2.18.2011

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", "http://www.mathforblondes.com/2010/11/wunderkindes-and-cretin-with-blondes.html" 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.

The Russian specialists of Photoshop compelled Paris Hilton to read a book "Discrete mathematics".

Comment - and to me reached!)

Nikolay Khyzhnjak - it is stranger work, this picture of trigonometric circle I only adorned))) it will be Needed to draw the trigonometric circle, by sight clear at first blush)))

Comment - you are a fool bald, explain normally! I am a hard blonde, I understand nothing!

Nikolay Khyzhnjak - If you are a hard fool, then medicine is here powerless. Mathematics - moreover. Now try once again, only already politely, to answer a question: that you are not clear exactly?

Comment - what if at being of value alpha equal 4pi, for me 720 degrees, how farther to find a sine and cosine alpha??

Comment - however I hate geometry, though I am a not blonde=)

Nikolay Khyzhnjak - concerning a hatred. It is your problems((( to Mathematics on a drum, blonde you or no, you love geometry or hate, in mathematics twice two however will equal four, and the laws of geometry will continue to work, as well as before. It is needed simply to reconcile oneself to with this fact. All people can not identically masterly own geometry. But it is desirable to know about basic concepts. In fact you are a reasonable creature, but not animal. It not needed the animal of knowledge, for life by him there are fully enough instincts.
Concerning 720 degrees now I will write on a separate page.

Comment - on my you engage in unavailing enough business. To teach blondes, it too same, that to try to drive away a tram... Although, good luck)

Comment - better will explain on an example - how to decide cos(x - pi/2)

Nikolay Khyzhnjak - Concerning educating of blondes. Blondes are a simply advertising trick))) Although in some places I understand their logic perfectly)))
In business-literature a phrase pleased me: "If not able to do business, teach other people, as needed to do business. If not able to teach people to business, teach teachers, as needed to teach people to business"))) As I in trigonometry small that I understand, I teach other and study. Blondes - it my level just)))
About the decision of examples. Examples without решебника I I am not able to decide. I am simpler to set about in a fuzz and ashes all trigonometric circle, what to decide a предложнный example in due form educational-bureaucratic art))) I will Find a decision - I will publish)))

Comment - is pi divided by 12 exists? If yes, then what is it equal to?

Nikolay Khyzhnjak - Yes, pi divided on 12 is a not mirage, it exists. For that, what finding, what it is equal to, is needed to divide 180 degrees on 12. 15 degrees ensue.
pi/12 = 180/12 = 15 degrees
next equality is Here used for translation of radian measure of corners in a degree:

Comment - Thank you, then yet: (Pi/6+2Pi)/2=13Pi/6. If is it calculated right, then how to bring this expression over? and whether it is possible to simplify him not taking away Pi.

Nikolay Khyzhnjak - It is calculated not correctly. After equal sign the result of calculation of expression is writtenin in brackets, and he needs to be yet divided in half. In the total it must turn out 13pi/12. With the value of corner already nothing will do, unless to write down as 1 and 1/12 pi. And trigonometric functions with the corner of such size it is possible to lead on the special formulas. Every trigonometric function, has the formulas of coersion. All trigonometric functions of any corners can be driven away in a stall measuring from 0 to 90 degrees. The formulas of bringing stray sheep over we will consider in the nearest time)))

2.09.2011

Mathematics of blondes and not only...

Today for blondes we will conduct a lesson of biological mathematics. Surprisingly, but the fact:

If to count up quantity of hair on a head of the person (and it is an abstract mathematics without any prejudices) results of calculation will appear such: at their blondes nearby 150,000 (hundred fifty thousand), chestnut hair on a head we will count about 110,000 (hundred ten thousand), black hair it will appear about 100,000 (hundred thousand), red - 90,000 (ninety thousand). Blondes! You possess the thickest hair-do among all people! And, it not a leisure invention of the bald author of a blog "Mathematics for blondes", and the fact confirmed with mathematics.

The number of the top eyelashes makes from 150 to 200, bottom - from 50 to 100 (horror, but even at blondes number of the bottom eyelashes in 2 - 3 times are less, than number of the top eyelashes - here even MaхFactor is powerless);

It is long the eyelashes, measured by the most ordinary ruler for measurement are long eyelashes, for the top eyelashes yields result 8 - 12 millimetres, for bottom equals 6 - 8 millimetres (here already is where to clear up not only Maksfaktoru, but also to false eyelashes);

The size of day growth of hair on a head makes 0.5 - 0.7 millimetres (specially for blondes I will write with letters these terrible numbers - the five tenth and seven tenth millimetre) is less than the tiniest division into your school ruler almost twice, and here day growth of hair of a beard already on the one tenth millimetre is more - from 0.6 to 0.8 for a day (it does not threaten blondes the same as to me - growth of hair on a head).

And some more the sad facts from life of the hairy. A day hair fall on a head - from 50 to 120 pieces (it is interesting, why I bald???!!! Where the mathematics looks!). Life expectancy of hair makes from 2 till 4 years. Life expectancy of eyelashes makes only 150 (hundred fifty!) days. You represent?! Already in half a year on your eyes there will be no familiar eyelash - all new! Horror!!!

Photo from Ledi Truth.

Sine and cosine 11 degrees and 32.7 minutes how to calculate?

When I went to school, I had to use the table of Bradisa for a finding of sine, cosine, tangents and cotangents. Has already forgotten, as this table to use. But today we live in a computer century, and what such the computer? Correctly, it is such big calculator. And in each big calculator there should be a calculator small. Here this calculator also needs to use. At me operating system Windows XP, on the screen the glory, below, is a button "Start-up". Press this button, then in the menu choose "All programs", from all programs choose "Standard". In standard programs the calculator of sine necessary to us and cosine hides.

The calculator usually has no sine and cosine. It is necessary to press a button "Kind" in top panel the calculator and to choose "Engineering". In the engineering calculator there are buttons necessary to us a sine "sin", cosine "cos" and a tangent "tg".

After that it is necessary to track that in the calculator decimal notation and degrees for corners would be included. For this purpose it is necessary to press "Dec" and "Degrees" as on a picture it is shown. Our tool for a finding of sine and cosine is ready. Now we will start directly process of extraction of useful trigonometrical minerals.

If it was not possible to you extract the calculator from the computer, do not despair! Specially for you I have placed in this blog "Mathematics for blondes" the calculator free of charge which you can use directly here and now!

web 2.0 scientific calculator

At first it is necessary to translate minutes in degrees. For this purpose 32.7 we divide on 60. It is As a result received 0.545 degrees. On 60 we divide because in one degree of 60 minutes. To received циферке it is added 11 degrees which at us already are, and it is received 11.545 degrees. Here from such corner on the calculator it is possible to take already sine and косинусы. For this purpose it is necessary to press simply a button "sin" or "cos".

All process press buttons looks so:

32.7 / 60 + 11 = sin

As a result in a calculator window there will be number 0.20013750391127021629780041181162

For math it registers so:

sin (11° 32.7') = sin 11.545° = 0.2001

For a cosine of the angle of 11 degrees of 32.7 minutes value is equal almost to unit and will register so:

cos (11° 32.7') = cos 11.545° = 0.9798

For a tangent all is carried out precisely also, only right at the end instead of a button "sin" the button "tg" is pressed. Here with cotangent, apparently, a problem. There is no such button in the calculator! But we clever, also remember that trigonometrical function cotangent is return trigonometrical function in relation to a tangent (so much clever words for once - already most terribly!). In practice it looks very simply: at first we find a tangent as it is described above. When numbers a tangent have appeared in a calculator window, we press a button "1/х". Numbers a tangent will exchange on numbers cotangent. And this additional magic button is called "number, the return entered". For the sake of a trick, enter number 2, press this magic button and you will have number 0.5 that equally 1/2.

For transfer of seconds in minutes, seconds as need to be divided on 60, as in each minute of 60 seconds. For transfer in the degrees, the received minutes it is necessary to divide once again on 60:

1" = 0.016667' = 0.00027778°

Here, apparently, all how to calculate a sine and косинус 11 degrees and 32,7 minutes. If someone still had questions, write to comments. If to someone laziness most to press buttons in the calculator, it is not necessary to write to comments! I understand, not imperial this business - to be picked the calculator. Then go on a site of the decision of problems, they will execute any your mathematical whim, naturally, for your money.

For the favourite blondes I can give some small helps. The sine of 6 degrees of 30 minutes needs to be typed on the calculator as 6,5 degrees, then to press a sine button.

Now example it is more difficult, with seconds: cosine 6 degrees of 7 minutes of 9 seconds. 9 seconds we divide on 60, we add 7 minutes, again we divide on 60, we add 6 degrees. Number 6.11916666 should turn out... Degrees. Now we press a button cosine "cos". An order press buttons the such:

9 / 60 + 7 / 60 + 6 = cos

For math recalculation of degrees, minutes and seconds in degrees for 6 degrees of 7 minutes of 9 seconds can be written down so:

(9 : 60 + 7) : 60 + 6 = 6.11916666...

In a general view for a corner in x degrees, y minutes, z seconds the transfer formula in degrees will look so:

(z : 60 + y) : 60 + x = degrees

I hope, this formula is useful to you.

How to find a value:

How do you find the sin of degrees and minutes on a calculator - here is a calculator it is written, how on him to calculate a sine

At me of 720 degrees how further to find a sine and cosine of corner?

"And what if at a finding of value of corner equal 4pi, at me of 720 degrees how further to find a sine and cosine of corner??" - such question has been set in comments. Really, how to find trigonometrical function if a corner the alpha is more than 360 degrees?

Present that all corners 360 degrees there are more or 2 pi is a ball of a yarn. To learn values of trigonometrical functions for such corners, this ball of a yarn needs to be unwound at first. One coil of a yarn equals to a corner in 360 degrees or 2 pi. To unwind a ball it is necessary until value of a corner does not become less than 360 degrees or 2 pi. After that under the table of values of trigonometrical functions or on a trigonometrical circle we find value of the necessary trigonometrical function.

720° - 2 x 360° = 0°

The same focus with corners in radians will look so:

4π - 2 x 2π = 0

For a corner of 720 degrees or 4 pi it turns out that a sine and cosine same, as well as for a corner of 0 degrees. The decision can be written down so:

sin 720° = sin 4π = sin 0 = 0

cos 720° = cos 4π = cos 0 = 1

The sine of 720 degrees or 4 pi is equal to a sine of zero of degrees and is equal to zero. Cosine 720 degrees or 4 pi it is equal cosine zero of degrees and it is equal to unit.

What for all these troubles with such big corners are necessary? I have knowingly mentioned a yarn ball. Let's try together it not only to untangle, but also we will look, as it is reeled up and what for in general is necessary.

Mathematics - the blonde searches x

The blonde searches x is a classical picture at which all Internet laughed. We have laughed also. And now we will try to understand, whether so it is ridiculous. What is told in the task? "Find х". What has the blonde made? It has found x and has shown it. Where it is told, it is necessary what to search for VALUE x? Remember the childhood when mum spoke to you "Show on a picture rabbit". You stuck with a finger in rabbit on a picture, and your mum admired the cleverest child.

The task "Find х" it is possible to understand on a miscellaneous. The blonde, from the engineering point of view, has carried out the task excellent - with the least expenses of time and forces. Give for a minute we will glance in any office. The director speaks to the manager on purchases (now fashionably all to call managers): "Find..." - also names the goods. What, the manager removes all money from accounts department and buys the named goods for all sum? No. Though it also is the manager on purchases, it simply collects all information on the named goods. It transfers this information to the director. And already the director solves, where, how many and at what price the goods need to be bought. Or while not to buy at all. So all normal managers arrive.

Really, surrounding conditions impose some stereotype of behaviour. As at the trained animals. If you the manager on purchases, without direct instructions what exactly to buy, buy it is necessary nothing. That to you would not speak. If you pass the decision of the equations in school, means on command "find х" you should search for value "x", instead of a dagger on a picture. So all arrive clever (and it would be desirable to write "little monkeys") pupils.

And blondes - they others. They well remember, to that them learnt in the childhood. They consider the children's decision as more simple. Really, what for to be soared with any decision when it is possible to show simply on a picture this most whether икс, whether a dagger. You consider yourselves already adult, and laugh at the naive children's decision of the blonde? And let's look at you, the adult, values x's finding, at other lesson...

... And so, you at a geography lesson, stand near with a pointer in hands of a card and the teacher speaks to you: "Find Honduras". You feverishly rummage eyes on a card in search of Honduras, but it anywhere is not present! Here to you nothing can already help - neither ability to find value x, nor knowledge of the Pythagorean theorem... Everything that from you is required it it is stupid to stick with a pointer into this ill-starred Honduras. You perplexed look in a class where tens fingers show in the most different corners of globe and you are even more lost. An unseeing sight having stared in a card, you suddenly understand that the teacher specially for you has brought today on this lesson a card WITHOUT Honduras!!! Doomed having lowered a pointer and eyes, at the very bottom of a card, you distinctly see an inscription tiny letters - even in a magnifier it is impossible to read it! But you already precisely know that there is written "printed in printing house "Revenge of the teacher". Circulation 1 copy (without Honduras)"...

2.08.2011

How many will be metres and decimeters?

"How many there will be 22 metres of 2 decimeters a minus of 1 metre of 8 decimeters?" Difficulty of this problem consists that for a designation of one number is used at once two units of measurement: metres and decimeters. If to translate told on normal language of blondes it almost the same that "I have bought yesterday a phrase two jackets and one has presented to the girl-friend" to replace with a phrase "I yesterday has bought two jackets and four sleeves, and then one jacket and two sleeves has presented to the girl-friend". It is natural that last phrase will confuse you at once: "And sleeves that, separately from jackets are on sale?". Certainly, no. Sleeves are sewn to jackets, here we only consider them separately. What for? What you to confuse. Problems in the mathematician always get confused, as a ball of threads. You should untangle and receive them the right answer. It develops ingenuity, here. And simple writing off from the answer book develops dullness. Now we will start the decision.

Means, a problem at us such: how many will be, if from 22 metres and 2 decimeters to take away 1 metre and 8 decimeters? The main thing what it is necessary to know at the decision of this problem - how many decimeters in one metre? Or how many metres in one decimeter.

1 metre = 10 decimeters

1 decimeter = 0.1 metres

One metre is equal to ten decimeters. One decimeter is equal the one tenth metre. Here now we have everything what to solve a problem. Is the whole three ways of the decision of this problem - in metres, in decimeters, in two units of measurement at once.

We solve a problem in metres. For this purpose decimeters reduced (this that number from which subtract) and subtracted (this that number which subtract) it is translated in metres.

2 decimeters = 0.2 metres

8 decimeters = 0.8 metres

We add these tails to our numbers in metres:

22 metres + 0.2 metres = 22.2 metres

1 metre + 0.8 metres = 1.8 metres

Now it is possible to execute subtraction and to receive a difference of these metres. It is possible to translate the answer at once in metres and decimeters:

22.2 metres - 1.8 metres = 20.4 metres = 20 metres of 4 decimeters

The second way is to solve the same problem in decimeters. For this purpose we will translate metres in decimeters:

22 metres = 220 decimeters

1 metre = 10 decimeters

To the received decimeters it is added those decimeters which at us are on a statement of the problem:

220 decimeters + 2 decimeters = 222 decimeters

10 decimeters + 8 decimeters = 18 decimeters

We carry out subtraction and we translate decimeters again in metres and decimeters:

222 decimeters - 18 decimeters = 204 decimeters = 20 metres of 4 decimeters

It was the mathematics. Now for blondes. What to understand that we did, let's replace metres with bows, and decimeters on ruches. Thus we will agree that it is possible to make ten ruches of one bow, and it is possible to make one bow of ten ruches – well to sew. We copy a statement of the problem: «you have 22 bows and 2 ruches. And 8 ruches you should give 1 bow to the girl-friend». With bows all is simple – we take one and we give. And how to be with ruches? They obviously do not suffice us. We will look, as we solved this problem math in two ways. From outside it looks so.

The first way. You take all bows and untie them, smooth all ruches. Take a needle with a thread and sew all it in one long tape. After you have stopped to sew, measure the necessary quantity from a tape and cut off scissors. Solemnly hand over to the gone nuts girlfriend a tape piece. Then you with the girlfriend cut the tapes, do of them bows and ruches. As a result you with the girlfriend, all in bows and ruches, stand, as two marriageable silly women.

The second way. You untie all bows and cut them on ruches. Do a heap of ruches and from this heap select 18 pieces for the girl-friend. After that solemnly hand over to the gone nuts girl-friend a small group of ruches. Then together begin to smooth ruches and to sew from them bows. Thus your girl-friend thinks: «In the silly woman! Because of this ugly creature I should sew now a bow». You think: «In the silly woman! Because of this ugly creature I at first cut, and now should sew already twenty bows». To tell the truth, I do not envy you.

As you can see, math correct ways from outside look is not absolutely practical. How any blonde will arrive in this situation? As the two-nuclear processor. Takes one bow from a small group of bows and will make of it ten ruches. Now you have one small group from twenty one bows and the second small group from twelve ruches. From the first small group you take one bow, eight ruches a beret from the second small group. Hand over to the happy girl-friend one bow and eight ruches. Then you admire the treasures and is simple blah-blah-blah instead of puffing with scissors or needles. It also is the third way of the decision of a problem. Coming back to our metres and decimeters, math it can be written down so:

22 metres and 2 decimeters = 21 metre and 12 decimeters

(21 metre and 12 decimeters) – (1 metre and 8 decimeters) = (21 metre – 1 metre) and (12 decimeters – 8 decimeters) = 20 metres and 4 decimeters

We with two different units of measure carry out two mathematical actions of subtraction simultaneously, in the same way, as two-nuclear processors Intel for computers work. I think, after such decision in your writing-book mathematicians will start to go nuts. Though so all calculations practically become. In the mathematician it is accepted to say that it is necessary to execute at first subtraction of decimeters, and already then subtraction of metres. Basically, mathematicians are right. So it is heavier to get confused. At subtraction of decimeters you can always steal one metre from metres if decimeters do not suffice you. Then from the remained metres subtract metres. Only do not forget about one metre if you have already dissipated it in decimeters.