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Showing posts with label cosine. Show all posts
Showing posts with label cosine. Show all posts

## 3.02.2017

### Inverse transformations

Last time we transformed the law of cosines and a Pythagorean theorem to the sum of a line segments. Now we will execute inverse transformations.

 Inverse transformations

In inverse transformations I made everything very simply. The minus sign from a formula disappeared. The problem is that we are not able to measure angles correctly. Than differ a angle 0 degrees from a angle of 180 degrees?

 Measurement of a angle
It is possible to assume that if a line segment one, then a angle is equal 0 degrees. If a line segments two, then a angle is equal 180 degrees.

The transformations executed by us show that the mathematics has no separate areas of mathematics: "arithmetic", "algebra", "geometry" or "trigonometry". The mathematics is a single whole.

The mathematics is DNA of the nature. Further we will continue to study a cosine gene in a cosine law.

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## 2.27.2017

### We use the law of cosines

I was always interested in a question: how the Pythagorean theorem turns into the sum of a line segments? What I speak about? Here you look.

 The Pythagorean theorem and the sum of a line segments
In geometry everything is very prime. The first time we draw a right triangle and we write down a Pythagorean theorem. The second time we draw two a line segments and we write down the sum of a line segments.

 Right triangle and two a line segments
How one formula turns into other formula? To see it, we use the law of cosines. We will draw the picture, we will write down conditions, we will execute transformations.

 Triangle and the law of cosines
 Right triangle and the Pythagorean theorem
 The law of cosines and sum of two a line segments
We use the law of cosines and turned the Pythagorean theorem into the sum of two a line segments. Further we will consider an inverse transformation.

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## 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

## 4.01.2016

### Law of cosines in general

To present the law of cosines in general need to go back to the beginning. If the expression for one side of the triangle the length multiplied by the same hand n-times, the equation does not change.

 Conversion formulas

If you add equal to three sides of the triangle, we obtain the law of cosines in general.

 Law of cosines in general

The law of cosines in general terms describes the relationship between the sides and angles of a triangle in a multidimensional space. If the given equation is satisfied, then the triangle is in the Euclidean space. Options description triangle in curvilinear spaces require further study. For the correct application of the law of cosines in solid geometry, spherical triangles side length should be unit of measurement in the same units of measurement, which is measured in planimetrics.

## 3.28.2016

### Degenerate triangle

Finally, we will check the law of cosines to the perimeter in the degenerate triangle. There can be two options. If we decrease the base of an isosceles triangle to zero, we obtain the two overlapping segments. The sum of the angles of this degenerate triangle is 180 degrees.

 Degenerate triangle

If we combine the upper vertex of the triangle with the base, we get the second type of degenerate triangles. This segment equal to the sum of the other two segments. The sum of the angles are also equal to 180 degrees.

 Degenerate triangle

A degenerate triangle - is the lower limit of the application of the law of cosines. After that we will look at the law of cosines in general.

## 3.24.2016

### Isosceles obtuse triangle

Now we will check how the law of cosines in obtuse triangle. For example, consider an isosceles obtuse triangle.

 Isosceles obtuse triangle

Ups! The perimeter of the triangle may be square roots. The law of cosines is not working? Do not jump to conclusions. The secret is revealed very simple. Let us express the base of the triangle through the sides and see what happens. Double-paste the resulting equation in our result. First time turvy-topsy, second time topsy-turvy.

 The perimeter of the triangle

The law of cosines to the perimeter works flawlessly. We have completed our review of the degenerate triangle.

P.S. 05/31/16 The онлайн игрыcalculator of triangles confirms the received result.

 To calculate a triangle

## 3.16.2016

### Right triangle

Now we will check the law of cosines to the perimeter of the example of a right triangle. At one corner of the right triangle is 90 degrees, the cosine of this angle is zero. Cosine of the other angles are obtained by dividing the length of the hypotenuse to the length of the adjacent side. In general, the law of cosines check looks like.

 Right triangle

Now we substitute in formula for the length of the sides of a right triangle and values of cosines.

 Right triangle

These values are equal to the perimeter of the triangle. Cosine law allows to check the triangle on the break. That's what happens when one of the sides does not reach the top of the triangle.

 Broken triangle

Equality is performed, but the result is not equal to the sum of the lengths of a broken line or the perimeter of the triangle. You can then proceed to test the law of cosines in the isosceles obtuse triangle.

## 3.15.2016

### Check the law of cosines

Check the law of cosines for the perimeter, we start with an equilateral triangle. All sides of an equilateral triangle are equal. All angles are equal to 60 degrees. The perimeter of the triangle is equal to three times the length of the sides.

 Equilateral triangle

Equality is performed. Now we check the law of cosines in a right triangle.

## 3.14.2016

### Law of cosines for perimeter

Carefully look at the proof of the law of cosines and make some corrections.

 Analysis of proof

Now we can write the law of cosines for the perimeter of the triangle.

 Law of cosines for perimeter

The result is a very simple and beautiful formula that describes the entire triangle. The law of cosines shows a relationship between the angles and the lengths of the sides of a triangle.

 The geometry of the law of cosines

Law of cosines for perimeter describes the perimeter of the triangle made up of one-dimensional Euclidean spaces. For multidimensional spaces cosine law has a different view.

## 3.13.2016

### Law of cosines

The triangle has three sides and three corners. Appearance cosine theorem depends on the received angles and sides of the triangle symbols. Here's how it looks in Wikipedia.

 Law of cosines

Three angle of the triangle gives three options for the formula of one triangle. In law of cosines can use a one formula, and three variants of symbols.

 Three variants of symbols

These two options allow to describe all sides and angles of the triangle. The traditional problems of mathematics we are taught to find one of the triangle elements.

Question: Can one formula with one variant of symbols to describe all the elements of the triangle?

Here's how to do it using the cosine theorem.

The proof of the theorem of cosines in the trigonometric form looks like.

 The proof using trigonometry

If you change the "minus" sign in the "plus" sign, we get the cosine theorem for the perimeter of the triangle.

Law of cosines in general form (in Russian).

## 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.

## 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.

## 10.12.2010

### 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:)

## 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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