How to memorize trigonometric functions?

How to memorize trigonometric functions? This question is set to itself very many at the study of trigonometric functions. On own experience I know that this business is not simple. Through many years I succeeded to carry two moments only, that behave to the trigonometric functions. First, that well burned into my memory, this determination of trigonometric functions generally: a trigonometric function is this relation something to something. Thus in memory at once there is a shot, at that I do not remember that stands in a numerator forgot that was in a denominator. From all trigonometric relations I remember only a tangent: a tangent is a sine on a cosine. A sine costs on the first place, he means in the numerator of shot. A cosine stands on the second place, he means in the denominator of shot. Sines with cosines I generally on life always mixed up. To the last time...

I succeeded to find a that failing element that so was not enough in the days of my educating. Today even I can faultlessly dab with a finger in a cosine to tell that on what in him divided.

The portrait of tangent appeared this magic stick. Yes, yes, not surprised - exactly portrait, at a look to that faultlessly know a tangent. This portrait of tangent is only work by oil of genius artist of beginning twenty the first century, that on auction of Sothebys in 2137 was sold for one hundred million dollars. Oddly enough, but a picture is written was on reverse side of canvas. Where do I it know from? I am a mathematician and I am able to see the future. By the way, yesterday I spoke with this artist, he asserts that such picture did not yet write, but an idea very pleased him. Specially for you I publish reproduction of this masterpiece here.

Masterpiece. Tangent. Mathematics For Blondes

Now we have all for creation of our own trigonometric Bible that will need to be learned by heart. Do not be frightened, in our Bible there will be only one line:

tangent - is sine by cosine

One-only picture from this Bible I already showed you. Now, as the most real preacher, I will teach you all of it it is correct to read. I think, the best method of educating will be a gape-seed of trigonometric comics. Let in the near time by it and we will engage.


Why are sines and cosines needed?

Why are sines and cosines needed? Really, interesting question. In comments to the trigonometric circle of sines and cosines such question appeared:

where will sin and cos be useful in life?
p.s why are they needed sines cosines?

Let us will call a spade a spade. To swingeing majority from you they will be never useful. Unless, when will your children go out into school and will begin to study trigonometric functions, they too will put question you "Why are sines and cosines needed?" and, in addition, will ask to explain, what is it.

Money we use every day already not alone thousand years and perfectly we do without every sines, cosines and other elegant mathematical pieces. I assure you, and through millions of years in the count of money nothing will change. Not because we are such dull, and because such are mathematical properties of money : it is impossible to increase roubles on roubles and with money in the second degree to hurry in a motor show to buy "Lamborghini".

On a kitchen, in culinary recipes, you will meet neither sines nor cosines too. If to give a glance soberly on our everyday life, then all our everyday mathematics remains somewhere at the level of knowledge of Ancient Greece. We are enough with a head.

So why are sines and cosines needed? As compared to Ancient Greece, we have very much different pieces about that ancient Greeks could not dream even today. Even their Gods did not ride on machines, did not use mobile communication, did not communicate on the Internet. But we have all of it and we use this constantly. Did all this extraordinary riches undertake from where? He was created by us. At first scientists did the scientific opening. Then engineers, on the basis of done by the scientists of opening, created every useful things. We use these things today, not having not the least concept about that is into these things and what scientific laws are fixed in basis of their work. So, if there were not sines and cosines, there would not be all these useful things.

Sines and cosines are used most effectively scientists and engineers. I will not say that they continuously trigonometric functions are used only. No, they use them rarely, but well-aimed. Sines and cosines often are in the formulas of different calculations an engineer or scientific.

Often with sines and cosines it is necessary to clash to the geodesists. They have the special instruments for an exact goniometry. Through sines and cosines corners can be converted into lengths or coordinates of points on an earth surface.

The teachers of mathematics on the sort of the duties constantly deal with trigonometric functions. This year they told about sines and cosines to you, the next year the teachers of mathematics will tell the same to other students. Such for them work - to teach.

Schoolchildren and students study trigonometric functions on the lessons of mathematics. Personally I got through tortures sines and cosines at school, техникуме, institute.

Adults sometimes engage in sines and cosines then, when their toschoolchildren need a help at preparation of homeworks.

All! Sines and cosines do not need other generally! In everyday life most people they are not used hardly ever. If I wrong, remedy me.

So why then generally to teach these sines and cosines? Well, firstly, such is the school program. Secondly, if you in life may need apply a sine or cosine, you know already, what is it and where it is needed to search information about them. The knowledge gained at school will fully have you, what is independent in everything to understand.

So what such the sines, cosines and other trigonometric functions? It is a mathematical instrument it is needed that to be able to use. That we this instrument we do not use hardly ever, talks not that studying them is not necessary, and that efficiency of application of the knowledge gained by us is practically equal to the zero. But it is quite another theme already.


How to calculate the area of surface of direct three-cornered prism

How to calculate the area of surface of direct three-cornered prism? For this purpose the best of all to imagine a three-cornered prism in all her beauty. How does a three-cornered prism look? Look at a picture

Three-cornered prism. Area of surface of three-cornered prism. Mathematics for blondes.
Now next question: do you have a tube of lipstick in form three-cornered prism? If you answered this question (and or no) though as, means now you already know about what three-cornered prism and as she looks. Here those two triangles, from above and from below, are named the grounds of prism. Three rectangles on each side are named the verges of prism. As a triangle lies in founding of prism, verges for us exactly three. If there will be a pentagon in founding, verges it will be five. And if is there 1234-cornered in founding of prism? Correctly, verges there will be 1234 things. With the construction of direct prism we understood, after it it is possible to take up the mathematical calculations of area of surface of three-cornered prism.

As a prism is a geometrical body, her structure can be investigated on the example of body of blonde. That is included in the area of surface of prism? If you will come a heel on something sharp or knocked by the обо-что top of the blond head, it will be badly you. That is why both heels and top of head enter in the complement of surface of your body. Just overhead and lower triangles are included in the area of surface of three-cornered prism. If to dab with you a finger in a side, you will say "Ouch"! because a side belongs to your body. Here those three rectangles, that are from the sides of three-cornered prism, make the area of her side.

By the result of our scientific researches of bodies of blonde and three-cornered prism, we came to the conclusion, that the area of surface of three-cornered prism consists of areas overhead and lower grounds (they are equal) and area of side. The area of triangle needs to be found on one of formulas. I remind that the area of triangle found on a formula at the calculation of area of surface of prism needs to be taken two times, id est to increase her on two.

The area of side of prism is determined as a sum of areas of rectangular verges. It is needed to increase length of every side of triangle (that triangle that is in founding) on the height of prism and lay down three got areas together. It is possible to act simpler: to increase the perimeter of triangle (a perimeter of triangle is a sum of lengths of all his parties) on the height of prism. But adding up not to avoid you in any case: either to the increase (we fold long parties) or after an increase (we fold the areas of rectangular verges). Another trouble that you can lie in wait on a way to the area of surface of three-cornered prism is this absence of values of all lengths of parties. It is not mortal for the calculation of area of triangle. And for the area of side of prism you will have preliminary to find lengths of all three parties of triangle, applying the mathematical knowledge.


About symmetry of mathematical actions

About symmetry of mathematical actions - is my first official publication. To all appearances, my flaming speech under the name "Mathematics forever!" remained unnoticed. It is clear. Reading a like, I would say that a next idiot rushed about on all Internet with the ridiculous idea. But... All, that is here written, I write exceptionally for you and publish here in an only copy, unlike other authors of raving ideas. With my article about symmetry of mathematical actions you are first can become familiar right here and now. I am herein anything interesting or no - decide. In brackets I will give some comments (specially for you) that in the printed variant of the article are absent.

Annotation: Rules of symmetry of mathematical actions allow to apply a commutative law to all mathematical actions: to addition, deduction, multiplication and division. (An annotation is this obligatory condition for the publication of the article. Such are rules of the bureaucratic playing science)

Changes in the surrounding world are expressed by mathematical actions. Quantitative changes are expressed by addition and deduction. Quality changes are expressed by an increase and division. No quantitative changes can cause the change of quality.

Quantitative changes reflect the change of amount of the separately taken unit. Addition and deduction are symmetric mathematical actions reflecting the quantitative changes of any unit. Addition and deduction are mirror symmetric relatively neutral element are points zero.

An increase and division similarly are symmetric mathematical actions reflecting the quality changes of units. An increase and division are back symmetric relatively neutral element are points one.

Rules of symmetry of mathematical actions:

1. Any mathematical action is begun with a neutral element.

2. A sign of mathematical action is the inalienable attribute of number before that he stands.
(This fragment is distinguished by me by fat text specially for you)

Application of these rules allows to apply a commutative law to all mathematical actions reflecting quality or quantitative changes.

0 + 3 + 7 + 4 = 0 + 7 + 3 + 4 = 14

0 – 3 – 7 – 4 = 0 – 7 – 3 – 4 = –14

0 + 3 – 7 – 4 = 0 – 7 + 3 – 4 = –8

1 х 3 х 7 х 4 = 1 х 7 х 3 х 4 = 84

1 : 3 : 7 : 4 = 1 : 7 : 3 : 4 = 1/84

1 х 3 : 7 : 4 = 1 : 7 х 3 : 4 = 3/28

A commutative law can not be used in the cases of the mixed implementation of mathematical actions reflecting quality and quantitative changes in one mathematical expression.

The change of the mathematical operating on symmetric gives a symmetric result, here the point of symmetry is a neutral element. Application of commutative law does not influence on a result.

0 – 3 – 7 – 4 = 0 – 7 – 3 – 4 = –14

0 + 3 + 7 + 4 = 0 + 7 + 3 + 4 = 14

0 – 3 + 7 + 4 = 0 + 7 – 3 + 4 = 8

1 : 3 : 7 : 4 = 1 : 7 : 3 : 4 = 1/84

1 х 3 х 7 х 4 = 1 х 7 х 3 х 4 = 84

1 : 3 х 7 х 4 = 1 х 7 : 3 х 4 = 28/3

Running the numbers in the mathematical operating on symmetric relatively neutral element of number gives a symmetric result.

0 + (–3) + (–7) + (–4) = 0 + (–7) + (–3) + (–4) = –14

0 – (–3) – (–7) – (–4) = 0 – (–7) – (–3) – (–4) = 14

0 + (–3) – (–7) – (–4) = 0 – (–7) + (–3) – (–4) = 8

1 х 1/3 х 1/7 х 1/4 = 1 х 1/7 х 1/3 х 1/4 = 1/84

1 : 1/3 : 1/7 : 1/4 = 1 : 1/7 : 1/3 : 1/4 = 84

1 х 1/3 : 1/7 : 4 = 1 : 1/7 х 1/3 : 1/4 = 28/3

Simultaneous change of the mathematical operating on symmetric and running the numbers on symmetric relatively neutral element of number abandons a result without changes.

0 – (–3) – (–7) – (–4) = 0 – (–7) – (–3) – (–4) = 14

0 + (–3) + (–7) + (–4) = 0 + (–7) + (–3) + (–4) = –14

0 – (–3) + (–7) + (–4) = 0 + (–7) – (–3) + (–4) = –8

1 : 1/3 : 1/7 : 1/4 = 1 : 1/7 : 1/3 : 1/4 = 84

1 х 1/3 х 1/7 х 1/4 = 1 х 1/7 х 1/3 х 1/4 = 1/84

1 : 1/3 х 1/7 х 4 = 1 х 1/7 : 1/3 х 1/4 = 3/28

The neutral elements of mathematical actions it is not accepted to write at the decision of mathematical problems and examples, as they do not influence on a result. Before application of commutative law introduction of neutral elements allows correctly to apply a commutative law.

All of it is written, certainly, not for blondes, and for mathematicians. In the future we yet not once will call to this article. And while... you know any more mathematician about symmetry of mathematical actions.


Mathematics forever!

Mathematics forever!

You can congratulate me, my first scientific publication went out in light. There are "The papers of independent authors" in a magazine, producing № 18, to the page 110 the little article is printed under the modest name "About symmetry of mathematical actions".

At the last time (in a blog in Russian language) I promised you to show that needs to be done in order that impossible in mathematics became possible. In this little article, only on two pages of text, it is shown, as a commutative law works at deduction and division. If someone wants to look, can pass on this page (in Russian language), there is reference for a flush-off. Nothing difficult in this article is present - the half of text is occupied by examples on addition, deduction, increase and division. All at the level of middle classes of school. Such mathematics any blonde will understand.

Here so, unnoticed, we with you began to live in a completely another epoch - Epoch of Great Mathematical Opening. Doing these mathematical opening coming to you, I can only broadly speaking explain to you, what mathematics, where in mathematics, opening is hidden and as they need to be searched. By the way, my article is the first step on a way to dividing by a zero. Mathematical actions are symmetric: if in mathematics there is multiplying by a zero, means there is under an obligation to be dividing by a zero. If dividing by a zero is impossible, means multiplying is impossible by a zero. The third variant (that is known by us all) can not be. When I will show you, where and as there is dividing by a zero, you will understand that mathematics is this not паханное field on that we engage in the most primitive collector. To throw open and take the crop on the mathematical field, at a desire, any of you can.


Four-valued mathematical tables for blondes

Four-valued mathematical tables of Bradis - this was the basic mathematical reference book of soviet schoolchildren, students, engineers to appearance of calculators.

A trigonometric table for blondes is done by me to the navigation more informatively saturated in a plan. It what you did not lose way and did not entangle trigonometric functions.

Trigonometric table of sines and cosines
- from 0 to 90 degrees punctually to the minute corner.

Trigonometric table tangent cotangent in degrees - from 0 to 90 degrees punctually to the minute corner.

Trigonometric table in radians - sin, cos, tan.

We will do justice to work of Bradis and we will remember a bit history. His book "Tables of the four-valued logarithms and natural trigonometric sizes" went out in 1921. This book was repeatedly reprinted, but already under more simple name "The Four-valued mathematical tables". This bestseller looked approximately so.

Four-valued mathematical tables of Bradis for blondes. The basic mathematical reference book of soviet schoolchildren, students, engineers
It is possible to say without a false modesty, that on these tables all Soviet Union was built, a man started to fly in space, a soviet nuclear club was created et cetera. Schoolchildren, engineers, scientists - all used the tables of Bradis. We will remember those distant times - the Internet is not present, mobile telephones are not present, computers and calculators are not present. Even televisions were not then! There were only books. In many books formulas were written for a calculation, and necessary for calculations numerical values were taken from the tables of Bradis. And what did numbers multiply by then? Not on accounts... Well and time was! As it was then possible normally to live??? But lived somehow.

An interesting question arises up. What mathematical tables did Americans build famous sky-scrapers on and created the nuclear club? In fact did not they steal for us tables of Bradis? Omniscient Wikipedia is quiet on this occasion, and information about the table of Bradis in Wikipedia in English language I did not find. There is there a mathematical reference book of Abramowitz, Milton and Irene A. Stegun with tables, but he is published was only in 1964. And that did Americans have to these tables?

I used the table of Bradis once. This whole art, to search the value of trigonometric functions punctually to the minute corner. Fortunately, today we have calculators.

Unfortunately, work of great soviet writer of mathematical tables Bradis me does not interest. Therefore the four-valued mathematical table into language of blondes transferred will not be. In the simplified kind the table of Bradis will be presented in the tables of sine, cosine, tangent and cotangent.


Trigonometric circle - comments 2

Trigonometric circle - comments 2. This continuation of publication of comments to the page the "Trigonometric circle", left by visitors in the Russian version of blog "Mathematics For Blondes".

Comment - I am very thankful you, I skipped these themes and there is nothing could understand, and now though that I understand)))

Comment - I in general understood nothing)

Nikolay Khyzhnjak - Nothing, it is not needed to be disordered. All do not can everything it is identically good to understand. A trigonometric circle someone understands well, someone can not understand in general.

Comment - Class site))))))

Nikolay Khyzhnjak - Concerning a site not quite true: here almost is not that it must be are mathematicians. Simple, clear, evident. I will try to reform)))

Comment - Your table very pleased me all is simple, evidently and clear. Thank you!!!

Nikolay Khyzhnjak - It is my little strokes to stranger work. The trigonometric circle I will draw together with you straight in this blog. I hope, soon this we will take up)))

Comment - I thank you. I do not present although, how all of it I will teach... But for a crib it very well))

Nikolay Khyzhnjak - And really do the salespeople of estimations require cramming yet?))) Personally I prefer to understand mathematics, but not to learn by rote ridiculous rules)))

Comment - cos pi/12 as possible to present?

Nikolay Khyzhnjak - cos pi/12 = cos (180/12) degrees = cos 15 degrees = 0,9659 Finding this value is possible on the table of cosines.

Comment - cos corner in intervals from pi to 3pi/2 does take on negative values??

Nikolay Khyzhnjak - Yes, in the third fourth at a cosine negative values, as well as in fourth, from 3pi/2 to 2pi.

Comment - how to find the cosine of pi on 12

Nikolay Khyzhnjak - It is hardly higher written in comments, that on the table of cosines it is needed to find a cosine 15 degrees.

Comment - you are dimwits

Comment - Thank you for the name "Mathematics for blondes". I 30 years engage in that I translate school mathematics into the language of "mathematics for blondes". Actuality grew with introduction of single state examination. I add to the line of tangents and cotangents (students understand it).

Nikolay Khyzhnjak - Accept my sincere condolences))) Unlike you, I have a right to be quiet. As a teacher of mathematics, you such right do not have. you are under an obligation to explain ALL themes, I can simply not talk about separate moments.
Yes, lines of tangents and cotangents simply are under an obligation to be on a trigonometric circle. Although I and lazy, but I will try in the near time to draw such picture.
very interesting me, how do you explain negative and complex numbers in language of blondes?

Comment - creator, for you alike at with a head badly, if you consider blondes fools.

Nikolay Khyzhnjak - I think that my articles "Mathematics - the blonde searches x", "" testify to reverse.

Comment - thank you that who did everything I am a not blonde and not girl but all is even

Nikolay Khyzhnjak - I write for all, who wants to read it)))

Comment - good site, to the author respect and respect, though I and not blonde, I badly understand in mathematics, but all is normally written with a clear language, thank you

Nikolay Khyzhnjak - Thank you for a comment. Not everybody can understand mathematics - it firstly. Secondly, someone must and mathematicians to feed, dress, write their portraits...

Comment - seems, you similarly are blond

Nikolay Khyzhnjak - When I became a mathematician, I, seems, began to understand logic of thinking of blondes. Paradox, but for blondes, alike, mathematical character of thinking.