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3.12.2023

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:

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 in degrees. sin and cos 0, 30, 45, 60, 90 degrees. The most unusual table of sines and cosines. Mathematics For Blondes.
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.

3.11.2023

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. Mathematics For Blondes.
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. Mathematics For Blondes.
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.

3.03.2023

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. Mathematics For Blondes.
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. Mathematics For Blondes.
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. Mathematics For Blondes.
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.

2.21.2023

Where does math end?

I was offered to sell my Russian site for 45.000 rubles. This prompted me to a very interesting question: "Where does mathematics end?". And here's my answer: "Math ends where money begins". Try any mathematical theory to apply to money and you will learn a lot of interesting things about this theory.

I love math trolling. The sentence contained the phrase: "The price may be revised upwards". My response was: "$45.000 Price may be revised down". From the point of view of mathematics, the equality 45.000=45.000 is beyond doubt. As soon as I add different units of money to this equality, it disappears.

Where does math end? Mathematics For Blondes.
Where does math end?

We use numbers to measure money. Numbers are written on coins and banknotes. Have you heard of banknote or a coin with the number "zero"? Zero is not a number. The more I study this question, the more I am convinced of the correctness of my statement.

Zero dollars. Mathematics For Blondes.
Zero dollars

I'm not even talking about the minus sign on money. Through the prism of money, I considered set theory. The result is very interesting.

2.12.2023

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

The answer to the question "why is the sine of 30 degrees equal to half one?" 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. The side is equal to one.  Mathematics For Blondes.
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. Mathematics For Blondes.
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. Mathematics For Blondes.
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. Mathematics For Blondes.
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.

11.26.2022

Sums classification

Annotation

If you take any sum, then all other sums can be divided into three groups: identical sums, decomposition sums, and other sums. The classification criteria are the terms and the result of the addition.

Identical sums

Identical sums are combined into a separate group by the commutativity property. Sums from this group have pairwise identical terms and an equal number of terms. The result of adding these sums is the same. Here is an example of identical sums.

Identical sums.  Mathematics For Blondes.
Identical sums

The number of sums in this group is determined by the number of terms. For infinite sums, it is equal to infinity.

Decomposition sums

If the results of adding a different number of terms or terms of different sizes are the same, then such sums form a decomposition group.

Decomposition sums.  Mathematics For Blondes.
Decomposition sums

Any of these sums can be obtained by decomposing the result of addition into terms using linear angular functions. Having thus obtained two terms, any of them can also be decomposed into terms, and so on. An example expansion for the first three sums:

Decomposing into terms. The linear angular functions.  Mathematics For Blondes.
Decomposing into terms

An example of decomposing a number into three terms shows that different decomposition options can give the same sum, which underlies the associative properties of addition. The more terms the sum contains, the more different expansion options there can be. The expansion into terms can be continued indefinitely. Different angles and different decomposition algorithms make it possible to obtain different variants of infinite sums. The theory of limits allows you to determine the result of addition based on the analysis of terms. The expansion into terms allows the summation result to be represented as an infinite series of terms.

For example, let's decompose the unit into an infinite sum according to the following principle: the expansion is performed at an angle of 45°, each second term is decomposed into two terms.

Decomposition into an infinite number of terms. The linear angular functions.  Mathematics For Blondes.
Decomposition into an infinite number of terms

The curly brackets indicate the sum of the invisible compensating group of terms, which complements the result of addition to an integer unit. It can be assumed that any sum, even an infinite divergent series, in trigonometric form is equal to one.

Other sums

If the result of adding any amount differs from the result of adding the amount in question, then this amount has nothing to do with the amount in question and belongs to the “other sums” group.

Let's assume that for the sum a+b=c there is another addition result d that is not equal to c. That is, a+b=d. We represent these two expressions using linear angle functions, and then we translate them into a trigonometric form:

Different addition results. The linear angular functions. Mathematics For Blondes.
Different addition results

The assumption that there are different addition results for the same sum takes us beyond the boundaries of mathematics, where the basic trigonometric relations stop working:

Beyond the boundaries of mathematics.  The basic trigonometric relations stop working. Mathematics For Blondes.
Beyond the boundaries of mathematics

Why can't the same sum of terms have two different addition results? You can understand this by considering the reverse process - the transformation of trigonometric functions into specific mathematical sums. I will cover this in more detail in a separate post.

In conclusion of the conversation about the presence of two different results of addition in divergent series, I will give an example from physics. In the earth's crust (convergent series) there are natural caves (the sum of the convergent series). Using special mechanisms (convergence according to Cesaro, etc.) we can get artificial tunnels (the sum of a convergent series). There are no natural caves in the seas and oceans (divergent series, the sum of the series is missing). The use of special mechanisms (convergence according to Cesaro, etc.) allows us to obtain artificial tunnels (the sum of a divergent series) in the water column. Based on this mathematically proven theory, it is possible to design a network of tunnels for roads and railways that encircle the entire earth's surface. Such a theory is quite possible if we do not understand the difference between solids (convergent series) and liquids (divergent series).

Conclusion

There are no mathematical methods that allow you to get a different addition result for the sum in question.

11.22.2022

Shift in arithmetic

Annotation

The use of a shift in arithmetic leads to the addition of terms with different units of measurement.

Consider the results of a shift in arithmetic addition. A shift is possible when performing addition in a column. For example, let's take a three-digit number and add to it the same number without a shift, with a shift by one position and with a shift by two positions. At the same time, we will observe the rule adopted when shifting infinite series - we discard the numbers of the second term that go beyond the boundaries of the first number.

Addition with a shift in a column. Mathematics For Blondes.
Addition with a shift in a column

Shifting by a different number of positions leads to different results in the same way as when shifting infinite series. You can write the same expressions in a line with the subsequent indication of the units of measurement for each of the terms.

Addition with a shift in a line. Mathematics For Blondes.
Addition with a shift in a line

Conclusion

The shift leads to a change in the unit of measurement of the shifted term. The shift is a violation of the basic law of addition - you cannot add terms with different units of measurement.