Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, this can be represented as a rectangle in which one side denotes lettuce, the other side denotes water. The sum of these two sides will denote borscht. The diagonal and area of such a "borscht" rectangle are purely mathematical concepts and are never used in borscht recipes.
How do lettuce and water turn into borscht in terms of mathematics? How can the sum of two segments turn into trigonometry? To understand this, we need linear angle functions.
You won't find anything about linear angle functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work whether we know they exist or not.
Linear angular functions are the laws of addition. See how algebra turns into geometry and geometry turns into trigonometry.
Is it possible to do without linear angular functions? You can, because mathematicians still manage without them. The trick of mathematicians lies in the fact that they always tell us only about those problems that they themselves can solve, and never tell us about those problems that they cannot solve. See. If we know the result of the addition and one term, we use subtraction to find the other term. Everything. We do not know other problems and we are not able to solve them. What to do if we know only the result of the addition and do not know both terms? In this case, the result of addition must be decomposed into two terms using linear angular functions. Further, we ourselves choose what one term can be, and the linear angular functions show what the second term should be in order for the result of the addition to be exactly what we need. There can be an infinite number of such pairs of terms. IN Everyday life we do very well without decomposing the sum, subtracting is enough for us. But at scientific research the laws of nature, the decomposition of the sum into terms can be very useful.
Another law of addition that mathematicians don't like to talk about (another one of their tricks) requires the terms to have the same unit of measure. For lettuce, water, and borscht, these may be units of weight, volume, cost, or unit of measure.
The figure shows two levels of difference for math. The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the area of units of measurement, which are shown in square brackets and are indicated by the letter U. This is what physicists do. We can understand the third level - the differences in the scope of the described objects. Different objects can have the same number of the same units of measure. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same notation for the units of measurement of different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or in connection with our actions. letter W I will mark the water with the letter S I will mark the salad with the letter B- Borsch. Here's what the linear angle functions for borscht would look like.
If we take some part of the water and some part of the salad, together they will turn into one serving of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals will turn out. What then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we do not understand what, it is not clear why, and we understand very poorly how this relates to reality, because of the three levels of difference, mathematicians operate on only one. It will be more correct to learn how to move from one unit of measurement to another.
And bunnies, and ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar problem for adults. What do you get when you add bunnies and money? There are two possible solutions here.
First option. We determine the market value of the bunnies and add it to the available cash. We got the total value of our wealth in terms of money.
Second option. You can add the number of bunnies to the number of banknotes we have. We will receive the quantity movable property in pieces.
As you can see, the same addition law allows you to get different results. It all depends on what exactly we want to know.
But back to our borscht. Now we can see what will happen for different values of the angle of the linear angle functions.
The angle is zero. We have salad but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. Zero borsch can also be at zero salad (right angle).
For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This is because addition itself is impossible if there is only one term and the second term is missing. You can relate to this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so discard your logic and stupidly cram the definitions invented by mathematicians: "division by zero is impossible", "any number multiplied by zero equals zero" , "behind the point zero" and other nonsense. It is enough to remember once that zero is not a number, and you will never have a question whether zero is a natural number or not, because such a question generally loses all meaning: how can one consider a number that which is not a number. It's like asking what color to attribute an invisible color to. Adding zero to a number is like painting with paint that doesn't exist. They waved a dry brush and tell everyone that "we have painted." But I digress a little.
The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but little water. As a result, we get a thick borscht.
The angle is forty-five degrees. We have equal amounts of water and lettuce. This is the perfect borscht (may the cooks forgive me, it's just math).
The angle is greater than forty-five degrees but less than ninety degrees. We have a lot of water and little lettuce. Get liquid borscht.
Right angle. We have water. Only memories remain of the lettuce, as we continue to measure the angle from the line that once marked the lettuce. We can't cook borscht. The amount of borscht is zero. In that case, hold on and drink water while it's available)))
Here. Something like this. I can tell other stories here that will be more than appropriate here.
The two friends had their shares in the common business. After the murder of one of them, everything went to the other.
The emergence of mathematics on our planet.
All these stories are told in the language of mathematics using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, let's return to the trigonometry of borscht and consider projections.
Saturday, October 26, 2019
Wednesday, August 7, 2019
Concluding the conversation about , we need to consider an infinite set. Gave in that the concept of "infinity" acts on mathematicians, like a boa constrictor on a rabbit. The quivering horror of infinity deprives mathematicians common sense. Here is an example:
The original source is located. Alpha denotes a real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take as an example an infinite set natural numbers, the considered examples can be presented in the following form:
To visually prove their case, mathematicians have come up with many different methods. Personally, I look at all these methods as the dances of shamans with tambourines. In essence, they all come down to the fact that either some of the rooms are not occupied and new guests are settled in them, or that some of the visitors are thrown out into the corridor to make room for the guests (very humanly). I expressed my view on such decisions in the form fantasy story about the blonde. What is my reasoning based on? Moving an infinite number of visitors takes an infinite amount of time. After we have vacated the first guest room, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will already be from the category of "the law is not written for fools." It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.
What is an "infinite hotel"? An infinity inn is an inn that always has any number of vacancies, no matter how many rooms are occupied. If all the rooms in the endless hallway "for visitors" are occupied, there is another endless hallway with rooms for "guests". There will be an infinite number of such corridors. At the same time, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, on the other hand, are not able to move away from the banal domestic problems: God-Allah-Buddha - there is always only one, the hotel - it is one, the corridor - only one. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to "shove the unpushed".
I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers exist - one or many? There is no correct answer to this question, since we ourselves invented numbers, there are no numbers in Nature. Yes, Nature knows how to count perfectly, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers exist. Consider both options, as befits a real scientist.
Option one. "Let us be given" a single set of natural numbers, which lies serenely on a shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take a unit from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:
I have written down the operations in algebraic notation and in set theory notation, listing the elements of the set in detail. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same unit is added.
Option two. We have many different infinite sets of natural numbers on the shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:
The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If one infinite set is added to another infinite set, the result is a new infinite set consisting of the elements of the first two sets.
The set of natural numbers is used for counting in the same way as a ruler for measurements. Now imagine that you have added one centimeter to the ruler. This will already be a different line, not equal to the original.
You can accept or not accept my reasoning - this is your own business. But if you ever run into mathematical problems, consider whether you are on the path of false reasoning, trodden by generations of mathematicians. After all, mathematics classes, first of all, form a stable stereotype of thinking in us, and only then they add mental abilities to us (or vice versa, they deprive us of free thinking).
pozg.ru
Sunday, August 4, 2019
I was writing a postscript to an article about and saw this wonderful text on Wikipedia:
We read: "... rich theoretical background Mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base.
Wow! How smart we are and how well we can see the shortcomings of others. Is it weak for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, personally I got the following:
The rich theoretical basis of modern mathematics does not have a holistic character and is reduced to a set of disparate sections, devoid of a common system and evidence base.
I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole cycle of publications to the most obvious blunders of modern mathematics. See you soon.
Saturday, August 3, 2019
How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.
May we have many BUT consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter but, the subscript with a number will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sexual characteristic" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set BUT on gender b. Notice that our "people" set has now become the "people with gender" set. After that, we can divide the sexual characteristics into male bm and women's bw gender characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, it does not matter which one is male or female. If it is present in a person, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.
After multiplication, reductions and rearrangements, we got two subsets: the male subset bm and a subset of women bw. Approximately the same way mathematicians reason when they apply set theory in practice. But they do not let us in on the details, but give us the finished result - "a lot of people consists of a subset of men and a subset of women." Naturally, you may have a question, how correctly applied mathematics in the above transformations? I dare to assure you that in fact the transformations are done correctly, it is enough to know the mathematical justification of arithmetic, Boolean algebra and other sections of mathematics. What it is? Some other time I will tell you about it.
As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.
As you can see, units of measurement and common math make set theory a thing of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. The mathematicians did what the shamans once did. Only shamans know how to "correctly" apply their "knowledge". This "knowledge" they teach us.
Finally, I want to show you how mathematicians manipulate .
Monday, January 7, 2019
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:
Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet been able to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not indefinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells of a flying arrow:
A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
I will show the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, and there are without a bow. After that, we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.
Now let's do a little trick. Let's take "solid in a pimple with a bow" and unite these "whole" by color, selecting red elements. We got a lot of "red". Now a tricky question: are the received sets "with a bow" and "red" the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.
This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid pimply with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a bump), decorations (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics. Here's what it looks like.
The letter "a" with different indices denotes different units of measurement. In parentheses, units of measurement are highlighted, according to which the "whole" is allocated at the preliminary stage. The unit of measurement, according to which the set is formed, is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dances of shamans with tambourines. Shamans can "intuitively" come to the same result, arguing it with "obviousness", because units of measurement are not included in their "scientific" arsenal.
With the help of units of measurement, it is very easy to split one or combine several sets into one superset. Let's take a closer look at the algebra of this process.
TABLE OF VALUES OF TRIGONOMETRIC FUNCTIONS
Table of values trigonometric functions compiled for angles of 0, 30, 45, 60, 90, 180, 270, and 360 degrees and their corresponding angles in radians. Of the trigonometric functions, the table shows the sine, cosine, tangent, cotangent, secant and cosecant. For the convenience of solving school examples, the values \u200b\u200bof trigonometric functions in the table are written as a fraction with the preservation of the signs of extracting the square root from numbers, which very often helps to reduce complex mathematical expressions. For tangent and cotangent, the values of some angles cannot be determined. For the values of the tangent and cotangent of such angles, there is a dash in the table of values of trigonometric functions. It is generally accepted that the tangent and cotangent of such angles is equal to infinity. On a separate page are formulas for reducing trigonometric functions.
The table of values for the trigonometric function sine shows the values \u200b\u200bfor the following angles: sin 0, sin 30, sin 45, sin 60, sin 90, sin 180, sin 270, sin 360 in degree measure, which corresponds to sin 0 pi, sin pi / 6 , sin pi / 4, sin pi / 3, sin pi / 2, sin pi, sin 3 pi / 2, sin 2 pi in radian measure of angles. School table of sines.
For the trigonometric cosine function, the table shows the values for the following angles: cos 0, cos 30, cos 45, cos 60, cos 90, cos 180, cos 270, cos 360 in degree measure, which corresponds to cos 0 pi, cos pi to 6, cos pi by 4, cos pi by 3, cos pi by 2, cos pi, cos 3 pi by 2, cos 2 pi in radian measure of angles. School table of cosines.
The trigonometric table for the trigonometric function tangent gives values \u200b\u200bfor the following angles: tg 0, tg 30, tg 45, tg 60, tg 180, tg 360 in degree measure, which corresponds to tg 0 pi, tg pi / 6, tg pi / 4, tg pi/3, tg pi, tg 2 pi in radian measure of angles. The following values of the trigonometric functions of the tangent are not defined tg 90, tg 270, tg pi/2, tg 3 pi/2 and are considered equal to infinity.
For the trigonometric function cotangent in the trigonometric table, the values of the following angles are given: ctg 30, ctg 45, ctg 60, ctg 90, ctg 270 in degree measure, which corresponds to ctg pi / 6, ctg pi / 4, ctg pi / 3, tg pi / 2, tg 3 pi/2 in radian measure of angles. The following values of trigonometric cotangent functions are not defined ctg 0, ctg 180, ctg 360, ctg 0 pi, ctg pi, ctg 2 pi and are considered equal to infinity.
The values of the trigonometric functions secant and cosecant are given for the same angles in degrees and radians as sine, cosine, tangent, cotangent.
The table of values of trigonometric functions of non-standard angles shows the values of sine, cosine, tangent and cotangent for angles in degrees 15, 18, 22.5, 36, 54, 67.5 72 degrees and in radians pi/12, pi/10, pi/ 8, pi/5, 3pi/8, 2pi/5 radians. The values of trigonometric functions are expressed in terms of fractions and square roots to simplify the reduction of fractions in school examples.
Three more monsters of trigonometry. The first is the tangent of 1.5 degrees and a half, or pi divided by 120. The second is the cosine of pi divided by 240, pi/240. The longest is the cosine of pi divided by 17, pi/17.
The trigonometric circle of the values of the sine and cosine functions visually represents the signs of the sine and cosine depending on the magnitude of the angle. Especially for blondes, the cosine values are underlined with a green dash in order to be less confused. The conversion of degrees to radians is also very clearly presented, when radians are expressed through pi.
This trigonometric table represents the values of sine, cosine, tangent, and cotangent for angles from 0 zero to 90 ninety degrees at one degree intervals. For the first forty-five degrees, the names of trigonometric functions must be looked at at the top of the table. The first column contains degrees, the values of sines, cosines, tangents and cotangents are written in the next four columns.
For angles from forty-five degrees to ninety degrees, the names of the trigonometric functions are written at the bottom of the table. The last column contains degrees, the values of cosines, sines, cotangents and tangents are written in the previous four columns. You should be careful, because the names of trigonometric functions in the lower part of the trigonometric table are different from the names in the upper part of the table. Sines and cosines are interchanged, just like tangent and cotangent. This is due to the symmetry of the values of trigonometric functions.
The signs of trigonometric functions are shown in the figure above. The sine has positive values from 0 to 180 degrees or from 0 to pi. The negative values of the sine are from 180 to 360 degrees or from pi to 2 pi. Cosine values are positive from 0 to 90 and 270 to 360 degrees, or 0 to 1/2 pi and 3/2 to 2 pi. Tangent and cotangent have positive values from 0 to 90 degrees and from 180 to 270 degrees, corresponding to values from 0 to 1/2 pi and from pi to 3/2 pi. Negative tangent and cotangent are 90 to 180 degrees and 270 to 360 degrees, or 1/2 pi to pi and 3/2 pi to 2 pi. When determining the signs of trigonometric functions for angles greater than 360 degrees or 2 pi, the periodicity properties of these functions should be used.
The trigonometric functions sine, tangent and cotangent are odd functions. The values of these functions for negative angles will be negative. Cosine is an even trigonometric function - the cosine value for negative angle will be positive. When multiplying and dividing trigonometric functions, you must follow the rules of signs.
The table of values for the trigonometric function sine shows the values \u200b\u200bfor the following angles
DocumentA separate page contains casting formulas trigonometricfunctions. IN tablevaluesfortrigonometricfunctionssinusgivenvaluesfornextcorners: sin 0, sin 30, sin 45 ...
The proposed mathematical apparatus is a complete analogue of the complex calculus for n-dimensional hypercomplex numbers with any number of degrees of freedom n and is intended for mathematical modeling of nonlinear
Document... functions equals functions Images. From this theorem should, what for finding the coordinates U, V, it is enough to calculate function... geometry; polynar functions(multidimensional analogues of two-dimensional trigonometricfunctions), their properties, tables and application; ...
-
Table of values of trigonometric functions
Note. This table of values of trigonometric functions uses the sign √ to denote square root. To denote a fraction - the symbol "/".
see also useful materials:
For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, a sine of 30 degrees - we are looking for a column with the heading sin (sine) and we find the intersection of this column of the table with the line "30 degrees", at their intersection we read the result - one second. Similarly, we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin (sine) column and the 60 degree row, we find the value sin 60 = √3/2), etc. In the same way, the values of sines, cosines and tangents of other "popular" angles are found.
Sine of pi, cosine of pi, tangent of pi and other angles in radians
The table of cosines, sines and tangents below is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the 60 degree angle in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.
The number pi uniquely expresses the dependence of the circumference of a circle on the degree measure of the angle. So pi radians equals 180 degrees.
Any number expressed in terms of pi (radian) can be easily converted to degrees by replacing the number pi (π) with 180.
Examples:
1. sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and is equal to zero.2. cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and is equal to minus one.3. Tangent pi
tg π = tg 180 = 0
thus, the tangent of pi is the same as the tangent of 180 degrees and is equal to zero.Table of sine, cosine, tangent values for angles 0 - 360 degrees (frequent values)
angle α
(degrees)angle α
in radians(via pi)
sin
(sinus)cos
(cosine)tg
(tangent)ctg
(cotangent)sec
(secant)cause
(cosecant)0 0 0 1 0 - 1 - 15 π/12 2 - √3 2 + √3 30 π/6 1/2 √3/2 1/√3 √3 2/√3 2 45 π/4 √2/2 √2/2 1 1 √2 √2 60 π/3 √3/2 1/2 √3 1/√3 2 2/√3 75 5π/12 2 + √3 2 - √3 90 π/2 1 0 - 0 - 1 105 7π/12 - - 2 - √3 √3 - 2 120 2π/3 √3/2 -1/2 -√3 -√3/3 135 3π/4 √2/2 -√2/2 -1 -1 -√2 √2 150 5π/6 1/2 -√3/2 -√3/3 -√3 180 π 0 -1 0 - -1 - 210 7π/6 -1/2 -√3/2 √3/3 √3 240 4π/3 -√3/2 -1/2 √3 √3/3 270 3π/2 -1 0 - 0 - -1 360 2π 0 1 0 - 1 - If in the table of values of trigonometric functions, instead of the value of the function, a dash is indicated (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle, the function does not have a definite value. If there is no dash, the cell is empty, so we have not yet entered the desired value. We are interested in what requests users come to us for and supplement the table with new values, despite the fact that the current data on the values of cosines, sines and tangents of the most common angle values is enough to solve most problems.
Table of values of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numerical values "as per Bradis tables")angle value α (degrees) value of angle α in radians sin (sine) cos (cosine) tg (tangent) ctg (cotangent) 0 0 15 0,2588
0,9659
0,2679
30 0,5000
0,5774
45 0,7071
0,7660
60 0,8660
0,5000
1,7321
7π/18
Table of basic trigonometric functions for angles 0, 30, 45, 60, 90, ... degrees
From the trigonometric definitions of the functions $\sin$, $\cos$, $\tan$, and $\cot$, one can find their values for angles $0$ and $90$ degrees:
$\sin0°=0$, $\cos0°=1$, $\tan 0°=0$, $\cot 0°$ not defined;
$\sin90°=1$, $\cos90°=0$, $\cot90°=0$, $\tan 90°$ is not defined.
In the school course of geometry when studying right triangles find the trigonometric functions of the angles $0°$, $30°$, $45°$, $60°$ and $90°$.
The found values of trigonometric functions for the specified angles in degrees and radians respectively ($0$, $\frac(\pi)(6)$, $\frac(\pi)(4)$, $\frac(\pi)(3) $, $\frac(\pi)(2)$) for ease of memorization and use are entered in a table called trigonometric table, table of basic values of trigonometric functions etc.
When using reduction formulas, the trigonometric table can be expanded to an angle of $360°$ and $2\pi$ radians respectively:
Applying the periodicity properties of trigonometric functions, each angle that differs from the already known one by $360°$ can be calculated and recorded in a table. For example, the trigonometric function for the angle $0°$ will have the same value for the angle $0°+360°$, and for the angle $0°+2 \cdot 360°$, and for the angle $0°+3 \cdot 360°$ and etc.
Using a trigonometric table, you can determine the values of all angles of a unit circle.
In the school geometry course, it is supposed to memorize the basic values of trigonometric functions collected in a trigonometric table for the convenience of solving trigonometric problems.
Using a table
In the table, it is enough to find the necessary trigonometric function and the value of the angle or radian for which this function needs to be calculated. At the intersection of the row with the function and the column with the value, we get the desired value of the trigonometric function of the given argument.
In the figure you can see how to find the value $\cos60°$ which is equal to $\frac(1)(2)$.
The extended trigonometric table is used similarly. The advantage of using it is, as already mentioned, the calculation of the trigonometric function of almost any angle. For example, you can easily find the value $\tan 1 380°=\tan (1 380°-360°)=\tan(1 020°-360°)=\tan(660°-360°)=\tan300°$:
Bradis tables of basic trigonometric functions
The ability to calculate the trigonometric function of absolutely any angle value for an integer value of degrees and an integer value of minutes gives the use of Bradis tables. For example, find the value $\cos34°7"$. The tables are divided into 2 parts: the table of $\sin$ and $\cos$ values and the table of $\tan$ and $\cot$ values.
Bradis tables make it possible to obtain an approximate value of trigonometric functions with an accuracy of up to 4 decimal places.
Using Bradis Tables
Using the tables of Bradys for sines, we find $\sin17°42"$. To do this, in the column on the left of the table of sines and cosines we find the value of degrees - $17°$, and in the top line we find the value of minutes - $42"$. At their intersection, we get the desired value:
$\sin17°42"=0.304$.
To find the value of $\sin17°44"$, you need to use the correction on the right side of the table. In this case, to the value of $42"$, which is in the table, you need to add a correction for $2"$, which is equal to $0.0006$. We get:
$\sin17°44"=0.304+0.0006=0.3046$.
To find the value of $\sin17°47"$, we also use the correction on the right side of the table, only in this case we take the value of $\sin17°48"$ as a basis and subtract the correction for $1"$:
$\sin17°47"=0.3057-0.0003=0.3054$.
When calculating the cosines, we perform similar actions, but we look at the degrees in the right column, and the minutes in the bottom column of the table. For example, $\cos20°=0.9397$.
There are no corrections for tangent values up to $90°$ and small angle cotangent. For example, let's find $\tan 78°37"$, which according to the table is $4,967$.
- In contact with 0
- Google Plus 0
- OK 0
- Facebook 0
