Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, it can be thought of as a rectangle, with one side representing lettuce and the other side representing water. The sum of these two sides will indicate borscht. The diagonal and area of such a “borscht” rectangle are purely mathematical concepts and are never used in borscht recipes.
How do lettuce and water turn into borscht from a mathematical point of view? How can the sum of two line segments become trigonometry? To understand this, we need linear angular functions.
You won't find anything about linear angular functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.
Linear angular functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.
Is it possible to do without linear angular functions? It’s possible, because mathematicians still manage without them. The trick of mathematicians is that they always tell us only about those problems that they themselves know how to solve, and never talk about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. All. We don’t know other problems and we don’t know how to solve them. What should we do if we only know the result of the addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angular functions. Next, we ourselves choose what one term can be, and linear angular functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. In everyday life, we get along just fine without decomposing the sum; subtraction is enough for us. But in scientific research into the laws of nature, decomposing a sum into its components can be very useful.
Another law of addition that mathematicians don't like to talk about (another of their tricks) requires that the terms have the same units of measurement. For salad, water, and borscht, these could be units of weight, volume, value, or unit of measure.
The figure shows two levels of difference for mathematical . The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the field of units of measurement, which are shown in square brackets and indicated by the letter U. This is what physicists do. We can understand the third level - differences in the area of the objects being described. Different objects can have the same number of identical units of measurement. How important this is, we can see in the example of borscht trigonometry. If we add subscripts to the same unit designation for different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or due to our actions. Letter W I will designate water with a letter S I'll designate the salad with a letter B- borsch. This is what linear angular functions for borscht will look like.
If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What were we taught to do then? We were taught to separate units of measurement from numbers and add numbers. Yes, any one number can be added to any other number. This is a direct path to the autism of modern mathematics - we do it incomprehensibly what, incomprehensibly why, and very poorly understand how this relates to reality, because of the three levels of difference, mathematicians operate with only one. It would be more correct to learn how to move from one unit of measurement to another.
Bunnies, ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar problem for adults. What do you get when you add bunnies and money? There are two possible solutions here.
First option. We determine the market value of the bunnies and add it to the available amount of money. We got the total value of our wealth in monetary terms.
Second option. You can add the number of bunnies to the number of banknotes we have. We will receive the amount of movable property in pieces.
As you can see, the same addition law allows you to get different results. It all depends on what exactly we want to know.
But let's get back to our borscht. Now we can see what will happen for different angle values of linear angular functions.
The angle is zero. We have salad, but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. There can be zero borscht with zero salad (right angle).
For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This happens because addition itself is impossible if there is only one term and the second term is missing. You can feel about this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so throw away your logic and stupidly cram the definitions invented by mathematicians: “division by zero is impossible”, “any number multiplied by zero equals zero” , “beyond the puncture point zero” and other nonsense. It is enough to remember once that zero is not a number, and you will never again have a question whether zero is a natural number or not, because such a question loses all meaning: how can something that is not a number be considered a number? It's like asking what color an invisible color should be classified as. Adding a zero to a number is the same as painting with paint that is not there. We waved a dry brush and told everyone that “we painted.” But I digress a little.
The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but not enough water. As a result, we will get thick borscht.
The angle is forty-five degrees. We have equal quantities of water and salad. This is the perfect borscht (forgive me, chefs, it's just math).
The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You will get liquid borscht.
Right angle. We have water. All that remains of the salad are memories, as we continue to measure the angle from the line that once marked the salad. We can't cook borscht. The amount of borscht is zero. In this case, hold on and drink water while you have it)))
Here. Something like this. I can tell other stories here that would be more than appropriate here.
Two friends had their shares in a common business. After killing one of them, everything went to the other.
The emergence of mathematics on our planet.
All these stories are told in the language of mathematics using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, let's return to borscht trigonometry and consider projections.
Saturday, October 26, 2019
I watched an interesting video about Grundy series One minus one plus one minus one - Numberphile. Mathematicians lie. They did not perform an equality check during their reasoning.
This echoes my thoughts about .
Let's take a closer look at the signs that mathematicians are deceiving us. At the very beginning of the argument, mathematicians say that the sum of a sequence DEPENDS on whether it has an even number of elements or not. This is an OBJECTIVELY ESTABLISHED FACT. What happens next?
Next, mathematicians subtract the sequence from unity. What does this lead to? This leads to a change in the number of elements of the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we added one element equal to one to the sequence. Despite all the external similarity, the sequence before the transformation is not equal to the sequence after the transformation. Even if we are talking about an infinite sequence, we must remember that an infinite sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.
By putting an equal sign between two sequences with different numbers of elements, mathematicians claim that the sum of the sequence DOES NOT DEPEND on the number of elements in the sequence, which contradicts an OBJECTIVELY ESTABLISHED FACT. Further reasoning about the sum of an infinite sequence is false, since it is based on a false equality.
If you see that mathematicians, in the course of proofs, place brackets, rearrange elements of a mathematical expression, add or remove something, be very careful, most likely they are trying to deceive you. Like card magicians, mathematicians use various manipulations of expression to distract your attention in order to ultimately give you a false result. If you cannot repeat a card trick without knowing the secret of deception, then in mathematics everything is much simpler: you don’t even suspect anything about deception, but repeating all the manipulations with a mathematical expression allows you to convince others of the correctness of the result obtained, just like when -they convinced you.
Question from the audience: Is infinity (as the number of elements in the sequence S) even or odd? How can you change the parity of something that has no parity?
Infinity is for mathematicians, like the Kingdom of Heaven is for priests - no one has ever been there, but everyone knows exactly how everything works there))) I agree, after death you will be absolutely indifferent whether you lived an even or odd number of days, but... Adding just one day into the beginning of your life, we will get a completely different person: his last name, first name and patronymic are exactly the same, only the date of birth is completely different - he was born one day before you.
Now let’s get to the point))) Let’s say that a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must lose parity. We don't see this. The fact that we cannot say for sure whether an infinite sequence has an even or odd number of elements does not mean that parity has disappeared. Parity, if it exists, cannot disappear without a trace into infinity, like in a sharpie’s sleeve. There is a very good analogy for this case.
Have you ever asked the cuckoo sitting in the clock in which direction the clock hand rotates? For her, the arrow rotates in the opposite direction to what we call “clockwise”. As paradoxical as it may sound, the direction of rotation depends solely on which side we observe the rotation from. And so, we have one wheel that rotates. We cannot say in which direction the rotation occurs, since we can observe it both from one side of the plane of rotation and from the other. We can only testify to the fact that there is rotation. Complete analogy with the parity of an infinite sequence S.
Now let's add a second rotating wheel, the plane of rotation of which is parallel to the plane of rotation of the first rotating wheel. We still can't say for sure in which direction these wheels rotate, but we can absolutely tell whether both wheels rotate in the same direction or in the opposite direction. Comparing two infinite sequences S And 1-S, I showed with the help of mathematics that these sequences have different parities and putting an equal sign between them is a mistake. Personally, I trust mathematics, I don’t trust mathematicians))) By the way, to fully understand the geometry of transformations of infinite sequences, it is necessary to introduce the concept "simultaneity". This will need to be drawn.
Wednesday, August 7, 2019
Concluding the conversation about, we need to consider an infinite set. The point is that the concept of “infinity” affects mathematicians like a boa constrictor affects a rabbit. The trembling horror of infinity deprives mathematicians of common sense. Here's an example:
The original source is located. 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 the infinite set of natural numbers as an example, then the considered examples can be represented in the following form:
To clearly prove that they were right, mathematicians came up with many different methods. Personally, I look at all these methods as shamans dancing with tambourines. Essentially, they all boil down to the fact that either some of the rooms are unoccupied 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 fantasy 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 time. Of course, the time factor can be stupidly ignored, but this will be in the category of “no law is written for fools.” It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.
What is an “endless hotel”? An infinite hotel is a hotel that always has any number of empty beds, regardless of how many rooms are occupied. If all the rooms in the endless "visitor" corridor are occupied, there is another endless corridor with "guest" rooms. There will be an infinite 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 are not able to distance themselves from banal everyday problems: there is always only one God-Allah-Buddha, there is only one hotel, there is only one corridor. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to “shove in the impossible.”
I will demonstrate the logic of my reasoning to you using 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; numbers do not exist in Nature. Yes, Nature is great at counting, but for this she uses other mathematical tools that are not familiar to us. I’ll tell you what Nature thinks another time. Since we invented numbers, we ourselves will decide how many sets of natural numbers there are. Let's consider both options, as befits real scientists.
Option one. “Let us be given” one 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 nowhere to take them. We cannot add one to this set, since we already have it. What 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 one from the shelf and add it to what we have left. As a result, we will again get an infinite set of natural numbers. You can write down all our manipulations like this:
I wrote down the actions in algebraic notation and in set theory notation, with a detailed listing 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 is subtracted from it and the same unit is added.
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. Let's 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. This is what we get:
The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If you add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.
The set of natural numbers is used for counting in the same way as a ruler is for measuring. Now imagine that you added one centimeter to the ruler. This will be a different line, not equal to the original one.
You can accept or not accept my reasoning - it is your own business. But if you ever encounter mathematical problems, think about whether you are following the path of false reasoning trodden by generations of mathematicians. After all, studying mathematics, first of all, forms a stable stereotype of thinking in us, and only then adds to our mental abilities (or, conversely, deprives us of free-thinking).
pozg.ru
Sunday, August 4, 2019
I was finishing a postscript to an article about and saw this wonderful text on Wikipedia:
We read: "... the rich theoretical basis of the mathematics of Babylon 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 difficult 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 in nature and is reduced to a set of disparate sections, devoid of a common system and evidence base.
I won’t 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 mistakes of modern mathematics. See you soon.
Saturday, August 3, 2019
How to divide a set into subsets? To do this, you need to enter a new unit of measurement that is present in some of the elements of the selected set. Let's look at an example.
May we have plenty A consisting of four people. This set is formed on the basis of “people.” Let us denote the elements of this set by the letter A, the subscript with a number will indicate the serial number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A based on gender b. Notice that our set of “people” has now become a set of “people with gender characteristics.” After this we can divide the sexual characteristics into male bm and women's bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, no matter which one - 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 use regular school mathematics. Look what happened.
After multiplication, reduction and rearrangement, we ended up with two subsets: the subset of men Bm and a subset of women Bw. Mathematicians reason in approximately the same way when they apply set theory in practice. But they don’t tell us the details, but give us the finished result - “a lot of people consist of a subset of men and a subset of women.” Naturally, you may have a question: how correctly has the mathematics been applied in the transformations outlined above? I dare to assure you that essentially everything was 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 will tell you about this.
As for supersets, you can combine two sets into one superset by selecting the unit of measurement present in the elements of these two sets.
As you can see, units of measurement and ordinary mathematics make set theory a relic of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. Mathematicians acted as shamans once did. Only shamans know how to “correctly” apply their “knowledge.” They teach us this “knowledge”.
In conclusion, I want to show you how mathematicians manipulate
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, 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 catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. 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 of Zeno tells about a flying arrow:
A 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 a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one 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 whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (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.
I'll show you the process with an example. We select the “red solid in a pimple” - this is our “whole”. At the same time, we see that these things are with a bow, and there are without a bow. After that, we select part of the “whole” and form a set “with a bow”. This is how shamans get their food by tying their set theory to reality.
Now let's do a little trick. Let’s take “solid with a pimple with a bow” and combine these “wholes” according to color, selecting the red elements. We got a lot of "red". Now the final question: are the resulting sets “with a bow” and “red” the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so it will be.
This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid with a pimple and a bow." The formation took place in four different units of measurement: color (red), strength (solid), roughness (pimply), decoration (with a bow). Only a set of units of measurement allows us 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. The units of measurement by which the “whole” is distinguished at the preliminary stage are highlighted in brackets. The unit of measurement by which the set is formed is taken out of brackets. The last line shows the final result - an 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, and not the dancing of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing that it is “obvious,” because units of measurement are not part of their “scientific” arsenal.
Using units of measurement, it is very easy to split one set or combine several sets into one superset. Let's take a closer look at the algebra of this process.
The trigonometric circle is one of the basic elements of geometry for solving equations with sine, cosine, tangent and cotangent.
What is the definition of this term, how to build this circle, how to determine a quarter in trigonometry, how to find out the angles in a constructed trigonometric circle - we will talk about this and much more further.
Trigonometric circle
The trigonometric form of a number circle in mathematics is a circle having a single radius with a center at the origin of the coordinate plane. As a rule, it is formed by a space of formulas for sine with cosine, tangent and cotangent on a coordinate system.
The purpose of such a sphere with n-dimensional space is that thanks to it trigonometric functions can be described. It looks simple: a circle, inside of which there is a coordinate system and multiple right-angled triangles formed from this circle using trigonometric functions.
What is sine, cosine, tangent, cotangent in a right triangle
A right-angled triangle is one in which one of the angles is 90°. It is formed by the legs and hypotenuse with all the meanings of trigonometry. The legs are two sides of the triangle that are adjacent to the 90° angle, and the third is the hypotenuse, it is always longer than the legs.
The sine is the ratio of one of the legs to the hypotenuse, the cosine is the ratio of the other leg to it, and the tangent is the ratio of two legs. Relationship symbolizes division. Tangent is also the division of an acute angle by sine and cosine. A cotangent is the opposite ratio of a tangent.
The formulas for the last two ratios are as follows: tg(a) = sin(a) / cos(a) and ctg(a) = cos(a) / sin(a).
Constructing a unit circle
The construction of a unit circle comes down to drawing it with a unit radius at the center of the coordinate system. Then, to construct, you need to count the angles and, moving counterclockwise, go around the whole circle, putting down the coordinate lines corresponding to them.
The construction begins after drawing a circle and setting a point in its center by placing the OX coordinate system. Point O on top of the coordinate axis is the sine, and X is the cosine. Accordingly, they are the abscissa and ordinate. Then you need to take measurements ∠. They are carried out in degrees and radians.
It is easy to translate these indicators - a full circle is equal to two pi radians. The angle from zero counterclockwise comes with a + sign, and ∠ from 0 clockwise comes with a - sign. Positive and negative values of sine and cosine are repeated every revolution of the circle.
Angles on a trigonometric circle
In order to master the theory of the trigonometric circle, you need to understand how ∠ are counted on it and in what way they are measured. They are calculated very simply.
The circle is divided by the coordinate system into four parts. Each part forms ∠ 90°. Half of these angles are 45 degrees. Accordingly, two parts of a circle are equal to 180°, and three parts are 360°. How to use this information?
If it is necessary to solve the problem of finding ∠, they resort to theorems about triangles and the basic Pythagorean laws associated with them.
Angles are measured in radians:
- from 0 to 90° — angle values from 0 to ∏/2;
- from 90 to 180° — angle values from ∏/2 to ∏;
- from 180 to 270° - from ∏ to 3*∏/2;
- last quarter from 270 0 to 360 0 - values from 3*∏/2 to 2*∏.
To find out a specific measurement, convert radians to degrees or vice versa, you should resort to a cheat sheet.
Converting angles from degrees to radians
Angles can be measured in degrees or radians. It is required to be aware of the connection between both meanings. This relationship is expressed in trigonometry using a special formula. By understanding the relationship, you can learn how to quickly control angles and move from degrees to radians back.
In order to find out exactly what one radian is equal to, you can use the following formula:
1 rad. = 180 / ∏ = 180 / 3.1416 = 57.2956
Ultimately, 1 radian is equal to 57°, and there are 0.0175 radians in 1 degree:
1 degree = (∏ /180) rad. = 3.1416 / 180 rad. = 0.0175 rad.
Cosine, sine, tangent, cotangent on a trigonometric circle
Cosine with sine, tangent and cotangent on a trigonometric circle - functions of alpha angles from 0 to 360 degrees. Each function has a positive or negative value depending on the magnitude of the angle. They symbolize the relationship to right triangles formed in a circle.
The sign of the trigonometric function depends solely on the coordinate quadrant in which the numerical argument is located. Last time we learned to convert arguments from a radian measure to a degree measure (see lesson “ Radian and degree measure of an angle”), and then determine this same coordinate quarter. Now let's actually determine the sign of sine, cosine and tangent.
The sine of angle α is the ordinate (y coordinate) of a point on a trigonometric circle that occurs when the radius is rotated by angle α.
The cosine of angle α is the abscissa (x coordinate) of a point on a trigonometric circle, which occurs when the radius is rotated by angle α.
The tangent of the angle α is the ratio of sine to cosine. Or, which is the same thing, the ratio of the y coordinate to the x coordinate.
Notation: sin α = y ; cos α = x ; tg α = y : x .
All these definitions are familiar to you from high school algebra. However, we are not interested in the definitions themselves, but in the consequences that arise on the trigonometric circle. Take a look:
Blue color indicates the positive direction of the OY axis (ordinate axis), red indicates the positive direction of the OX axis (abscissa axis). On this "radar" the signs of trigonometric functions become obvious. In particular:
- sin α > 0 if angle α lies in the I or II coordinate quadrant. This is because, by definition, sine is an ordinate (y coordinate). And the y coordinate will be positive precisely in the I and II coordinate quarters;
- cos α > 0, if angle α lies in the 1st or 4th coordinate quadrant. Because only there the x coordinate (aka abscissa) will be greater than zero;
- tan α > 0 if angle α lies in the I or III coordinate quadrant. This follows from the definition: after all, tan α = y : x, therefore it is positive only where the signs of x and y coincide. This happens in the first coordinate quarter (here x > 0, y > 0) and the third coordinate quarter (x< 0, y < 0).
For clarity, let us note the signs of each trigonometric function - sine, cosine and tangent - on separate “radars”. We get the following picture:
Please note: in my discussions I never spoke about the fourth trigonometric function - cotangent. The fact is that the cotangent signs coincide with the tangent signs - there are no special rules there.
Now I propose to consider examples similar to problems B11 from the trial Unified State Exam in mathematics, which took place on September 27, 2011. After all, the best way to understand theory is practice. It is advisable to have a lot of practice. Of course, the conditions of the tasks were slightly changed.
Task. Determine the signs of trigonometric functions and expressions (the values of the functions themselves do not need to be calculated):
- sin(3π/4);
- cos(7π/6);
- tg(5π/3);
- sin (3π/4) cos (5π/6);
- cos (2π/3) tg (π/4);
- sin (5π/6) cos (7π/4);
- tan (3π/4) cos (5π/3);
- ctg (4π/3) tg (π/6).
The action plan is this: first we convert all angles from radian measures to degrees (π → 180°), and then look at which coordinate quarter the resulting number lies in. Knowing the quarters, we can easily find the signs - according to the rules just described. We have:
- sin (3π/4) = sin (3 · 180°/4) = sin 135°. Since 135° ∈ , this is an angle from the II coordinate quadrant. But the sine in the second quarter is positive, so sin (3π/4) > 0;
- cos (7π/6) = cos (7 · 180°/6) = cos 210°. Because 210° ∈ , this is the angle from the third coordinate quadrant, in which all cosines are negative. Therefore cos(7π/6)< 0;
- tg (5π/3) = tg (5 · 180°/3) = tg 300°. Since 300° ∈ , we are in the IV quarter, where the tangent takes negative values. Therefore tan (5π/3)< 0;
- sin (3π/4) cos (5π/6) = sin (3 180°/4) cos (5 180°/6) = sin 135° cos 150°. Let's deal with the sine: because 135° ∈ , this is the second quarter in which the sines are positive, i.e. sin (3π/4) > 0. Now we work with cosine: 150° ∈ - again the second quarter, the cosines there are negative. Therefore cos(5π/6)< 0. Наконец, следуя правилу «плюс на минус дает знак минус», получаем: sin (3π/4) · cos (5π/6) < 0;
- cos (2π/3) tg (π/4) = cos (2 180°/3) tg (180°/4) = cos 120° tg 45°. We look at the cosine: 120° ∈ is the II coordinate quarter, so cos (2π/3)< 0. Смотрим на тангенс: 45° ∈ — это I четверть (самый обычный угол в тригонометрии). Тангенс там положителен, поэтому tg (π/4) >0. Again we got a product in which the factors have different signs. Since “minus by plus gives minus”, we have: cos (2π/3) tg (π/4)< 0;
- sin (5π/6) cos (7π/4) = sin (5 180°/6) cos (7 180°/4) = sin 150° cos 315°. We work with the sine: since 150° ∈ , we are talking about the II coordinate quarter, where the sines are positive. Therefore, sin (5π/6) > 0. Similarly, 315° ∈ is the IV coordinate quarter, the cosines there are positive. Therefore cos (7π/4) > 0. We have obtained the product of two positive numbers - such an expression is always positive. We conclude: sin (5π/6) cos (7π/4) > 0;
- tg (3π/4) cos (5π/3) = tg (3 180°/4) cos (5 180°/3) = tg 135° cos 300°. But the angle 135° ∈ is the second quarter, i.e. tg(3π/4)< 0. Аналогично, угол 300° ∈ — это IV четверть, т.е. cos (5π/3) >0. Since “minus by plus gives a minus sign,” we have: tg (3π/4) cos (5π/3)< 0;
- ctg (4π/3) tg (π/6) = ctg (4 180°/3) tg (180°/6) = ctg 240° tg 30°. We look at the cotangent argument: 240° ∈ is the III coordinate quarter, therefore ctg (4π/3) > 0. Similarly, for the tangent we have: 30° ∈ is the I coordinate quarter, i.e. the simplest angle. Therefore tan (π/6) > 0. Again we have two positive expressions - their product will also be positive. Therefore cot (4π/3) tg (π/6) > 0.
Finally, let's look at some more complex problems. In addition to figuring out the sign of the trigonometric function, you will have to do a little math here - exactly as it is done in real problems B11. In principle, these are almost real problems that actually appear in the Unified State Examination in mathematics.
Task. Find sin α if sin 2 α = 0.64 and α ∈ [π/2; π].
Since sin 2 α = 0.64, we have: sin α = ±0.8. All that remains is to decide: plus or minus? By condition, angle α ∈ [π/2; π] is the II coordinate quarter, where all sines are positive. Therefore, sin α = 0.8 - the uncertainty with signs is eliminated.
Task. Find cos α if cos 2 α = 0.04 and α ∈ [π; 3π/2].
We act similarly, i.e. take the square root: cos 2 α = 0.04 ⇒ cos α = ±0.2. By condition, angle α ∈ [π; 3π/2], i.e. We are talking about the third coordinate quarter. All cosines there are negative, so cos α = −0.2.
Task. Find sin α if sin 2 α = 0.25 and α ∈ .
We have: sin 2 α = 0.25 ⇒ sin α = ±0.5. We look at the angle again: α ∈ is the IV coordinate quarter, in which, as we know, the sine will be negative. Thus, we conclude: sin α = −0.5.
Task. Find tan α if tan 2 α = 9 and α ∈ .
Everything is the same, only for the tangent. Extract the square root: tan 2 α = 9 ⇒ tan α = ±3. But according to the condition, the angle α ∈ is the I coordinate quarter. All trigonometric functions, incl. tangent, there are positive, so tan α = 3. That's it!
Trigonometric circle. Unit circle. Number circle. What it is?
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
Very often terms trigonometric circle, unit circle, number circle poorly understood by students. And completely in vain. These concepts are a powerful and universal assistant in all areas of trigonometry. In fact, this is a legal cheat sheet! I drew a trigonometric circle and immediately saw the answers! Tempting? So let's learn, it would be a sin not to use such a thing. Moreover, it is not at all difficult.
To successfully work with the trigonometric circle, you need to know only three things.
If you like this site...
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You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Lesson type: systematization of knowledge and intermediate control.
Equipment: trigonometric circle, tests, task cards.
Lesson objectives: systematize the studied theoretical material according to the definitions of sine, cosine, tangent of an angle; check the degree of knowledge acquisition on this topic and application in practice.
Tasks:
- Generalize and consolidate the concepts of sine, cosine and tangent of an angle.
- Form a comprehensive understanding of trigonometric functions.
- To promote students’ desire and need to study trigonometric material; cultivate a culture of communication, the ability to work in groups and the need for self-education.
“Whoever does and thinks for himself from a young age,
Then it becomes more reliable, stronger, smarter.
(V. Shukshin)
DURING THE CLASSES
I. Organizational moment
The class is represented by three groups. Each group has a consultant.
The teacher announces the topic, goals and objectives of the lesson.
II. Updating knowledge (frontal work with the class)
1) Work in groups on tasks:
1. Formulate the definition of sin angle.
– What signs does sin α have in each coordinate quadrant?
– At what values does the expression sin α make sense, and what values can it take?
2. The second group is the same questions for cos α.
3. The third group prepares answers to the same questions tg α and ctg α.
At this time, three students work independently at the board using cards (representatives of different groups).
Card No. 1.
Practical work.
Using the unit circle, calculate the values of sin α, cos α and tan α for angles of 50, 210 and – 210.
Card No. 2.
Determine the sign of the expression: tg 275; cos 370; sin 790; tg 4.1 and sin 2.
Card number 3.
1) Calculate:
2) Compare: cos 60 and cos 2 30 – sin 2 30
2) Orally:
a) A series of numbers is proposed: 1; 1.2; 3; , 0, , – 1. Among them there are redundant ones. What property of sin α or cos α can these numbers express (Can sin α or cos α take these values).
b) Does the expression make sense: cos (–); sin 2; tg 3: ctg (– 5); ; ctg0;
cotg(–π). Why?
c) Is there a minimum and maximum value of sin or cos, tg, ctg.
d) Is it true?
1) α = 1000 is the angle of the second quarter;
2) α = – 330 is the angle of the IV quarter.
e) The numbers correspond to the same point on the unit circle.
3) Work at the board
No. 567 (2; 4) – Find the value of the expression
No. 583 (1-3) Determine the sign of the expression
Homework: table in notebook. No. 567(1, 3) No. 578
III. Acquiring additional knowledge. Trigonometry in the palm of your hand
Teacher: It turns out that the values of the sines and cosines of angles are “located” in the palm of your hand. Reach out your hand (either hand) and spread your fingers as far apart as possible (as in the poster). One student is invited. We measure the angles between our fingers.
Take a triangle where there is an angle of 30, 45 and 60 90 and apply the vertex of the angle to the hillock of the Moon in the palm of your hand. The Mount of the Moon is located at the intersection of the extensions of the little finger and thumb. We combine one side with the little finger, and the other side with one of the other fingers.
It turns out that there is an angle of 90 between the little finger and the thumb, 30 between the little and ring fingers, 45 between the little and middle fingers, and 60 between the little and index fingers. And this is true for all people without exception.
little finger No. 0 – corresponds to 0,
unnamed No. 1 – corresponds to 30,
average No. 2 – corresponds to 45,
index number 3 – corresponds to 60,
large No. 4 – corresponds to 90.
Thus, we have 4 fingers on our hand and remember the formula:
Finger no. |
Corner |
Meaning |
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This is just a mnemonic rule. In general, the value of sin α or cos α must be known by heart, but sometimes this rule will help in difficult times.
Come up with a rule for cos (angles do not change, but are counted from the thumb). A physical pause associated with the signs sin α or cos α.
IV. Checking your knowledge of knowledge and skills
Independent work with feedback
Each student receives a test (4 options) and the answer sheet is the same for everyone.
Test
Option 1
1) At what angle of rotation will the radius take the same position as when turning through an angle of 50?
2) Find the value of the expression: 4cos 60 – 3sin 90.
3) Which number is less than zero: sin 140, cos 140, sin 50, tg 50.
Option 2
1) At what angle of rotation will the radius take the same position as when turning by an angle of 10.
2) Find the value of the expression: 4cos 90 – 6sin 30.
3) Which number is greater than zero: sin 340, cos 340, sin 240, tg (– 240).
Option 3
1) Find the value of the expression: 2ctg 45 – 3cos 90.
2) Which number is less than zero: sin 40, cos (– 10), tan 210, sin 140.
3) Which quarter angle is angle α, if sin α > 0, cos α< 0.
Option 4
1) Find the value of the expression: tg 60 – 6ctg 90.
2) Which number is less than zero: sin(– 10), cos 140, tg 250, cos 250.
3) Which quadrant angle is angle α, if ctg α< 0, cos α> 0.
A |
B |
IN |
G |
D |
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AND |
Z |
AND |
L |
M |
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ABOUT |
P |
R |
WITH |
T |
U |
F |
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Sh |
YU |
I |
(the key word is trigonometry)
V. Information from the history of trigonometry
Teacher: Trigonometry is a fairly important branch of mathematics for human life. The modern form of trigonometry was given by the greatest mathematician of the 18th century, Leonhard Euler, a Swiss by birth who worked in Russia for many years and was a member of the St. Petersburg Academy of Sciences. He introduced well-known definitions of trigonometric functions, formulated and proved well-known formulas, we will learn them later. Euler’s life is very interesting and I advise you to get acquainted with it through Yakovlev’s book “Leonard Euler”.
(Message from the guys on this topic)
VI. Summing up the lesson
Game "Tic Tac Toe"
The two most active students are participating. They are supported by groups. The solutions to the tasks are written down in a notebook.
Tasks
1) Find the error
a) sin 225 = – 1.1 c) sin 115< О
b) cos 1000 = 2 d) cos (– 115) > 0
2) Express the angle in degrees
3) Express the angle 300 in radians
4) What is the largest and smallest value the expression can have: 1+ sin α;
5) Determine the sign of the expression: sin 260, cos 300.
6) In which quarter of the number circle is the point located?
7) Determine the signs of the expression: cos 0.3π, sin 195, ctg 1, tg 390
8) Calculate:
9) Compare: sin 2 and sin 350
VII. Lesson reflection
Teacher: Where can we meet trigonometry?
In what lessons in 9th grade, and even now, do you use the concepts of sin α, cos α; tg α; ctg α and for what purpose?