10.15.2010

Legend about Sine and Cosine (continuation)

Beginning of Legend about Sine and Cosine

An anxious new that Sine and Cosine go to Mathematics, flew around the whole worlds quickly. Gods were in confusion. To this case they worked out all nascent problems. But now a problem arose up in the world of Numbers, and Numbers are not subject to Gods. Only Mathematics can establish order among Numbers.

Nobody of Gods never saw Mathematics all at once. Nobody knew exactly, that in their ideas about Mathematics is reality, and that by the figment of the imagination. All Gods sooner or later tried to know at Numbers, how correctly they understand one or another moment in the mathematical reasoning. In reply to all questions of Number enigmatically smiled and always pronounced a the same phrase: time will "Come, you will answer the question". Really, time passed and all became into the places. Some deductions were confirmed by life, some was dispersed, as fog. During all life of Gods their idea about Mathematics changed constantly.

And Gods got an exceptional case face to face to see Mathematics. She does not can to say no in attention to the faithful servants - to Sine and Cosine. Gods have a right to be present as the interested observers. From the decision of Mathematics will depend, in what world to live Gods farther.

When Sine and Cosine appeared before Mathematics, all looks of Gods were tied down only to Her. But Gods were waited by great disappointment - each of them saw that character which was by it created only. To not the least new detail.

What was seen to Mathematician by Sine and Cosine? Will pass many to time, before someone from Gods, possibly, will answer this question.

"That did bring you over to me?" - Mathematics asked at Sine and Cosine.

"We want, that you said, who of us main," - they pronounced.

"There are not main in family of Numbers. All Numbers are equal," - Mathematics repeated all the known truth.

"It is but as Unit?!" - Sine and Cosine exclaimed amicably.

"It is Unit such is exact Number, as all other numbers. Yes, at Unit the special status. But this status needs to be deserved. You know perfectly, to what Numbers and what this status is appropriated for," - Mathematics answered imperturbably.

"Then say to us, who is main from us two? We all the time argue and can not decide in any way," - stubbornly repeated over and over again Sine and Cosine.

"Reasonable creatures always can agree inter se. Unfortunately, you such are not," - Mathematics pronounced with bitter taste.

All Gods stopped beating in a fright. To declare reasonable creature unreasonable is means to deprive his rights on life. Did mathematics take away capital punishment to two representatives of immortal family of Numbers?! Such decision was not laid in consciousness of Gods. To give Numbers in power of Death is equipotent to capital punishment to the whole surrounding world!

"Do you want to be equal to unit? Do you want, that one of you always was more main than other? Do you want, that I stopped your dispute?" - Mathematics continued after short meditation. Anger was heard in voice of Mathematics, - "You are two capricious babies, which did not manage to become adults. Your place is in a corner," - Mathematics was slightly softened, Gods breathed with relief, - "Here my decision:"

sin2α + cos2α = 1

Gods stopped beating in bewilderment. They looked around on parties, trying to understand, what Mathematics did. Seemed, the world had remained former, but almost elusive differences were felt in everything. Gradually to Gods sense of this event began to come. Mathematics sentenced Sine and Cosine to perpendicular symmetry! Such usual to all World of Symmetry disappeared forever. The new world - World of Perpendicular Symmetry appeared.

Сontinuation will be

10.14.2010

A sine is 0 degrees, sin 0

The sine of zero of degrees equals a zero. On the picture of sine it looks so:

A sine is 0 degrees, sin 0 pi. Mathematics for blondes.

Will you, certainly, ask: "And where, actually, sine on this picture"? And there is not he, he in a tiny hole from a zero hid. As a baby mouse in a burrow. To see a baby mouse, it is needed to trick out of him from a burrow. In the animated cartoons assert, the smell of cheese very helps on this business. Sine will not beckon cheese. But there is one piece which on a sine operates smoothly. This magic lure is named for sines is a corner. Not that which babies are put in, and that which in degrees or radians is measured. Let more attentive on we watch this hunt on sines.

We do not have a corner (a corner equals a zero) - there is not a sine.

sin 0° = sin = 0

Now we will try to beckon a sine the littlest value of corner. We will look, as a sine will react on a corner in the zero of degrees, zero of minutes, one thousandth seconds:

sin (0° 0' 0,001") = 0,00000000484813681109

Do you see, the tag of curious spout appeared from a burrow? We will try to increase tenfold our lure and we will take a corner in one hundredth seconds.

sin (0° 0' 0,01") = 0,00000004848136811095

The amount of zeros before numbers grew short on one, and five appeared in the end. Certainly, in a burrow hidden yet very much цифер, which at a desire can be seen. It rather is a boa with a long tail from numbers. Yet tenfold we will increase a corner.

sin (0° 0' 0,1") = 0,00000048481368110954

Did you notice that numbers not nearly changed after zeros? It does not mean that a sine, as well as corner, increases exactly tenfold. Somewhere there, in distance from a comma, numbers change - a baby mouse moves a tail, but we it do not see. We watch only the first twenty numbers after a comma.

Here now for us unique possibility to admire the sine of one second in all his beauty (more precisely, his first two ten of numbers) appeared:

sin (0° 0' 1") = 0,00000484813681107637

We will look farther, as the first ten numbers change after a comma for 10, 20, 30, 40 and 50 seconds (quite naturally, that a superfluous tail we round off) :

sin (0° 0' 10") = 0,0000484814
sin (0° 0' 20") = 0,0000969627
sin (0° 0' 30") = 0,0001454441
sin (0° 0' 40") = 0,0001939255
sin (0° 0' 50") = 0,0002424068

It is possible to consider that for the sake of one minute a sine already fully abandons the burrow and begins quickly to hurry round us. You only will look at a sine 10, 20, 30, 40 and 50 minutes:

sin (0° 1') = 0,0002908882 
sin (0° 10') = 0,0029088780
sin (0° 20') = 0,0058177314
sin (0° 30') = 0,0087265355
sin (0° 40') = 0,0116352658
sin (0° 50') = 0,0145438977

I hope, you understand now, that when a corner arrives at all one degree, a sine becomes quite great. A little baby mouse grows into an adult mouse. Look, as quickly the sizes of sine change for corners in 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 degrees:

sin 1° = 0,017452
sin 2° = 0,034899
sin 3° = 0,052336
sin 4° = 0,069756
sin 5° = 0,087156
sin 6° = 0,104528
sin 7° = 0,121869
sin 8° = 0,139173
sin 9° = 0,156434
sin 10° = 0,173648

Some values of sine of corner alpha on the requests of visitors:

sin 17° = 0,292372

If you watching was not yet tired of a sine, then I suggest to pass to the page a sine 30 degrees. He can be not only seen there but also touch hands, at a desire.

10.12.2010

That will be, if to increase a cosine on a sine?

In someone an inquisitive mind... Woke up If sine of corner alpha to increase on the cosine of corner alpha, a number, equal to the half of sine two alpha, will ensue. This statement hatches from the functions of multiple corners, where sine two alpha equals the doubled product of sine alpha on a cosine alpha. The picture of this trigonometric miracle I will show later:)

10.11.2010

Sine is 45 degrees, sin 45

Sine 45 degrees, or sine, pi divided on 4 (four), equal unit, divided by a root from 2 (two). Simpler all sin to represent 45 degrees on a picture:

Sine is 45 degrees, sin 45, sine of pi on 4, sin  pi/4. A value of sine of 45 degrees. Mathematics for blondes.
Sine is 45 degrees, sin 45

In decimal fractions value of sine 45 degrees will be equal zero whole seven thousands seventy one tenthousandth:

sin 45° = sin π/4 = 0,7071

Anymore about trigonometric equalities of sine of corner alpha it is possible to look on a page, sanctified to the sine 30 degrees. I hope, you will understand, what numbers change in equalities, and what letters and symbols remain unchanging. You will compare the portraits of sines on pictures, value of corners and value of sines of these corners.

Now a bit searching phrases are for blondes, that they did not lose way in the searching systems. On this page you will find answers for such queries: sin 45 degrees, sin пи/4, what a sine is equal to 45 degrees, value of sines, value of sin, sin пи on 4, sin pi/4. Sine 45 degrees equal 1 divided by a root from 2. Value of sine, a sine what is equal to, how to find a sine, sine of acute angle - specially for blondes this page is created. Sine of number of pi on 4 radians, sine alpha of corner 45 degrees. Calculations of sine, even looking out a sine on a picture. Exhibit as a sine 45 degrees it is settled to touch hands. Before procedure to wash down being of sine of hand with soap.

Trigonometric sine - it just one of a few functions from family Functions Trigonometric. Where is pi divided by four? - in a district 45 degrees. Above there is a picture, it is drawn there, how to him to pass. Sine of pi divided by four - really, sexual belonging of number of pi somehow is not strongly paraded in mathematics. So nevertheless, pi - it he, she or it? The strangest, all answers correct. He is a corner of pi, she is pi, it is a number of pi. sin 45 what equal to? - to unit, divided by a root from two.

Devidend of fraction as named?

The devidend of fraction is named numerator. A numerator is always written above a fractional line. It is needed nowhere to peep in search of numerator of fraction, type under a bed. A numerator of fraction is always on the most prominent position, from above fractional line, as a prince on a pea. As a numerator there can be numbers or mathematical expressions.