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Showing posts with label sine. Show all posts
Showing posts with label sine. Show all posts

## 2.25.2017

### Law of sines

The law of sines (sine law, sine formula, or sine rule) is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles.

 The law of sines
Where:
a, b, c - are the lengths of the sides of a triangle;
α, β, γ - are the opposite angles;
T - are the area of triangle;
R - are the radius of the triangle's circumcircle.

How to use this monster? Use dress-making courses. Cut the law of sines on a part.

 Cut the law of sines on a part
Make a necessary formula of two parts. Use properties of proportions.

 Example of use of the law of sine

## 2.08.2017

### 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.

## 8.03.2016

### Terminating trigonometric functions

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Three main types of trigonometric functions

Lesson 4

TERMINATING TRIGONOMETRIC FUNCTIONS

The sine and cosine are rather well studied, their values cannot be more unit. If to divide elements of a rectangle into length of diagonal, lengths of the parties will accept values of a sine and a cosine. Also the parties of a rectangle are diagonal projections in the perpendicular directions.

 Sine and cosine

Names of all trigonometric functions depend on the line of the beginning of measurement of a angle. The same line defines the direction of projection. Dependence between legs and a hypotenuse in a rectangular triangle, known as "Pythagorean theorem" (for length units of measure, not the bound to a hypotenuse) or "Pythagorean trigonometric identity" (when hypotenuse length is accepted to a unit of measure of length), is an integral part of properties of a rectangle.

If we project simple diagonal on the parties of a rectangle, then we will receive two projections of diagonal expressed through different angles. If we project the same parties on diagonal, then we will receive the diagonal length as the sum of two projections of the parties.

 Pythagorean trigonometric identity

The Pythagorean theorem is a dependence between diagonals and the parties of a rectangle.

At the following lesson we will consider
The infinite trigonometric functions

## 11.12.2015

### The sine values

The unit circle can be considered not only for individual trigonometric functions, but also under the microscope each function separately. Here are the sine values for different angles.

 sin 0

 sin 30
 sin 45
 sin 60
 sin 90
 sin 120
 sin 135
 sin 150
 sin 180
 sin 210
 sin 225
 sin 240
 sin 270
 sin 300
 sin 315
 sin 330
 sin 360

The unit circle is like a movie that shows us mathematics. Here you see some pictures of the cult TV series.

## 11.11.2015

### Unit circle

Mathematicians consider themselves clever and all the rest. But not all of us are as smart as mathematics. The unit circle mathematics invented for themselves. For those who are just beginning to study mathematics, I suggest a simpler version - separately cosines and sines separately.

If we remove from the unit circle all that relates to the sine, we get the unit circle cosines.

 Unit circle cosines

If we remove from the unit circle everything about cosines, we get a unit circle sinuses.

 Unit circle sinuses

Now, you will not confuse the values of sines and cosines.

## 9.29.2012

### Sine x with minus

We must find a sine x with minus, a value is equal -0.8453

sin x = -0.8453

On the table of sines we find the value of corner in degrees and minutes for x, equal +0.8453. This corner is in limits from 0 to 90 degrees and equal 54 degrees 42 minutes.

sin a = +0.8453

a = 57 degrees 42 minutes

Farther we look a trigonometric circle and find a corner a. The positive values of sine are located in the overhead half of circle, negative values will be situated in the underbody of trigonometric circle.

One value of sine always has two values of corner. We will find these corners for the negative value of sine.

The first corner we will get, if to the corner a we will add 180 degrees. The second corner we will get, if from 360 degrees we will take away a corner a.

x1 = 180 degrees + a

x1 = 180 degrees + 57 degrees 42 minutes

x1 = (180 + 57) degrees 42 minutes

x1 = 237 degrees 42 minutes

x2 = 360 degrees - a

x2 = 360 degrees - 57 degrees 42 minutes

x2 = 359 degrees 60 minutes - 57 degrees 42 minutes

x2 = (359 - 57) degrees (60 - 42) minutes

x2 = 302 degrees 18 minutes

This was explanation, how to find a sine x with minus. Now a decision needs to be written down in accordance with bureaucratic rules that I do not know. It looks approximately so.

sin x = -0.8453

x = arcsin (-0.8453)

x1 = 237 degrees 42 minutes

x2 = 302 degrees 17 minutes

## 2.28.2012

### Modulus of sine

Me it was asked to show method to simplify trigonometric expression, containing the sum of sine and modulus of sine of corner, knowing that corner alpha is finished in 4 fourths. This expression looks so:

|sinA|+sinA

At once I will say honestly, that I a concept do not have, as such expressions are simplified. But about the module of sine I can tell and that will turn out in the total, then. All of you know well, that a sine, as well as all trigonometric functions, can take on positive and negative values. So, sine in the Chinese sticks, that in mathematics read as a "module of sine of corner A", can not have negative values, only positive. When mathematicians are fastidious to touch to the negative numbers, they apply these chopsticks (or modulus of number), as condom at sex, that to be not infected by minus. They rescue the life these, as all numbers in the module from negative grow into positive.

Well now a bit about sign life of sine of corner of А. Sine - it for us upwards and downward on an axis Y-mill from unit to minus units. When corner A from 0 to 180 degrees take on values, all sines of these corners are positive. In this case the chopsticks of the module are the superfluous measure of caution and they can be cast aside. In this range of values of corner A our expression will assume an air:

|sinA|+sinA = sinA+sinA = 2sinA (0 < A < 180)

If value of corner A to increase farther, from 180 to 360 degrees, values of sines of these corners will be negative, id est with a sign " minus". In this case the module begins to play the rock role the fate of our mathematical expression. The value of sine with the module remains positive, and the value of sine without the module becomes negative, as well as it is fixed to all decent sines. What will we get, if from a number we will take away a such is exact number? Correctly, zero. Our expression dies out, as dinosaurs. By the way, if all people always will use condoms during sex, humanity here will die out fully. Effect of the modulus. We will look, that takes place with our expression in this case:

|sinA|+sinA = sinA-sinA = 0 (180 < A < 360)

Application of formulas of bringing trigonometric functions over will give an exactly such result. Thus the module compels us to cast aside in garbage all signs minus, got as a result of transformations.

At corners 0, 180, 360 et cetera degrees our expression will equal a zero, as a zero the values of sine of these corners are equal to.

As all of it it is correct to write down in complete accordance with the rules of mathematical bureaucracy, I do not know. But sense of what be going on, I hope, clear you and you without effort will design this expression in the best kind.

## 2.14.2012

This trigonometric table is made for the values of corners in radians. Radians are here given as decimal fractions within two signs after a comma. Value of sine, cosine and tangent given within four signs after a comma. It is such small trigonometric table in radians.

## Table of trigonometric values in degrees:sin costan cot

In this trigonometric table the value of corner in radians closes on a 3,15 radian, that corresponds hardly anymore 180 degrees in the degree measure of corners. Here you will not find the value of tangent, equal to unit, value of sine, equal to unit and value of cosine, equal to the zero. In the radian measure of corners to get these values unassisted number of Pi it is impossible. And as a self number of Pi is an endless shot not having the exact meaning, expediency of goniometry in radians is very doubtful. Radians - it, put it mildly, strange unit of measurement.

Values of corner in radians are in blue columns mark the letter of "X". In three columns the values of sin x are given on the right, cos x and tg x for corners in radians. Value of cotangent, secant and cosecant to the table not driven, as these trigonometric functions are reverse shots to driven to the table. For the receipt of values of ctg x, sec x and cosec x in radians, it is needed to divide unit into a tangent, cosine or sine of corresponding corner in radians.

## 1.10.2012

### Trigonometric table of sines and cosines

The trigonometric table of values of sines and cosines within one minute is counted on blondes. For comfort of the use for sines and cosines distinguished this table of value of corners by different colors. For sines the blue color of cells is accepted with degrees and by minutes. For cosines the green background of cells is accepted. A yellow background is distinguish the values of minutes that if necessary is added or subtracted from tabular values.

Usually it is not accepted so in detail to give a navigation on a trigonometric table. Firstly, a table is counted on experience users by mathematics. Secondly, publishers from old times produce these tables in a blackly white variant and in every way save printer's ink on a navigation on a trigonometric table.

I hope, such registration will not give to lose way you even in the middle of this table and you will not entangle sines with cosines at the search of their values. By the way, the trigonometric table of sines and cosines presents the values of these trigonometric functions for corners from 0 to 90 degrees. For other values of corners can use a trigonometric circle as a crib.

If you need more exact calculation of values of trigonometric functions, then can take advantage of calculator. How to use the table of Брадиса, we will understand next time. Specially for those, whoever knows.

## 2.09.2011

### 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.

## 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:

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

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.

## 8.15.2010

### Sine is 30 degrees, sin 30

Sine 30 degrees are evened by the one second or zero whole five tenth.

sin 30° = 1/2 or sin 30° = 0,5

In the radian measure sine of corners 30 degrees correspond a sine π/6:

sin 30° = sin π/6

Oddly enough, but justly and reverse equality, which asserts that sine π/6 (sin pi/6) equal a sine 30 degrees:

sin π/6 = sin 30°

A sine pi / 6 is evened similarly one second or zero whole five tenth.

sin π/6 = 1/2 or sin π/6 = 0,5

It was for blondes. For brunettes and bald academicians of mathematical sciences it is possible to write down all of it in a general view, let untangle:

sin 30° = sin π/6 = 1/2 = 0,5

For complete happiness here obviously the picture of sine 30 degrees is not enough. Surprise! And she:

 Sine is 30 degrees

I hope, the first part of task I decided and I succeeded to explain to the blondes, what the sine of thirty degrees is equal to. It is now needed to decide the second half of task, with which all academy of mathematical sciences, together taken, is unable to manage even. It is needed in the Internet to find blondes which search a sine 30 degrees. Will I try to be armed with logic of blondes and I will transfer below searching queries which blondes can enter in the searching systems at the search of answer for a question: what is a sine equal to 30 degrees? And so, searching queries, dilute my comments, in order the searching systems did not separate my creation from blondes.

Sine - it mathematicians collate the knowledge with the Internet. Sine, cosine - the authority of mathematicians appeared, to check, as mathematicians collated the knowledge. A sine of corner - is botanists spectacled, future of Bills Gatesis, comb the clever turnips and try to remember the school course of mathematics. Sine of degrees - smart schoolboys pelt the question on the way, what quicker cut with this nasty thing and continue a game. Table of sines, tangents - conscientious schoolboys and respectable brunettes scraped away all cognitions in trigonometry and try to collate them with the Internet. A value of sine - is mathematicians, after the long wandering on sites with blondes, at last understood how it is better to formulate a searching query. Values of cosines and sines - mathematicians remembered suddenly, that by a not sine single trigonometry lives by. Is a sine what equal to? - and the first signs of presence of blondes in the Internet with their appealing spontaneity of intercourse, even with a computer. Sines and cosines of corners. Table of values of sines. The sine of corner is equal - it mothers try to check up, as their children did lessons. How to find a sine? - it is already the typical question of the confused blonde. Sine of acute angle. Sine and cosine 30 - are mothers hardly, but already begin to understand that requires to be found in a task. A sine of number - is poor mothers, they are even unaware that corners can be measured by radians. A sine is alpha - mothers remembered, by what letter once at school they designated corners. Calculator of sines - for help clever dads come helpless mothers. How to find the sine of corner? - clever dads found the calculator of sine, it is now needed to know, how to use this thing. Geometry is a sine, cosine. To get the table of sines. Values of sines of corners - it children unstucked from the virtual games and try to prove to the clever dads, that dads not correctly press on the buttons of calculator, because the sine of corner can not equal three jars of beer. Calculations of sine - mathematicians-theorists try to steal job of mathematicians of the applied sciences performances. To calculate a sine - it the mathematicians of the applied sciences steal job of the colleagues performances. Sine of endlessness - it already physicists try to check up the calculations of mathematicians. Being of sine - blondes try to know, where sines live. Trigonometric sine - all discovered in surprise, that sines were not only in mathematics.

How many sine 30 - is a zero whole five tenth. How many sin 30 is equal - so much, zero five. How many will a sine be 30? - it is a question, certainly, interesting.. ышт 30 - it is possible so, it is only better to be commuted into English language. A table is sin alpha of 30 degrees - not quite table, but a sine and 30 degrees are exactly. Numbers from which sines natural are - mathematicians, a blonde put question. Are there variants of answers? Honour of full-dress uniform is put on kitty.

Who can explain why sine 30 degrees 1/2 is equal? - well, infant prodigies, blonde put question. Who will dare answer? A prize fund of competition is three dollars, I throw down on the mobile telephone number indicated by a winner or on the purse of WebMoney.

## 5.18.2010

### Trigonometric circle sine cosine

The trigonometric circle presents the values of trigonometric functions sine (sin) and cosine (cos) as co-ordinates of points of single circumference at the different values of corner alpha in degrees and radians.

 Trigonometric circle sine cosine
As I always become confused during translation of co-ordinates of points of circumference in sine and cosine, for simplicity all values of cosines (cos) for corners from 0 to 360 degrees (from 0 pi there is to 2 pi rad) are underline a green hyphen. Even at unsealing of this picture of trigonometric circle on a not coloured printer all values of cosine will be underline, and values of sine will be without underlining.

Opposite the indicated corners on a circumference points are located, and the co-ordinates of these points are indicated in parentheses. The coordinate of Х is writtenin the first.

Let us conduct a survey excursion on this corner of mathematical zoo. Foremost, it is needed to mark that is here present the Euclidean system of coordinates is Cartesian is one black horizontal line with the letter of Х near a pointer, second is a vertical with a letter Y. On the axis of Х, which is yet named abscise axis (this clever word of mathematics was thought of specially, what to tangle blondes) cosine live, - cos. On an axis Y, which is named y-axis (another clever word which in the mouths of blonde can become a killing weapon), sine live - sin. If to look at domestic life of these trigonometric functions, then it is not difficult to notice that sine always on a kitchen at a flag for vertical lines, and cosine - on a sofa before a television set on a horizontal.

In this system of coordinates a circumference is drawn by a radius, equal to unit. A centre of circumference is at the beginning of the system of co-ordinates - wherein abscise axis (axis of Х) and ordinates (axis of У) intersect in the centre of picture.

From the center of circumference thin hyphens which show corners 30 are conducted, 45, 60, 120, 135, 150, 210, 225, 240, 300, 315, 330 degrees. In the radian measure of corners this pi/6, pi/4, pi/3, 2pi/3, 3pi/4, 5pi/6, 7pi/6, 5pi/4, 4pi/3, 3pi/2, 5pi/3, 7pi/4, 11pi/6. With the axes of coordinates such values of corners coincide: 0, 90, 180, 270 degrees or 0 pi, pi/2, pi, 3pi/2. Using a picture, it is very simple to transfer corners from degrees in radians and from radian in degrees. Identical values in the different systems of goniometry are written on one line, representing this corner.

The lines of corners end with points on a single circumference. Near every point, in round собках, the coordinates of this point are writtenin. The coordinate of Х, which corresponds to the cosine of corner, forming this point, is writtenin the first. The coordinate is writtenin the second Y this point, that corresponds to the value of sine of corner. On a picture easily enough to find a sine and cosine of the set corner and vice versa, by set value of sine or cosine, it is possible easily to find the value of corner. Mainly, not to entangle a sine with a cosine.

I watch out for circumstance that if you by value search a sine or cosine corner, it is necessarily needed to finish writing the period of corner. Mathematicians very athrob behave to this appendicitis of trigonometric functions and at his absence can stick in two after, it would seem, right answer. What period at determination of corner by value to the trigonometric function? It is such piece which is thought of mathematicians specially in an order to be tangled and tangle other. Especially blondes. But about it we will talk somehow other time.

All, that it is collected in a small group on the picture of trigonometric circle of sine and cosine, it is possible attentively to consider on separate pictures with the portraits of sine 0, 30, 45 degrees (reference to the separate pages I will add as far as the increase of photo gallery of sine and cosine).

Automatic translation from Russian.

On this page you will find: cos sin tabel 30, 45, 60, 90. Function for student: sin60, sine cosine of 0 45 30 60 90 180 270 360. Table 2 pi to 360 values of trigonometric identities. Table of values of sine function, sin cos 60 30 45. Table of degrees to radians 0 to 2pi with sines and cosines. cos sin pi for student: 0pi, pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, 5pi/6, pi, 7pi/6, 5pi/4, 4pi/3, 3pi/2, 5pi/3, 7pi/4, 11pi/6, 2pi. sin pi/4 table values, sin = 3pi / 2/ Tabel trigonometri 360 derajat, the value of sin 3pi/2 - cos pi/3. Trigonometric circle illustration.

How to find a decision:

Sines and cosines are a circle - here picture in all trigonometric beauty.

Corner 120 degrees in radians - equal 2/3 pi or 2 pi divided by 3, it is very beautifully drawn on a picture.

Values of sines of cosines of corners are in radians - there are such on a picture, I hope, exactly those corners that you search.

Value of cosine of corner in 45 degrees - equal a root is square from two divided by two, can check on a picture.

Trigonometric circumference - I am not quite sure that the circumference presented on a picture is trigonometric, but something from trigonometry in this circumference there are certainly, for example, sines and cosines on a circumference is the outpoured trigonometry.

A trigonometric circle is a picture - I am here such. Indeed, most not beautiful picture, it is possible to draw much more beautiful and clearer. To me minus in reputation - why did not I until now draw him for blondes? you present a situation in the art gallery of the future : a tour guide explains to the group of schoolchildren "Before you known worldwide picture "Trigonometric Madonna with an unit segment on hands" is a picture of genius artist of Early Mathematical Renaissance ." age. Farther she names the name of this artist (he or she).This name can be your!

Circle of sines and cosines - a just the same circle quite by chance appeared here on a picture.

Corner 9 degrees how many it in pi - in pi it 1/20 or pi/20.
Decision: for translation of degrees in pi radian, it is needed to divide present for us degrees into 180 degrees (this 1 pi is a radian). 9/180 = 1/20 turns out for us.
Answer: 9 degrees = 1/20 pi.

Unit circle degrees and radians marked for units of pie - it here in blonde math.

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