taken from the same point at different points in time, but it is impossible to determine the distance from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, 4 July 2018

I have already told you that, with the help of which shamans try to sort "" reality. How do they do it? How does the formation of a set actually take place?

Let's take a closer look at the definition of a set: "a set of different elements, thought of as a single whole." Now feel the difference between the two phrases: "thinkable as a whole" and "thinkable as a whole." The first phrase is the end result, the set. The second phrase is preliminary preparation for the formation of a set. At this stage, reality is broken down into separate elements ("whole") from which a set will then be formed ("a single whole"). At the same time, the factor that makes it possible to unite the "whole" into a "single whole" is carefully monitored, otherwise the shamans will fail. After all, shamans know in advance what kind of multitude they want to demonstrate to us.

Let me show you the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, but there are no bows. After that we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

Now let's do a little dirty trick. Take "solid in a pimple with a bow" and combine these "wholes" by color, selecting the red elements. We got a lot of "red". Now a question to fill in: the resulting sets "with a bow" and "red" are the same set or are they two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We have formed a set of "red solid into a bump with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a pimple), ornaments (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics... This is what it looks like.

The letter "a" with different indices denotes different units of measurement. Units of measurement are highlighted in brackets, by which the "whole" is allocated at the preliminary stage. The unit of measurement, by which the set is formed, is taken out of the brackets. The last line shows the final result - the element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, not the dancing of shamans with tambourines. Shamans can "intuitively" arrive at the same result, arguing it "by the obviousness", because units of measurement are not included in their "scientific" arsenal.

It is very easy to use units to split one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Saturday, 30 June 2018

If mathematicians cannot reduce a concept to other concepts, then they do not understand anything in mathematics. I answer: how do the elements of one set differ from the elements of another set? The answer is very simple: numbers and units.

Today, everything that we do not take belongs to some set (as mathematicians assure us). By the way, have you seen on your forehead in the mirror a list of those sets to which you belong? And I have not seen such a list. I will say more - not a single thing in reality has a tag with a list of the sets to which this thing belongs. The multitudes are all the inventions of shamans. How do they do it? Let's look a little deeper in history and see what the elements of a set looked like before shamanic mathematicians pulled them apart into their sets.

A long time ago, when no one had ever heard of mathematics, and only trees and Saturn had rings, huge herds of wild set elements roamed physical fields (after all, shamans had not yet invented mathematical fields). They looked something like this.

Yes, do not be surprised, from the point of view of mathematics, all the elements of sets are most similar to sea urchins - from one point, like needles, units of measurement stick out in all directions. For those who, I remind you that any unit of measurement can be geometrically represented as a segment of arbitrary length, and a number as a point. Geometrically, any value can be represented as a bunch of segments sticking out in different directions from one point. This point is point zero. I won't draw this piece of geometric art (no inspiration), but you can easily imagine it.

What units of measurement form an element of the set? Anyone describing this element from different points of view. These are the ancient units of measurement that were used by our ancestors and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are also unknown units of measurement that our descendants will invent and which they will use to describe reality.

We figured out the geometry - the proposed model of the elements of the set has a clear geometric representation. What about physics? Units of measurement are the direct connection between mathematics and physics. If shamans do not recognize units of measurement as a full-fledged element of mathematical theories, this is their problem. I personally cannot imagine the real science of mathematics without units of measurement. That is why, at the very beginning of my story about set theory, I spoke of it as the Stone Age.

But let's move on to the most interesting thing - to the algebra of elements of sets. Algebraically, any element of a set is a product (the result of multiplication) of different quantities. It looks like this.

I deliberately did not use the conventions of set theory, since we were looking at a set element in its natural habitat prior to the emergence of set theory. Each pair of letters in brackets denotes a separate value, consisting of the number indicated by the letter " n"and units of measurement indicated by the letter" a". The indices next to the letters indicate that the numbers and units of measurement are different. One element of the set can consist of an infinite number of quantities (as far as we and our descendants have enough imagination). Each bracket is geometrically depicted as a separate segment. In the example with the sea urchin one bracket is one needle.

How do shamans form sets from different elements? In fact, by units or numbers. Not understanding anything in mathematics, they take different sea urchins and carefully examine them in search of that single needle, along which they form a set. If there is such a needle, then this element belongs to the set, if there is no such needle, it is an element not from this set. Shamans tell us fables about thought processes and a single whole.

As you may have guessed, the same element can belong to very different sets. Further I will show you how sets, subsets and other shamanic nonsense are formed. As you can see, "there cannot be two identical elements in a set", but if there are identical elements in a set, such a set is called a "multiset". Such logic of absurdity will never be understood by rational beings. This is the level of talking parrots and trained monkeys, who lack intelligence from the word "completely". Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the incompetent engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer would build other bridges.

No matter how mathematicians hide behind the phrase "chur, I'm in the house", or rather "mathematics studies abstract concepts," there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to the mathematicians themselves.

We studied mathematics very well and now we are sitting at the checkout, giving out salaries. Here comes a mathematician to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and hand the mathematician his “mathematical set of salary”. Let us explain the mathematics that he will receive the rest of the bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: "You can apply it to others, you can not apply it to me!" Further, we will begin to assure us that there are different banknote numbers on bills of the same denomination, which means that they cannot be considered the same elements. Okay, let's count the salary in coins - there are no numbers on the coins. Here the mathematician will start to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms in each coin is unique ...

And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science did not lie anywhere near here.

Look here. We select football stadiums with the same pitch. The area of ​​the fields is the same, which means we have got a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How is it correct? And here the mathematician-shaman-shuller takes a trump ace out of his sleeve and begins to tell us either about the set or about the multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "thinkable as not a single whole" or "not thinkable as a whole."

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

Changing wheel alignment values ​​and adjusting them

The Piri Reis map has been compared to the modern map projection. Thus, he came to the conclusion that a mysterious map is taking over the world, as seen from a satellite soaring high above Cairo. In other words, over the Great Pyramid. Surprisingly, Egyptologists are consistently defending these spaces, although a recent survey of one recently opened corridor was conducted that has yet to bring any breakthroughs.

It is also worth noting that unusual psychotronic effects were found in the pyramid, which, among other things, can affect human health. We are talking about spatial psychotronics, which creates both energetic and geomagnetic "anomalous zones", which are further investigated.

When turning the wheel, the tire deforms under the action of lateral forces. And in order to maintain the maximum contact with the road, the car 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 wheel will tilt strongly, and then it will lose grip.

Initially, the lateral tilt of the steering axis was applied by engineers to eliminate the shortcomings of the car's suspension. He got rid of such "ailments" of the car as a positive camber and a positive break-in shoulder.

During archaeological excavations, strange burial offerings in the form of birds with outstretched wings were also found. Later aerodynamic studies of these subjects showed that they are most likely ancient glider models. One of them was found with the inscription "Amun's gift." The god Amon in Egypt was worshiped as a god of the wind so associated with flight is obvious.

But how did the members of this ancient civilization come 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 possible for members of a technologically advanced civilization that has more than a thousand years ago, disappeared without a trace.

Many vehicles use a McPherson type suspension. It makes it possible to get a negative or zero break-in leverage. After all, the pivot of the wheel consists of a single lever support, which can be easily placed inside the wheel. But this suspension is not perfect either, because due to its design it is almost impossible to make the tilt angle of the pivot axis small. When cornering, it tilts the outer wheel at an unfavorable angle (like a positive camber), and the inner wheel simultaneously tilts in the opposite direction.

But such objects are still in short supply. They disintegrate, 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.

Great Pyramid in size and design precision has never been equal. 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. No stone weighs less than a ton.

As a result, the contact patch at the outer wheel is greatly reduced. And since the main load is on the outer wheel in the corner, the entire axle loses much grip. This, of course, can be partially offset by caster and camber. Then the grip of the outer wheel will be good, while that of the inner wheel will practically disappear.

Toe-in of car wheels

On the rear axle, with a positive toe-in of the wheels, the car will be more stable in straight-line motion, and if there is a negative toe-in, the car will behave inadequately and scour from side to side.

And some of over seventy tons. Inside the chambers are connected by corridors. Today, a rough stone pyramid, but as soon as it was processed to a mirror-like luster of the masonry. It is believed that the peak of the Great Pyramid was adorned 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 the symbolic gateway to the afterlife. Others believe that the pyramid was an astronomical observatory. Someone says that help is in the geographic dimension.

But it should be remembered that an excessive deviation of the vehicle toe-in from zero will increase the rolling resistance when driving in a straight line, while cornering it will be noticeable to a lesser extent.

Camber

If you look from the front of the car, and the wheels tilt inward, then this is negative camber, and if they deviate outward of the car, then this is already positive camber. Camber is necessary to maintain the grip of the wheel with the road surface.

One bizarre theory is that the Great Pyramid was on the granaries. However, experts today generally agree that the pyramids were much more than just a giant grave. Scientists argue that the massive pyramid technology may not have been accessible to humans 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 an accuracy that was never achieved.

And surprisingly, the Great Pyramid lies at 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 our time in the sixteenth century. The pyramids were built at the exact center of the earth. Also the height of the pyramid - seen from a great height, can be seen from the moon. Moreover, the shape of the pyramid is one of the best for reflecting radars. These reasons lead some researchers to believe that the Egyptian pyramids were built outside of their other purposes and for navigation by would-be 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 vehicles when starting off with slipping.

For those who want to understand what the 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 apartment with an area of ​​1.6 square kilometers. Its base extends up to 900 square meters and is almost a millimeter in horizontal position. Two and three quarters of a million stone blocks were used for construction, with the heaviest weight weighing up to 70 tons. They fit in in such a way that this fact is a mystery. However, the technical side of creating the pyramid remains a mystery, as it will be a serious problem for today's advanced technology.

A digression into history shows that sophisticated wheel alignment was used on various vehicles long before the advent of the automobile. Here are some more or less well-known examples.
It is no secret that the wheels of some horse-drawn carriages and other horse-drawn carriages, intended for "dynamic" driving, were installed with a large positive camber clearly visible to the eye. This was done so that the dirt flying from the wheels did not fall into the carriage and important riders, but was scattered around. Thus, pre-revolutionary guidelines on how to build a good cart recommended installing wheels with negative camber. In this case, with the loss of the dowel locking the wheel, it did not immediately jump off the axle. The driver had time to notice the damage to the "running gear", fraught with especially big troubles in the presence of several tens of poods of flour in the cart and the absence of a jack. In the design of gun carriages (again, vice versa), positive camber was sometimes used. It is clear that not with the aim of protecting the gun from dirt. So it was convenient for the servants to roll the gun over the wheels with their hands from the side, without fear of crushing their legs. But at the cart, its huge wheels, which helped to easily get over the ditches, were tilted in the other direction - to the cart. The resulting increase in track gauge contributed to an increase in the stability of the Central Asian "mobile", which was distinguished by a high center of gravity. How do these historical facts relate to the installation of wheels of modern cars? Yes, in general, not any. However, they do lead to a useful conclusion. It can be seen that the installation of the wheels (in particular, their camber) is not subject to any single regularity.

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

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

The pyramid consists of 203 layers of stone blocks weighing from 2.5 to 15 tons. Some of the blocks at the bottom of the pyramid at the base weigh up to 50 tons. Initially, the entire pyramid was covered with a fine white and polished limestone shell, but the stone was used for construction, especially after the 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 scheme is divided by the double height of the pyramid, we get the Ludolph number - 3. The deviation from the Ludolph 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 one speaks of individual convergence. It is the angle between the plane of rotation of the wheel and the axis of the vehicle when viewed from above. The total convergence (or simply convergence) of the wheels of one axle. as the name suggests, is the sum of the 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 - negative (toe-out). In the latter case, we can talk about wheel alignment.
In the adjustment data, sometimes the convergence is given not only in the form of an angular, but also a linear value. This is due to the fact. that the toe-in of the wheels is also judged by the difference in the distances between the rim flanges, measured at the level of their centers at the rear and in front of the axle.

Whatever the truth, maybe archaeologists will certainly recognize the skill of ancient builders, for example. Flinders Petrie concluded that the measurement errors were so small that he covered 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, pedantry of architects and builders, or unknown Egyptian magic, or the simple need to keep dimensions as close as possible to maximize the benefits of the pyramid?

In various sources, including serious technical literature, a version is often cited 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 represented as the base of the cone. If the wheels are installed with a positive camber angle (why is not important yet), they tend to "roll out" 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 represents the distance from the Sun, which is reported in millions of miles? The Egyptian cubit is exactly one ten-millimeter radius of the earth. The Great Pyramid expresses the ratio of 2p between the circumference and the radius of the Earth. Circle The square area of ​​a circle is 023 feet.

He also discusses the similarities between figures at Nazca, the Great Pyramid, and Egyptian hieroglyphic texts. Bowles notes that the Great Pyramid and the Nazca Plateau will be at the equator when the North Pole is located in southeastern Alaska. Using coordinates and spherical trigonometry, the book demonstrates a remarkable connection between three points - ancient sites.

The version, I must say, is not devoid of grace, but does not stand up to criticism. If only because it assumes an unambiguous relationship between the collapse and convergence. Following the proposed logic, wheels with a negative camber angle must be installed with a discrepancy, and if the camber angle is zero, then there should be no toe. In reality, this is not at all the case.

Of course, this connection also exists between the Great Pyramid, the Nazca platform and the axis of the "ancient line", regardless of where the North Pole is. 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 east 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 Earth's periphery. The distance from the 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-angled triangles with sides ranging from 15% to 20% - 25%.

Reality, as usual, obeys more complex and ambiguous patterns. When a tilted wheel rolls, there is indeed a lateral force in the contact patch, which is often called camber thrust. It arises as a result of the elastic deformation of the tire in the lateral direction and acts in the direction of the slope. The greater the angle of inclination of the wheel, the greater the camber thrust. It is precisely this that is used by drivers of two-wheeled vehicles - motorcycles and bicycles - when cornering. It is enough for them to tilt their steed in order to make it "prescribe" a curved trajectory, which can only be corrected by the steering control. Camber thrust also plays an important role when maneuvering vehicles, which will be discussed below. So it is unlikely that it should be deliberately compensated for by convergence. And the message itself is that due to the positive camber angle, the wheels tend to turn outward, i.e. in the direction of discrepancy, is incorrect. On the contrary, the design of the suspension of the steered wheels in most cases is such that, with a positive camber, its thrust tends to increase toe-in. So there is nothing to do with "compensation of side effects of camber." The nature and depth (and hence the result) of the influence depends on many circumstances: the driving wheel is either freely rolling, controlled, or not, 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 tending to turn the wheel relative to the suspension attachment points in the direction of divergence. If the car's suspension is rigid (for example, not split or torsion beam), then the effect will not be very significant. Nevertheless, it will certainly be, 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 clearances in their joints, wheel bearings, etc.
In the case of a suspension with high flexibility (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 to roll, 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 in the front suspension is complemented by a moment that tends to turn the wheel around the steering axis. The turning moment, the magnitude of which depends on the position of the pivot axis, affects the parts of the steering mechanism and, due to their pliability, also makes a significant contribution to the change in wheel toe in motion. Depending on the run-in shoulder, the contribution of the turning moment can be with a plus or minus sign. That is, it can either increase the toe-out of the wheels, or counteract it. If you do not take all this into account and initially install the wheels with zero toe, they will take a divergent position in motion. This will "flow" the consequences that are typical for cases of violation of the toe adjustment: increased fuel consumption, sawtooth tread wear and problems with handling, which will be discussed later.
The amount of resistance to motion depends on the speed of the vehicle. Therefore, the ideal solution would be variable toe, providing the same ideal wheel alignment at all speeds. Since this is difficult to do, the wheel is preliminarily "flattened" so as to achieve minimum tire wear at cruising speed. A wheel located on the drive axle is subjected to tractive 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 get that in this case the wheels in statics must be set with a discrepancy. A similar conclusion can be made with regard to steerable drive wheels.
The best criterion for truth is practice. If, with this in mind, you look at the adjustment data for modern cars, you might be disappointed not to find much difference in wheel alignment between rear and front wheel drive models. In most cases, both those and others will have this parameter positive. Unless among front-wheel drive vehicles, cases of "neutral" toe adjustment are more common. The reason is not that the logic described above is not correct. It's just that when choosing the value of convergence, along with the compensation of longitudinal forces, other considerations are taken into account, which amend the final result. One of the most important is ensuring optimal vehicle handling. With the growth of speeds and dynamics of vehicles, this factor is becoming more and more important.
Controllability is a multifaceted concept, so it is worth clarifying that wheel alignment most significantly affects the stabilization of the straight-line trajectory of the car and its behavior at the entrance to the turn. This influence can be clearly explained by the example of steerable wheels.

Suppose, in motion in a straight line, one of them is subject to a random disturbance from the unevenness of the road. The increased drag force turns the wheel in the direction of decreasing toe-in. Through the steering mechanism, the impact is transmitted to the second wheel, the convergence of which, on the contrary, increases. If initially the wheels have a positive toe-in, the resistance force decreases on the first and increases on the second, 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 indignation. A car with such a toe adjustment will scour the road, it will have to be constantly steered, which is unacceptable for an ordinary road car.
This "coin" has a downside, a positive side - negative toe allows you to get the fastest response from the steering. The slightest action of the driver immediately provokes a sharp change in trajectory - the car willingly maneuvers, easily "agrees" to turn. This toe adjustment is used all the time in motorsport.

Those who watch TV shows about the WRC championship have probably paid attention to how actively the same Loeb or Grönholm have to work at the wheel, even on relatively straight sections of the track. Toe-in of the rear axle has a similar effect on the behavior of the car - reducing the toe-in up to a small discrepancy increases the "mobility" of the axle. This effect is often used to compensate for understeer in vehicles such as 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 rubber and achieve optimal handling characteristics for the car. Moreover, it is noticeable that in recent years the latter has been prevalent.

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 should be no collapse. Most road cars do, however. 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 to break the wheels, i.e. give the camber angles positive values. Many, for sure, remember textbooks on the theory of the car, in which the installation of wheels with camber was explained by the desire to redistribute the load between the outer and inner wheel bearings. Like, with a positive camber angle, most of it falls on the inner bearing, which is easier to make more massive and durable. As a result, the durability of the bearing assembly is beneficial. The thesis is not very convincing, if only because, if it is true, then only for an ideal situation - a straight-line movement of a car on an absolutely flat road. It is known that during maneuvers and driving irregularities, even the smallest ones, the bearing assembly experiences dynamic loads that are an order of magnitude higher than static forces. Yes, and they are distributed not exactly as "dictated" by the positive camber.

Sometimes people try to interpret positive camber as an additional measure aimed at reducing the break-in shoulder. When it comes to our acquaintance with this important parameter of the steering wheel suspension, it becomes 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 straighter and less painful options for changing the break-in shoulder. Also, minimizing it is not always the goal of suspension designers.

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

To provide the vehicle with good stability, it is not enough to make the camber angles negative in statics. Suspension designers must ensure that the wheels remain at or near optimal orientation in all driving conditions. This is not easy to do, since during maneuvers, any changes in the position of the body, accompanied by displacement of the suspension elements (pecks, side rolls, etc.), lead to a significant change in the camber. Oddly enough, this problem is easier to solve on sports cars with their "furious" suspensions, characterized by high angular stiffness and short strokes. Here, the static values ​​of camber (and toe) differ least of all from how they look in dynamics.

Solutions exist to minimize camber changes and give these changes the desired “trend”. For example, it is desirable that in a corner the most loaded outer wheel would remain in the same optimal position - with a slight negative camber. For this, when the body rolls, the wheel should "roll over" on it even more, which is achieved by optimizing the geometry of the suspension guide elements. In addition, they try to reduce the body roll themselves by using anti-roll bars.
It's fair to say that suspension elasticity isn't always the enemy of stability and handling. In “good hands,” elasticity, on the other hand, favors them. For example, with the skillful use of the effect of "self-steering" 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 corner.

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

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 pivot axis strictly vertical? In contrast to the cases with collapse and convergence, the answer to this question is more unambiguous. Here they are practically unanimous, at least with regard to the longitudinal tilt angle - caster.

If you are already familiar with trigonometric circle , and you just want to refresh your memory of individual 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 associate with impassable thicket. Suddenly, so many values ​​of trigonometric functions, so many formulas ...

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

If you are constantly looking at the table with the values ​​of trigonometric formulas, let's get rid of this habit!

Will help us out! You will work with it several times, and then it will pop up in your head. Why 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, tell by looking in standard table of trigonometric formula values which is the sine of, say, 300 degrees, or -45.

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

And when solving trigonometric equations and inequalities without a trigonometric circle - in general, nowhere.

Introducing the trigonometric circle

Let's go in order.

First, let's write out the following series of numbers:

And now this:

And finally, like this:

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

But how beautiful it turned out! In which case, we will restore this "miraculous ladder".

And why do we need it?

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

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

From the "0-Start" ray, set aside the angles in the direction of the arrow (see Fig.).

We get the corresponding points on the circle. So if we project the points on each of the axes, then we will come out just at the values ​​from the above chain.

Why is that, you ask?

We will not analyze everything. Consider principle, which will allow you to cope with other, similar situations.

Triangle AOB - rectangular, in it. And we know that opposite the angle b lies a leg half the size of the hypotenuse (our hypotenuse = the radius of the circle, that is, 1).

Hence, AB = (and therefore OM =). And by the Pythagorean theorem

Hopefully, something is already becoming clear?

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

Likewise with the rest of the values ​​of the first quarter.

As you understand, the axis (ox) familiar to us will be cosine axis and the (oy) axis is sine axis ... later.

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

So, here he is, the Omnipotent, without which there is nowhere in trigonometry.

But how to use the trigonometric circle, we'll talk in.

Counting angles on a trigonometric circle.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very ..."
And for those who are "very even ...")

It is almost the same as in the previous lesson. There are axes, a circle, an angle, everything is chin-chinarem. Added numbers of quarters (in the corners of the large square) - from the first to the fourth. And then suddenly who does not know? As you can see, the quarters (they are also called the beautiful word "quadrants") are numbered counterclockwise. Added values ​​for the angle on the axes. Everything is clear, no problems.

And a green arrow has been added. With a plus. What does it mean? Let me remind you that the fixed side of the corner always nailed to the positive OX semiaxis. So, if we twist the movable side of the corner along the arrow with a plus, i.e. in ascending order of the quarter numbers, the angle will be considered positive. For example, the picture shows a positive angle of + 60 °.

If we postpone the corners in the opposite direction, clockwise, the angle will be considered negative. Move the cursor over the picture (or tap the picture on the tablet), you will see a blue arrow with a minus. This is the direction of the negative reading of the angles. A negative angle (- 60 °) is shown as an example. And you will also see how the numbers on the axes have changed ... I also translated them into negative angles. Quadrant numbering does not change.

This is where the first misunderstandings usually begin. How so !? And what if the negative angle on the circle coincides with the positive one !? And in general, it turns out that one and the same position of the moving side (or a point on the numerical circle) can be called both a negative angle and a positive one !?

Yes. Exactly. Let's say a positive angle of 90 degrees takes up on a circle exactly the same position as a negative angle of minus 270 degrees. A positive angle, for example + 110 ° degrees, takes exactly the same position as a negative angle of -250 °.

No problem. Anything is correct.) The choice of positive or negative calculus of the angle depends on the condition of the task. If the condition does not say anything in plain text about the sign of the angle, (like "determine the smallest positive angle ", etc.), then we work with values ​​convenient for us.

The exception (and how without them ?!) are trigonometric inequalities, but there we will master this trick.

Now a question for you. How did I know that the 110 ° angle position is the same as the -250 ° angle position?
I will hint that this is due to the full turnover. 360 ° ... Not clear? Then draw a circle. We draw on paper ourselves. Marking the corner about 110 °. AND consider how much remains until the full turnover. It will remain just 250 ° ...

Got it? And now - attention! If the angles 110 ° and -250 ° are on the circle same position, then what? Yes, that at the angles 110 ° and -250 ° exactly the same sine, cosine, tangent and cotangent!
Those. sin110 ° = sin (-250 °), ctg110 ° = ctg (-250 °), and so on. This is already really important! And in itself - there are a lot of tasks where you need to simplify expressions, and as a basis for the subsequent development of reduction formulas and other wisdom of trigonometry.

Obviously, I took 110 ° and -250 ° at random, purely for an example. All these equalities work for any angles that occupy the same position on the circle. 60 ° and -300 °, -75 ° and 285 °, and so on. I note right away that the corners in these pairs - various. But their trigonometric functions - the same.

I think you understand what negative angles are. It’s quite simple. Counterclockwise - positive count. On the way - negative. Consider an angle positive or negative depends on us... From our desire. Well, and also from the task, of course ... I hope you understand how to switch from negative angles to positive angles in trigonometric functions and vice versa. Draw a circle, an approximate angle, and see how much is missing to a full turn, i.e. up to 360 °.

Angles greater than 360 °.

Let's tackle angles that are greater than 360 °. And there are such? There are, of course. How to draw them on a circle? No problem! Let's say we need to figure out which quarter the 1000 ° angle will fall into? Easily! We make one full turn counterclockwise (the angle was given to us a positive one!). Unwound 360 °. Well, let's move on! Another turn - already got 720 °. How much is left? 280 °. Not enough for a full revolution ... But the angle is more than 270 ° - and this is the border between the third and fourth quarters. So our angle of 1000 ° falls into the fourth quarter. Everything.

As you can see, it is quite simple. Let me remind you once again that an angle of 1000 ° and an angle of 280 °, which we obtained by discarding "extra" full revolutions, are, strictly speaking, various corners. But the trigonometric functions at these angles exactly the same! Those. sin1000 ° = sin280 °, cos1000 ° = cos280 °, etc. If I were a sine, I would not notice the difference between these two angles ...

Why do you need all this? Why do we need to translate angles from one to another? Yes, all for the same.) In order to simplify expressions. Simplification of expressions is, in fact, the main task of school mathematics. Well, along the way, the head is training.)

Well, let's practice?)

We answer the questions. Simple at first.

1. In which quarter does the angle -325 ° fall?

2. Which quarter does the angle 3000 ° fall into?

3. In which quarter does the angle -3000 ° fall?

There is a problem? Or insecurity? We go to Section 555, Practical work with the trigonometric circle. There, in the first lesson of this very "Practical work ..." everything is detailed ... such questions of uncertainty to be shouldn't!

4. What sign does sin555 ° have?

5. What is the sign of tg555 °?

Have you identified? Fine! Doubt? It should be in Section 555 ... By the way, there you will learn how to draw tangent and cotangent on the trigonometric circle. A very useful thing.

And now the questions are wiser.

6. Reduce the expression sin777 ° to the sine of the smallest positive angle.

7. Reduce the expression cos777 ° to the cosine of the largest negative angle.

8. Reduce the expression cos (-777 °) to the cosine of the smallest positive angle.

9. Reduce the expression sin777 ° to the sine of the largest negative angle.

Are you puzzled by questions 6-9? Get used to it, on the exam, and not such formulations are found ... So be it, I will translate. Only for you!

The words "cast an expression to ..." mean to transform an expression so that its meaning hasn't changed and the appearance has changed in accordance with the assignment. So, in tasks 6 and 9 we should get a sine, inside which is smallest positive angle. Everything else does not matter.

I will give the answers in order (in violation of our rules). But what to do, there are only two signs, and only four quarters ... You will not run away in variants.

6.sin57 °.

7.cos (-57 °).

8.cos57 °.

9.-sin (-57 °)

I suppose the answers to questions 6-9 confused some. Especially -sin (-57 °), right?) Indeed, in the elementary rules for counting angles there is room for mistakes ... That is why I had to make a lesson: "How to determine the signs of functions and bring angles on a trigonometric circle?" Section 555. There tasks 4 - 9 are sorted out. Well disassembled, with all the pitfalls. And they are here.)

In the next lesson we will deal with the mysterious radians and pi numbers. Let's learn how to easily and correctly convert degrees to radians and vice versa. And we are surprised to find that this elementary information on the site already enough to solve some non-standard trigonometry problems!

If you like this site ...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Instant validation testing. Learning - with interest!)

you can get acquainted with functions and derivatives.