Trigonometry and division by zero

Why was I so surprised by the result of dividing zero by zero when transforming the formula for the area of a trapezoid? In this article I will try to explain to you.

I have been studying trigonometry for a long time. The trigonometric transformations that I will show you now are for me somewhere at the foundation of mathematics. In those lost pages of mathematics textbooks that preceded natural numbers. Or that we have yet to write. A question for all of you: without understanding what could the very concept of counting not have arisen?

Trigonometry

This is a small lyrical digression for those mathematicians who have long forgotten trigonometry. What self-respecting mathematician would engage in such a long-known nonsense as trigonometry? I remind them especially for them:

Trigonometry formulas

Schoolchildren know these trigonometric formulas well. They have been tested by practical application for hundreds of years. Many generations of mathematicians have applied these trigonometric transformations without any problems. I have no reason to doubt the correctness of these trigonometric formulas.

But these formulas contain one small problem, the solution of which has proven beyond the capabilities of all generations of mathematicians. This problem is called "division by zero". Now we will see how it looks in trigonometric formulas.

Division by zero

Now we will substitute the values of trigonometric functions for different angle values into the given trigonometric formulas. No special problems arise in this case, except for angles equal to 0 and 90 degrees. Let's see what happens:

Trigonometry and division by zero

Trigonometric formulas clearly show that:

1. Dividing zero by zero results in one.

2. Dividing a number (one) by zero is possible.

Statement number 1 has been known to me for a long time. If the formulas given are algebra, then I have no problems with the geometric and physical meaning of dividing zero by zero. Perhaps I have already written about this. I don't remember. If I haven't, I will.

For a long time I didn't believe in the possibility of dividing a number by zero. I recently solved this problem. Now I need to format it nicely and publish it. Unfortunately, all my time is taken up with earning money for a living, and not with solving mathematical problems.

By the way, I have believed in the possibility of division by zero for a very long time. It was this belief that made me take up trigonometry. I did not know trigonometry either at school or at the institute. It is the understanding of trigonometry that allows you to find a solution to the problem of division by zero. It is enough to know what and where to look for. You already know what to look for. Where to look, I will soon show you along with the basic principles of division by zero.

The order of performing mathematical operations
and the result of division by zero

Now let's get back to the little problem that dividing zero by zero has different results in different formulas: in trigonometric formulas and in the formula for transforming the area of ​​a trapezoid. It is quite possible that the result depends not on the mathematical operations, but on the order in which the multiplication and division are performed.

Let me remind you that according to the rules of mathematics, multiplication and division are performed in the order in which they are written in a mathematical expression. Who writes down a mathematical expression? A mathematician. On what basis? Now that's an interesting question. Ideally, the order of mathematical operations should be determined by the physical process that the mathematical expression describes. In fact, the mathematician will write them down in whatever order he wants. But let's see what I mean:

Mathematical operations and division by zero

In my opinion, this option fits better into the mathematical rules. But the determining factor will be the result of practical application of these theories in the future. Why did I designate the result of dividing one by zero as "x"? Whatever it is, as a result of multiplying by zero we get zero.

Conclusions

I have neither moral nor legal right to make authoritative statements. I have neither authority nor mathematical education and I have nothing to do with mathematics. Mathematics is my hobby. But this is my main advantage. I can say whatever I consider necessary. No one will kick me out of school or work for this. Therefore, as a rational being, I will tell you the following.

Mathematicians are absolutely right when they say that division by zero is impossible. Within the framework of number theory, no one has been able to solve this problem and is unlikely to be able to. Set theory has nothing to do with mathematics at all. What other tools do mathematicians have to solve the problem of division by zero? Geometry? Or something else?

I believe that the main mistake of mathematicians is their global approach to solving problems. Mathematicians always generalize and expand everything. The correct scientific approach also implies the ability to separate and narrow. Without this, science turns into a mediocre religion. Modern mathematicians are not even able to count the terms with their fingers when rearranging the terms in a sum. So what if this sum is infinite? The laws of mathematics are always the same for everyone. I still have many questions for mathematicians like "Why exactly this way and not otherwise?" I hope I will have time to formulate them.

More interesting math ideas on the page "My Math"

Sine and cosine table

This is the most unusual table of sines and cosines. The idea to make such a table came to me in the process of writing these articles:

Why is the sine of 30 degrees equal to half one?

What is the sine of 60 degrees?

Sine and cosine of 45 degrees

I strongly recommend that you read these articles. Then you will know how easy it is to calculate the values of sine and cosine yourself. I made one sine and cosine table from these three articles.

Sine and cosine table

The table indicates which sides of the triangle to take as a unit. For 0 and 90 degrees, triangles do not exist, these are ordinary segments.

The third column of the table shows how to use the Pythagorean theorem to calculate the desired value of the sine or cosine. For a cosine of 30 degrees and a sine of 60 degrees, the calculations are the same.

At the bottom of the table there are hints on how to calculate the tangent and cotangent values if you know the sine and cosine values. If you forgot how to divide one fraction by another, use one more hint. You need to multiply the first fraction by the reciprocal of the second fraction.

Sine and cosine of 45 degrees

We have already considered what the sine of 30 degrees is and what the sine of 60 degrees is equal to. Now we will see how to find the sine and cosine of 45 degrees and why they are equal.

Last time we considered an equilateral triangle. Now we will consider an isosceles right triangle. A very rare beast in a herd of triangles. Mathematicians have known it for many thousands of years, and it is simply boring for mathematicians to tinker with it.

Isosceles right triangle

We will consider a tired isosceles right triangle. Why is the triangle tired? He lay down on his side to rest.

Isosceles right triangle

The legs of this triangle are equal to one. Once again I repeat that both legs have the same length. We do not know the length of the hypotenuse of this triangle, but we can easily calculate it using the Pythagorean theorem.

Sine and cosine of 45 degrees

Smart people came to us and said that the sine and cosine of an angle are equal to the ratio of the legs to the hypotenuse. Since the legs of a right triangle with an angle of 45 degrees are equal, then the value of the sine of 45 degrees is equal to the value of the cosine of 45 degrees. We take the math in hand and calculate this value.

Sine and cosine of 45 degrees

Why did I multiply the numerator and denominator of a fraction by the square root of two? Small children do not like lumps in porridge. Mathematicians don't like square roots in denominators. Very capricious uncles and aunts.

What is the sine of 60 degrees?

Before we look for the answer to the question: "What is the sine of 60 degrees?", I strongly recommend reading the post "Why is the sine of 30 degrees equal to half one?".

Height of an equilateral triangle

I again take an equilateral triangle with a side equal to one. I draw height.

Height of an equilateral triangle

After that, I will not turn anything. I'll just remove half of the drawing.

Cosine 60 degrees

I have a right-angled triangle, the diagonal of which is equal to one.

Cosine 60 degrees

The base of this triangle is equal to 1/2 and is also equal to the cosine of an angle of 60 degrees.

Sine 60 degrees

The height of this triangle is equal to the sine of the angle of 60 degrees. We calculate the height using the Pythagorean theorem.

Sine 60 degrees

The sine of 60 degrees is equal to the square root of three divided by two. It is calculated in the same way as the cosine of 30 degrees.

Why is the sine of 30 degrees equal to 1/2?

The answer to the question "why is the sine of 30 degrees equal to 1/2?" can be searched in the history of mathematics. These are ancient Mesopotamia, ancient Greece and other ancient civilizations. I am not an expert in this area. Obviously, knowledge has evolved from simpler to more complex. I will show you the most obvious answer to this question as I see it.

Equilateral triangle

Equilateral triangle

I have drawn an equilateral triangle with sides equal to one. I so want. The sum of the angles of a triangle is 180 degrees. An equilateral triangle has three 60 degree angles. I don't know trigonometry yet.

Height of an equilateral triangle

Height of an equilateral triangle

I drew the height in an equilateral triangle. The height is always perpendicular to the base of the triangle. If the height is drawn through the vertex of such a triangle, it will divide it into two equal right-angled triangles. This is always the case in isosceles triangles. An equilateral triangle is a special case of an isosceles triangle, in which the base is equal to the sides.

What happened as a result? The height divided the angle at the apex into two equal angles 60=30+30, it divided the base into two equal segments 1=(1/2)+(1/2). I still haven't heard anything about trigonometric functions.

Sine 30 degrees

After that, someone came up with trigonometric functions. I was told that the sine of an angle in a right triangle is the ratio of the opposite leg to the hypotenuse. How do I find the sine value for a 30 degree angle? I just flip the picture 90 degrees and remove all unnecessary.

Sine 30 degrees

The hypotenuse is equal to one. Any number divided by one does not change. So the length of the opposite leg in my triangle is equal to the sine of the angle of 30 degrees, that is, 1/2.

Cosine 30 degrees

The cosine of 30 degrees I can easily find from the Pythagorean theorem. We take the Pythagorean theorem in our hands and count.

Cosine 30 degrees

The cosine of 30 degrees turned out to be equal to the square root of three, divided by two.

Here's how easy it is to calculate. No tables needed.

Trigonometry value table

Trigonometry value table for children

In trigonometry value table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.

Trigonometry value table

Why is this table of trigonometry values for kids (and mathematicians)? Firstly, the values of trigonometric functions are not given in numerical form, but in the form of mathematical expressions using square roots. Secondly, in this table it is indicated that the tangent of 90 degrees is undefined (I specially highlighted this cell in yellow).

The value of the tangent of 90 degrees is determined very simply, but mathematicians, as well as small children, do not understand what this means.

Division by zero

By definition the tangent of 90 degrees is equal to the sine of 90 degrees divided by the cosine of 90 degrees. Let's take a look at the result.

Division by zero

The tangent of 90 degrees is equal to one divided by zero. That's what the math says. To understand division by zero, you need to know mathematics much better than we know it. If you firmly believe that zero is a number, you will never understand division by zero. Here you can find some of my publications on mathematics, which are somehow related to division by zero.

Division by zero can be found in various places in mathematics: the sine theorem, the equation of the straight line y=kx and so on. I suggest you explore these formulas on your own. Write about your results in the comments. If necessary, I will show you under what conditions division by zero appears there.

What is division by zero, I will definitely tell you in the future. But before that, there will be many more interesting things.

Trigonometry value table for engineers

For engineers, a trigonometry table of values should contain numbers so that calculations can be done quickly.

Trigonometry value table for engineers

Division by zero is not as important for engineers as it is for physicists. Mathematicians have learned to bypass problem areas in engineering calculations. But physicists in their theories can meet division by zero much more often. If you already want to know what is the result of division by zero in your theoretical research, write in the comments, I will try to help you.

Borsch trigonometry. Addition.

I will not tell you recipes for cooking borsch, I will talk about mathematics. What is borsch? These are vegetables cooked in water according to a specific recipe. I will consider two source components (vegetable salad and water) and the final result is borsch. Geometrically, this can be thought of as a rectangle in which one side is a salad, the other side is water. The sum of these two sides is borsch. The diagonal and area of such a rectangle are mathematical concepts and are never used in recipes.

Borsch trigonometry

How do salad and water turn into borsch? How does the sum turn into trigonometry? To understand this, linear angular functions will help us.

The linear angular functions

There are no linear angular functions in mathematics textbooks. But without them there can be no mathematics. Mathematical laws work whether we know them or not.

The linear angular functions are the laws of addition. This is how algebra turns into geometry, and geometry turns into trigonometry.

Trigonometry addition

The main law of addition, which mathematicians do not like to talk about, requires that the summands have the same units of measurements. For salad, water and borsch it can be units of weight, volume, value or others.

Law of addition

The figure shows two levels of difference. The first level is differences in the field of numbers. This is what mathematicians do. The second level is the difference in the area of units of measurement (shown in square brackets). Physicists are doing this. We can understand the third level - the differences in the field of described objects. Different objects can describe the same number of identical units of measurements. How important this is, the trigonometry of borscht will show us. Here is how linear angular functions for borsch will look.

The law of addition for borsch

Part of the water and part of the salad give one serving of borsch. Now we can see what will happen when the angle of linear angular functions changes.

The angle is zero

The angle is zero. We have a salad but no water. You can’t cook borsch. The amount of borsch is also zero. This does not mean that zero borsch is equal to zero water. Zero borsch can be at zero salad (right angle).

The angle is greater than zero, but less than the right angle

The angle is greater than zero, but less than forty-five degrees. We have a lot of salad, but not enough water. The result is a thick borsch.

The angle is forty-five degrees. We have an equal amount of water and lettuce. Perfect borsch (forgive me the cook, it's just math).

The angle is more than forty-five degrees, but less than ninety degrees. We have a lot of water and a little salad. Get liquid borsch.

Right angle

Right angle. We have water. From the salad there were only memories. We cannot cook borscht. The amount of borscht is zero. In this case, drink water.

Something like this. If you're tired of borscht, check out a few other stories.

Percentage.

Percentage

Cell division.

Cell division

Two friends had their shares in the general business. After killing one of them, everything went to another.

The general business

The emergence of mathematics on our planet.

The emergence of mathematics

All these stories in the language of mathematics are presented using linear angular functions. I will tell you separately about the real place of these functions in the structure of mathematics. In the meantime, we return to the trigonometry of borscht and consider the projections.

Inverse transformations

Last time we transformed the law of cosines and a Pythagorean theorem to the sum of a line segments. Now we will execute inverse transformations.

Inverse transformations

In inverse transformations I made everything very simply. The minus sign from a formula disappeared. The problem is that we are not able to measure angles correctly. Than differ a angle 0 degrees from a angle of 180 degrees?

Measurement of a angle

It is possible to assume that if a line segment one, then a angle is equal 0 degrees. If a line segments two, then a angle is equal 180 degrees.

The transformations executed by us show that the mathematics has no separate areas of mathematics: "arithmetic", "algebra", "geometry" or "trigonometry". The mathematics is a single whole.

The mathematics is DNA of the nature. Further we will continue to study a cosine gene in a cosine law.

If you liked the publication and you want to know more, help me with working on by other publications.

We use the law of cosines

I was always interested in a question: how the Pythagorean theorem turns into the sum of a line segments? What I speak about? Here you look.

The Pythagorean theorem and the sum of a line segments

In geometry everything is very prime. The first time we draw a right triangle and we write down a Pythagorean theorem. The second time we draw two a line segments and we write down the sum of a line segments.

Right triangle and two a line segments

How one formula turns into other formula? To see it, we use the law of cosines. We will draw the picture, we will write down conditions, we will execute transformations.

Triangle and the law of cosines

Right triangle and the Pythagorean theorem

The law of cosines and sum of two a line segments

We use the law of cosines and turned the Pythagorean theorem into the sum of two a line segments. Further we will consider an inverse transformation.

Transformations of trigonometric functions

Transformations of trigonometric functions

This table shows how one trigonometric functions can be transformed to other trigonometric functions. Sin, cos, tan, cot, sec, csc - all these functions can be transformed.

Trigonometrical circle of a tangent

Last time I drew for you a unit circle of cotangent. Now you can look at a trigonometrical circle of tangent.

Trigonometrical circle of a tangent

 There is no sense here. There are angles and values of tangents. If you want to understand sense of a tangent and cotangent, look here.

Trigonometric table

Trigonometric table are crazy

This trigonometrical table is an example of mathematical marasmus. I am touched by exact values of a sine and cosine. No comments.

If mathematicians do not know what is trigonometric functions, let read here. If mathematicians are not able to divide into zero, let study. I like the idea of this table. I do not like its contents. I corrected this trigonometrical table.

Trigonometric table

The most popular angles are highlighted with blue color. 0, 30, 45, 60, 90 degrees most often occur in textbooks.Common fractions will be useful to pupils to fight against teachers. Decimal fractions will be useful physics and to engineers to calculations.

In this trigonometrical table there are no cotangents (cot, cotan, cotg, ctg, ctn). Anything terrible. There are useful formulas which will help you.

Useful formulas

Sin 0, 15, 22.5, 30, 45, 60, 67.5, 75, 90, 120, 135, 150, 180 degrees in this table.
Cos 0, Pi/12, pi/8, pi/6, pi/4, pi/3, 3/8 pi, 5/12 pi, pi/2, 2/3 pi, 3/4 pi, 5/6 pi, 1 pi radians.
Tan pi/2 radians or 90 degrees it makes sense.

Cotangent 210 degrees

Today we will consider a cotangent of 210 degrees. We will draw a unit disk and we will write values of cotangents. Here what turned out.

The unit circle and cotangent

Value of a cotangent of 210 degrees same, as at a cotangent of 30 degrees. We take mathematics in hand and we check. The cotangent is the cosine divided into a sine. Values of a sine and cosine of 30 and 210 degrees can be taken here.

Cotangent 210 degrees

 We already know how to divide fraction into fraction. Formulas of transformation of angles of trigonometric functions.

Formulas of transformation of angles

These formulas will be very useful to you.

Signs of trigonometric functions

All trigonometric functions can have different signs. Signs of trigonometric functions depend on the coordinate system. In mathematician the Cartesian coordinate system is accepted. Four quadrants of a Cartesian coordinate system define signs of trigonometric functions.

Signs of trigonometric functions

You do not speak to mathematicians, but remember. Signs are not property of trigonometric functions. Without coordinate system signs will not be. If to measure corners on another, signs will be others. In other coordinate system signs can be others.Signs of trigonometric functions are a property of the chosen coordinate system. Functions are considered in mathematician only in the Cartesian coordinate system.

Decomposition on items

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Addition

Lesson 16

Decomposition on items

If only the result of addition is known and items are unknown, then the sum can be spread out to items by means of the linear angular functions.

Decomposition of the sum on items

Transformation of a square of quantity to the work of two sums (see an example above), it is possible to execute with application of decomposition of result of multiplication to multipliers and items.

Transformation of a square to the work of the sums

Similitudes can be useful when studying various natural phenomena to their best comprehension. Let's review an example of reproduction of living beings.

Asexual reproduction of live organisms can be described by means of decomposition of the sum on items. As a result of division of an organism A two self-contained organisms B and C.

Asexual reproduction

Decomposition on items with corners about 45 degrees, is characteristic of unicells. For metaphytes the range of an angle of decomposition varies in wider limits (a vegetative reproduction, budding, fragmentation). The unit of measure at decomposition can be considered a physical body of an organism.

The beginning of life (zero) of similar organisms can be considered the moment of division of a parental organism. The termination of life (unit) can be considered characteristic division or death.

Sexual reproduction is described by means of multiplication. The moment of emergence of sexual reproduction can be described by means of the linear angular functions. At simultaneous reproduction of organisms A and B there was the common stream C which had signs of two parents.

Emergence of sexual reproduction

What have to be the angles of decomposition for emergence of the common offspring? The most probable candidates in "invented" sexual reproduction it is a larger cage and a virus. The virus breeds in a cage. Along with cell division there was a division of a virus. The result was a new body. Or two body - male M and female W.

Male and female

Molecule DNA which is available both for a cage, and for a virus could act as a basis for addition (unit of measure).

It is only one of a set of options of possible succession of events. From the moment of emergence of life on Earth until emergence of sexual reproduction in the Nature was enough time for the most different experiments.

Closing part

Further studying of properties of units of measure will help to understand better and more precisely to describe by mathematical methods various phenomena in the world around.

The separate ideas published in this work will be considered in more detail in the subsequent publications.

Gratitude

I express sincere gratitude to the parents and the daughter Inna for financial support of my working on with mathematics.

Addition

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
The linear angular functions

Lesson 15

Addition

As a result of addition of two different quantities the third quantity turns out. At addition of change occurs in a number domain, the area of units of measure does not change. Addition is possible only for parallel quantities with identical units of measure. Addition reflects the quantitative changes of quantities.

5а+3а=(5+3)а=8а

For realization of addition of two different quantities with units of measure in different scales (the corner of scale of units of measure is not equal to zero), it is necessary to change the scale of units of measure so that the scale corner between them equaled to zero. At the same time does not matter, the first item, the second or both changes at once.

It is impossible to put two identical numbers with different units of measure as the result does not make sense.

5а+5b=5(a+b)

Transformation of result of addition of pieces to the parties of a rectangle looks so.

Addition and rectangle

Items can be presented as the party of a rectangle, then a half of perimeter of a rectangle is result of addition. For any sum it is possible to define the linear angular functions if items are known.

At the following lesson we will consider
Decomposition on items

The linear angular functions

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Division

Lesson 14

The linear angular functions

If to consider terminating trigonometric functions as coordinates of points of a unit circle in a Cartesian coordinate system, then the linear angular functions are coordinates of points of the chord connecting circle cross points to a coordinate. The sum of coordinates of any point of this chord is always equal to unit.

The linear angular functions

In mathematician of a concept, similar to the linear angular functions, are used since ancient times - it is division whole on a part. In the modern world an analog are percent.

The explanation for readers of this website. For myself I called the linear angular functions "linos" and "loses". How I thought up these names? Took designation of a sine and cosine. Visually they quite well differ. In each designation I replaced the first letter with the Latin letter "l" from the word "line". It turned out quite nicely. But to solve to you. Whether these functions in mathematician will get accustomed and as they will be called - time will show. I just offer one more mathematical tool for the description of reality.

At the following lesson we will consider
Addition