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.

Chopsticks. Modulus is in mathematics. Mathematics For Blondes.
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

Trigonometric table in radians

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 cos
tan cot



Trigonometric table in radians. sin cos tan table in radians trig chart. Mathematics for blondes.
Trigonometric table in radians. sin cos tan table, sine cosine tangent chart, trig table radians. Mathematics for blondes.
Trigonometric table in radians. Tabel trigonometric sin cos tan chart. sin and cos and tan chart. Mathematics for blondes.
Trigonometric table in radians. Tabelul sin cos tan radian values chart. Tabela sen cos tg completa. Tabel cu valori trigonometrice. Mathematics for blondes.
Trigonometric table in radians. Tangent cosine sine chart. Tabelul trigonometric complet. Mathematics for blondes.
Trigonometric table in radians. Tabel trigonometric radian. Tabel trigonometrie radian. Mathematics for blondes.
Trigonometric table in radians. Trig value chart, sinus cosinus tabel. Mathematics for blondes.
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.

2.04.2012

Trigonometric table tangent cotangent in degrees

Trigonometric table tangent cotangent in degrees - it is the four-valued table of tangents and cotangents in degrees within one minute. A few examples are made at the beginning, as to use this table. For comfort in a table additional navigation elements are plugged with pointing by the name of the names of functions and values of minutes column-wise for every function. Value of degrees and minutes for every trigonometric function distinguished by different colors: for tangents - green, for cotangents - blue. Connotations for minutes are distinguished by yellow.

A trigonometric table for tangents and cotangents consists of two parts. The first part includes tangents from 0 to 75 degrees and cotangents from 15 to 90 degrees. Before to use this table, it is recommended on present examples to find the values of tangents and cotangents on a table and compare results. If all turned out correctly for you, means you can consider itself an experience user of table.

Trigonometric table tangent cotangent in degrees. Examples of the use of table. How to find a tangent and cotangent, tabel trigonometric tan cot. Mathematics for blondes. Tabelul trigonometric.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 0 to 15 degrees, a cotangent of value is from 75 to 90 degrees, trig chart tan, tg, cotan, cot, ctn, cotg, ctg. Mathematics for blondes. Table of values of trigonometric functions.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 15 to 30 degrees, a cotangent of value is from 60 to 75 degrees, trigonometry table tg, tan, cot, cotan, cotg, ctg, ctn. Mathematics for blondes.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 30 to 45 degrees, a cotangent of value is from 45 to 60 degrees, trigonometry table. Mathematics for blondes.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 45 to 60 degrees, a cotangent of value is from 30 to 60 degrees, tabel trigonometric. Mathematics for blondes. sin cos tan chart.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 60 to 75 degrees, a cotangent of value is from 15 to 30 degrees, trigonometry chart cotg, ctg, ctn, tg, tan, cot, cotan. sin cos tan chart. Mathematics for blondes.
At the use it is necessary to remember this table, that additional values one, two and three minutes for a tangent have a sign plus and at addition added and at deduction subtracted, for a cotangent they have a sign minus and at addition subtracted, and at deduction added. For verification compare the got result to the values of the same name trigonometric functions in the alongside located cells.

The second part of trigonometric table includes tangents from 75 to 90 degrees and cotangents from 0 to 15 degrees. To use this table some simpler, than by the first part.

Trigonometric table tangent cotangent in degrees. A tangent of value is from 76 to 79 degrees, a cotangent of value is from 11 to 14 degrees, tabel trigonometric tg, tan, cot, cotan, ctg, cotg, ctn. Mathematics for blondes. Table of values of trigonometric functions.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 79 to 82 degrees, a cotangent of value is from 8 to 11 degrees, trig chart, tabel tg ctg. Mathematics for blondes.Tbel tangenta.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 82 to 85 degrees, a cotangent of value is from 5 to 8 degrees, tg, tan, cot, cotan, cotg, ctg, ctn. Mathematics for blondes. Tabelul trigonometric.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 85 to 88 degrees, a cotangent of value is from 2 to 5 degrees,  trigonometry chart, tangent chart, tg ctg tabela. Mathematics for blondes. Trigonometric functions table cot, tan chart.
Trigonometric table tangent cotangent in degrees. A tangent of value is from 88 to 90 degrees, a cotangent of value is from 0 to 2 degrees, trig chart tan, tg, cot, cotg, ctg, ctn. Mathematics for blondes.
For a tangent 90 degrees and cotangent 0 degrees a value is not certain. It is so accepted to consider because task on dividing by a zero mathematicians until now so not succeeded to decide.

If you need a trigonometric table of sines and cosines, then she can be found on a separate page.

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.

Trigonometric table of sines and cosines is in degrees and minutes. Values of sines 0 to 15 degrees and cosines 75 to 90 degrees. Trigonometry chart, trigonometric table from 0 to 90 degrees.
Trigonometric table of sines and cosines is in degrees and minutes. Values of sines 15 to 30 degrees and cosines 60 to 75 degrees. Trigonometry chart, trigonometric table from 0 to 90 degrees.
Trigonometric table of sines and cosines is in degrees and minutes. Values of sines 30 to 45 degrees and cosines 45 to 60 degrees. Trigonometry chart, trigonometric table from 0 to 90 degrees.
Trigonometric table of sines and cosines is in degrees and minutes. Values of sines 45 to 60 degrees and cosines 30 to 45 degrees. Trigonometry chart, trigonometric table from 0 to 90 degrees.
Trigonometric table of sines and cosines is in degrees and minutes. Values of sines 60 to 75 degrees and cosines 15 to 30 degrees. Trigonometry chart, trigonometric table from 0 to 90 degrees.
Trigonometric table of sines and cosines is in degrees and minutes. Values of sines 75 to 90 degrees and cosines 0 to 15 degrees. Trigonometry chart, trigonometric table from 0 to 90 degrees.
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.

11.08.2011

Table of squares of natural numbers

Table of squares of natural numbers from 1 to 100. Exponent chart 2. Mathematics For Blondes
In the table of squares of natural numbers the squares of numbers are presented from 1 to 100. Index of exponent of number as a number two writtenin on the right in the top of that number that is involuted. If a number is erected in a square, it means that this number needs to be increased on a such is exact number.

For example, ten in a square equals one hundred, because if ten to multiplication on ten, for us will turn out one hundred. Five in the second degree equals twenty five, because all know that five at five "again twenty five". And two in a square it not that another, "as twice two four".

Why was such thought of is a exponent of number? For short of record of mathematical expressions, whatever to repeat several times multiplication of identical numbers. If at erection of number in the second exponent we save a quite small place, then at erection of numbers in great exponents the economy of place turns out rather significant.

By the feature of squares of numbers, as well as all other even degrees of number, there is that after involution negative number a positive number ensues. In fact minus on minus gives a plus at an multiplication, how many times we this focus did not repeat. A negative number is in the second, fourth, sixth, eighth et cetera degrees gives a positive number as a result.

10.27.2011

Multiplication and division by zero

As a zero is not a number, all mathematical operations on multiplying and dividing by a zero take place in area of units of measurements. In relation to the operation of divizion by zero of unit of measuring can be real and virtual. Units of measurements of length behave to the real units of measurements. All other units of measurements, probably, are virtual. Dividing by the zero of virtual units of measuring is impossible, as a result of division by zero of such units of measuring does not make sense.

In the special group it is necessary to distinguish natural unit of measurement of speeds (speed of light) and mathematical unit of corners (corner in 45 degrees). These units of measurements hatch through mathematical methods and their mathematical properties require a further study. More detailed study is required similarly by units of measurements of time.

Virtual units of measurements appear as a result of process that mathematically can be written down as dividing of zero by zero.

0/0=1а

where а – is virtual unit of measurement.

The described mathematical properties of virtual units of measurements allow to enter any units of measurements us and use them without influence on the surrounding world. These units of measurements are used both for description of surrounding reality and for everyday needs. The examples of virtual units of measurements can be units of measurements of money, temperature, many physical sizes or applied in a technique and commerce. The process of exit from everyday life of virtual units of measurements of measuring can be mathematically represented as multiplying by a zero. Mathematical properties of similar units of measurements are tested by practice of their use during many millenniums.

In multidimensional space, division by zero increases the amount of the spatial dimensions, multiplcanion by zero diminishes this amount.

In rectangular cartesian coordinates it will look so:

x/0 = xy
xy/0 = (x/0)y = x(y/0) = xyz


At multiplication by zero it is necessary to take into account project properties of space, as a result of such increase depends on that, which one component is multiplied by zero.

xyz*0 = 0 and xy or xz or yz
xy*0 = 0 and x or y

In physical equalizations, division by zero requires introduction of new unit of measurement to examined by equalization the physical co-operation expressed by a mathematical action by an increase (probably, another unit of measurement of length). For example, if to divide unit of measurement of length into a zero, unit of measurement of area will ensue. If to divide unit of measurement of area into a zero, unit of measurement of volume will ensue et cetera.

m/0 = m²
m²/0 = m³


Algebraically it can be presented in a next kind:

a/0 = ab
ab/0 = abc


where а, b, c - mutually perpendicular units of measurements of length.

At multiplication by a zero one of the components of co-operation, described by physical equalization, from co-operation is eliminated. The primary result of co-operation grows into a zero. Remaining components continue to co-operate.

m³*0 = 0 and
m²*0 = 0 and m

Algebraically it can be presented in a next kind:

abc*0 = 0 and ab or ac or bc
ab*0 = 0 and a or b

where а, b, c - mutually perpendicular units of measurements of length.

Expl for blondes: It only began division and multiplication by zero. More in detail we will consider it other time.

10.26.2011

Space evolution

Probably, in the process of the evolution, space generates different multidimensional universes with the even amount of dimension. Development originates from spaces with less of dimension to spaces with plenty of dimension. For basis of existence of space it is possible to accept principle of existence of speeds. Our 6-dimension Universe in the chain of space evolution will look like the following.

Space evolution. Mathematics for blondes. Nikolay Khyzhnjak.
Quite possible, that to Big Bang putting beginning of our Universe, there was a 4-dimension universe in that energy had one dimension of length, and a matter two dimension of length in that part of universe that corresponds our slower-than-light. A black hole, the consequence of that in 6-dimension space was Big Bang, giving beginning of our Universe, appeared in the process of evolution of this 4-dimension universe. A matter with two dimension of length in our Universe grew into 2-dimensional energy. Question about transformation of unidimensional on length energy in the process of transition through a black hole, remains open.

In the process of evolution of our Universe energy with two dimension of length partly passes to the matter with three dimension of length. A matter generates in our Universe black holes that give beginning to the new universes in the eightmeasured space. After Big Bang in the eightmeasured space our matter with three measuring of length grows into energy of the eightmeasured universe. Et cetera. The process of space evolution can develop for ever and ever.

During an evolution one 4-dimension universe generates the great number of 6-dimension universes. In turn every 6-dimension universe generates the great number of the 8-dimension universes. It look like the process of spawning. All 6-dimension universe, generated by one 4-dimension universe, can be in the mutually perpendicular dimension, that eliminates their cross-coupling on each other. Each of universes is mapped to all other universes as a point. The 8-dimension universes can be formed just.

In regard to faster-than-light part of universes of any type the theory of symmetry deserves attention in relation to velocity of light. Energy and matter of slower-than-light part freely move in three dimension of length and hardly fixed in the continuous stream of three dimension of time. Dark energy and dark matter freely move in three dimension of time and hardly fixed in the continuous stream of three dimension of length. Even if it not so, possibility of existence of universes of similar type does not need to be thrown down from accounts.

A transition process from a black hole to Big Bang requires an additional study. Mathematically he can be described by the operations of multiplication and division by zero. There are grounds to suppose that the trigger mechanism of gravitational collapse, resulting in appearance of black hole, is space geometry. This question will be considered additionally.