The table of sines and cosines is complete. Sine (sin x) and cosine (cos x) – properties, graphs, formulas
Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were derived by astronomers to create an accurate calendar and orientation by the stars. These calculations related to spherical trigonometry, while in the school course they study the ratio of sides and angles of a plane triangle.
Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationships between the sides and angles of triangles.
During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazwi introduced functions such as tangent and cotangent, and compiled the first tables of values for sines, tangents and cotangents. The concepts of sine and cosine were introduced by Indian scientists. Trigonometry received a lot of attention in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.
Basic quantities of trigonometry
The basic trigonometric functions of a numeric argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.
The formulas for calculating the values of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants are equal in all directions,” since the proof is given using the example of an isosceles right triangle.
Sine, cosine and other relationships establish the relationship between the acute angles and sides of any right triangle. Let us present formulas for calculating these quantities for angle A and trace the relationships between trigonometric functions:
As you can see, tg and ctg are inverse functions. If we imagine leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, we obtain the following formulas for tangent and cotangent:
Trigonometric circle
Graphically, the relationship between the mentioned quantities can be represented as follows:
The circle, in this case, represents all possible values of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will have a “+” sign if α belongs to the 1st and 2nd quarters of the circle, that is, it is in the range from 0° to 180°. For α from 180° to 360° (III and IV quarters), sin α can only be a negative value.
Let's try to build trigonometric tables for specific angles and find out the meaning of the quantities.
Values of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values of trigonometric functions for them are calculated and presented in the form of special tables.
These angles were not chosen at random. The designation π in the tables is for radians. Rad is the angle at which the length of a circle's arc corresponds to its radius. This value was introduced in order to establish a universal dependence; when calculating in radians, the actual length of the radius in cm does not matter.
Angles in tables for trigonometric functions correspond to radian values:
So, it is not difficult to guess that 2π is a complete circle or 360°.
Properties of trigonometric functions: sine and cosine
In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.
Consider the comparative table of properties for sine and cosine:
| Sine wave | Cosine |
|---|---|
| y = sin x | y = cos x |
| ODZ [-1; 1] | ODZ [-1; 1] |
| sin x = 0, for x = πk, where k ϵ Z | cos x = 0, for x = π/2 + πk, where k ϵ Z |
| sin x = 1, for x = π/2 + 2πk, where k ϵ Z | cos x = 1, at x = 2πk, where k ϵ Z |
| sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z | cos x = - 1, for x = π + 2πk, where k ϵ Z |
| sin (-x) = - sin x, i.e. the function is odd | cos (-x) = cos x, i.e. the function is even |
| the function is periodic, the smallest period is 2π | |
| sin x › 0, with x belonging to the 1st and 2nd quarters or from 0° to 180° (2πk, π + 2πk) | cos x › 0, with x belonging to the I and IV quarters or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk) |
| sin x ‹ 0, with x belonging to the third and fourth quarters or from 180° to 360° (π + 2πk, 2π + 2πk) | cos x ‹ 0, with x belonging to the 2nd and 3rd quarters or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk) |
| increases in the interval [- π/2 + 2πk, π/2 + 2πk] | increases on the interval [-π + 2πk, 2πk] |
| decreases on intervals [π/2 + 2πk, 3π/2 + 2πk] | decreases on intervals |
| derivative (sin x)’ = cos x | derivative (cos x)’ = - sin x |
Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with the signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs coincide, the function is even, otherwise it is odd.
The introduction of radians and the listing of the basic properties of sine and cosine waves allow us to present the following pattern:
It is very easy to verify that the formula is correct. For example, for x = π/2, the sine is 1, as is the cosine of x = 0. The check can be done by consulting tables or by tracing function curves for given values.
Properties of tangentsoids and cotangentsoids
The graphs of the tangent and cotangent functions differ significantly from the sine and cosine functions. The values tg and ctg are reciprocals of each other.
- Y = tan x.
- The tangent tends to the values of y at x = π/2 + πk, but never reaches them.
- The smallest positive period of the tangentoid is π.
- Tg (- x) = - tg x, i.e. the function is odd.
- Tg x = 0, for x = πk.
- The function is increasing.
- Tg x › 0, for x ϵ (πk, π/2 + πk).
- Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
- Derivative (tg x)’ = 1/cos 2 x.
Consider the graphic image of the cotangentoid below in the text.
Main properties of cotangentoids:
- Y = cot x.
- Unlike the sine and cosine functions, in the tangentoid Y can take on the values of the set of all real numbers.
- The cotangentoid tends to the values of y at x = πk, but never reaches them.
- The smallest positive period of a cotangentoid is π.
- Ctg (- x) = - ctg x, i.e. the function is odd.
- Ctg x = 0, for x = π/2 + πk.
- The function is decreasing.
- Ctg x › 0, for x ϵ (πk, π/2 + πk).
- Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
- Derivative (ctg x)’ = - 1/sin 2 x Correct
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
First of all, let me remind you of a simple but very useful conclusion from the lesson “What are sine and cosine? What are tangent and cotangent?”
This is the output:
Sine, cosine, tangent and cotangent are tightly connected to their angles. We know one thing, which means we know another.
In other words, each angle has its own constant sine and cosine. And almost everyone has their own tangent and cotangent. Why almost? More on this below.
This knowledge helps a lot in your studies! There are a lot of tasks where you need to move from sines to angles and vice versa. For this there is table of sines. Similarly, for tasks with cosine - cosine table. And, as you may have guessed, there is tangent table And table of cotangents.)
Tables are different. Long ones, where you can see what, say, sin37°6’ is equal to. We open the Bradis tables, look for an angle of thirty-seven degrees six minutes and see the value of 0.6032. It’s clear that there is absolutely no need to remember this number (and thousands of other table values).
In fact, in our time, long tables of cosines, sines, tangents, cotangents are not really needed. One good calculator replaces them completely. But it doesn’t hurt to know about the existence of such tables. For general erudition.)
And why then this lesson?! - you ask.
But why. Among the infinite number of angles there are special, which you should know about All. All school geometry and trigonometry are built on these angles. This is a kind of "multiplication table" of trigonometry. If you don’t know what sin50° is equal to, for example, no one will judge you.) But if you don’t know what sin30° is equal to, be prepared to get a well-deserved two...
Such special The angles are also quite good. School textbooks usually kindly offer memorization sine table and cosine table for seventeen angles. And, of course, tangent table and cotangent table for the same seventeen angles... I.e. It is proposed to remember 68 values. Which, by the way, are very similar to each other, repeat themselves every now and then and change signs. For a person without perfect visual memory, this is quite a task...)
We'll take a different route. Let's replace rote memorization with logic and ingenuity. Then we will have to memorize 3 (three!) values for the table of sines and the table of cosines. And 3 (three!) values for the table of tangents and the table of cotangents. That's all. Six values are easier to remember than 68, it seems to me...)
We will obtain all other necessary values from these six using a powerful legal cheat sheet - trigonometric circle. If you have not studied this topic, follow the link, don’t be lazy. This circle is not only needed for this lesson. He is irreplaceable for all trigonometry at once. Not using such a tool is simply a sin! You do not want? That's your business. Memorize table of sines. Table of cosines. Table of tangents. Table of cotangents. All 68 values for a variety of angles.)
So, let's begin. First, let's divide all these special angles into three groups.
First group of angles.
Let's consider the first group seventeen angles special. These are 5 angles: 0°, 90°, 180°, 270°, 360°.
This is what the table of sines, cosines, tangents, and cotangents looks like for these angles:
Angle x
|
0 |
90 |
180 |
270 |
360 |
Angle x
|
0 |
||||
sin x |
0 |
1 |
0 |
-1 |
0 |
cos x |
1 |
0 |
-1 |
0 |
1 |
tg x |
0 |
noun |
0 |
noun |
0 |
ctg x |
noun |
0 |
noun |
0 |
noun |
Those who want to remember, remember. But I’ll say right away that all these ones and zeros get very confused in the head. Much stronger than you want.) Therefore, we turn on logic and the trigonometric circle.
We draw a circle and mark these same angles on it: 0°, 90°, 180°, 270°, 360°. I marked these corners with red dots:
It is immediately obvious what is special about these angles. Yes! These are the angles that fall exactly on the coordinate axis! Actually, that’s why people get confused... But we won’t get confused. Let's figure out how to find trigonometric functions of these angles without much memorization.
By the way, the angle position is 0 degrees completely coincides with a 360 degree angle position. This means that the sines, cosines, and tangents of these angles are exactly the same. I marked a 360 degree angle to complete the circle.
Suppose, in the difficult stressful environment of the Unified State Examination, you somehow doubted... What is the sine of 0 degrees? It seems like zero... What if it’s one?! Mechanical memorization is such a thing. In harsh conditions, doubts begin to gnaw...)
Calm, just calm!) I will tell you a practical technique that will give you a 100% correct answer and completely remove all doubts.
As an example, let's figure out how to clearly and reliably determine, say, the sine of 0 degrees. And at the same time, cosine 0. It is in these values, oddly enough, that people often get confused.
To do this, draw on a circle arbitrary corner X. In the first quarter, it was close to 0 degrees. Let us mark the sine and cosine of this angle on the axes X, everything is fine. Like this:
And now - attention! Let's reduce the angle X, bring the moving side closer to the axis OH. Hover your cursor over the picture (or tap the picture on your tablet) and you’ll see everything.
Now let's turn on elementary logic! Let's look and think: How does sinx behave as the angle x decreases? As the angle approaches zero? It's shrinking! And cosx increases! It remains to figure out what will happen to the sine when the angle collapses completely? When does the moving side of the angle (point A) settle down on the OX axis and the angle becomes equal to zero? Obviously, the sine of the angle will go to zero. And the cosine will increase to... to... What is the length of the moving side of the angle (the radius of the trigonometric circle)? One!
Here is the answer. The sine of 0 degrees is equal to 0. The cosine of 0 degrees is equal to 1. Absolutely ironclad and without any doubt!) Simply because otherwise it can not be.
In exactly the same way, you can find out (or clarify) the sine of 270 degrees, for example. Or cosine 180. Draw a circle, arbitrary an angle in a quarter next to the coordinate axis of interest to us, mentally move the side of the angle and grasp what the sine and cosine will become when the side of the angle falls on the axis. That's all.
As you can see, there is no need to memorize anything for this group of angles. Not needed here table of sines... Yes and cosine table- too.) By the way, after several uses of the trigonometric circle, all these values will be remembered by themselves. And if they forget, I drew a circle in 5 seconds and clarified it. Much easier than calling a friend from the toilet and risking your certificate, right?)
As for tangent and cotangent, everything is the same. We draw a tangent (cotangent) line on the circle - and everything is immediately visible. Where they are equal to zero, and where they do not exist. What, you don’t know about tangent and cotangent lines? This is sad, but fixable.) We visited Section 555 Tangent and cotangent on the trigonometric circle - and there are no problems!
If you have figured out how to clearly define sine, cosine, tangent and cotangent for these five angles, congratulations! Just in case, I inform you that you can now define functions any angles falling on the axes. And this is 450°, and 540°, and 1800°, and an infinite number of others...) I counted (correctly!) the angle on the circle - and there are no problems with the functions.
But it’s precisely with the measurement of angles that problems and errors occur... How to avoid them is written in the lesson: How to draw (count) any angle on a trigonometric circle in degrees. Elementary, but very helpful in the fight against errors.)
Here's a lesson: How to draw (measure) any angle on a trigonometric circle in radians - it will be cooler. In terms of possibilities. Let's say, determine which of the four semi-axes the angle falls on
you can do it in a couple of seconds. I am not kidding! Just in a couple of seconds. Well, of course, not only 345 pi...) And 121, and 16, and -1345. Any integer coefficient is suitable for an instant answer.
And if the corner
Just think! The correct answer is obtained in 10 seconds. For any fractional value of radians with a two in the denominator.
Actually, this is what is good about the trigonometric circle. Because the ability to work with some corners it automatically expands to infinite set corners
So, we’ve sorted out five corners out of seventeen.
Second group of angles.
The next group of angles are the angles 30°, 45° and 60°. Why exactly these, and not, for example, 20, 50 and 80? Yes, somehow it turned out this way... Historically.) Further it will be seen why these angles are good.
The table of sines cosines tangents cotangents for these angles looks like this:
Angle x
|
0 |
30 |
45 |
60 |
90 |
Angle x
|
0 |
||||
sin x |
0 |
1 |
|||
cos x |
1 |
0 |
|||
tg x |
0 |
1 |
noun |
||
ctg x |
noun |
1 |
0 |
I left the values for 0° and 90° from the previous table to complete the picture.) So that you can see that these angles lie in the first quarter and increase. From 0 to 90. This will be useful to us later.
The table values for angles of 30°, 45° and 60° must be remembered. Memorize it if you want. But here, too, there is an opportunity to make your life easier.) Pay attention to sine table values these angles. And compare with cosine table values...
Yes! They same! Just arranged in reverse order. Angles increase (0, 30, 45, 60, 90) - and sine values increase from 0 to 1. You can check with a calculator. And the cosine values are are decreasing from 1 to zero. Moreover, the values themselves same. For angles of 20, 50, 80 this would not work...
This is a useful conclusion. Enough to learn three values for angles of 30, 45, 60 degrees. And remember that for the sine they increase, and for the cosine they decrease. Towards the sine.) They meet halfway (45°), that is, the sine of 45 degrees is equal to the cosine of 45 degrees. And then they diverge again... Three meanings can be learned, right?
With tangents - cotangents the picture is exactly the same. One to one. Only the meanings are different. These values (three more!) also need to be learned.
Well, almost all the memorization is over. You have (hopefully) understood how to determine the values for the five angles falling on the axis and learned the values for the angles of 30, 45, 60 degrees. Total 8.
It remains to deal with the last group of 9 corners.
These are the angles:
120°; 135°; 150°; 210°; 225°; 240°; 300°; 315°; 330°. For these angles, you need to know the table of sines, the table of cosines, etc.
Nightmare, right?)
And if you add angles here, such as: 405°, 600°, or 3000° and many, many equally beautiful ones?)
Or angles in radians? For example, about angles:
and many others you should know All.
The funniest thing is to know this All - impossible in principle. If you use mechanical memory.
And it’s very easy, in fact elementary - if you use a trigonometric circle. Once you get the hang of working with the trigonometric circle, all those dreaded angles in degrees can be easily and elegantly reduced to the good old fashioned ones:
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his 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 catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl 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 a hundred steps. Now Achilles is eight hundred steps 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 irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every 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 a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs 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 whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
TABLE OF VALUES OF TRIGONOMETRIC FUNCTIONS
The table of values of trigonometric functions is compiled for angles of 0, 30, 45, 60, 90, 180, 270 and 360 degrees and the corresponding angle values in vradians. Of the trigonometric functions, the table shows sine, cosine, tangent, cotangent, secant and cosecant. For the convenience of solving school examples, the values of trigonometric functions in the table are written in the form of a fraction while preserving the signs for extracting the square root of 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 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 there are formulas for reducing trigonometric functions.
The table of values for the trigonometric sine function shows the values for the following angles: sin 0, sin 30, sin 45, sin 60, sin 90, sin 180, sin 270, sin 360 in degrees, 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 degrees, which corresponds to cos 0 pi, cos pi by 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 tangent function gives values for 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 tangent functions are not defined tan 90, tan 270, tan pi/2, tan 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, tan 3 pi/2 in radian measure of angles. The following values of the 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 make it easier to reduce fractions in school examples.
Three more trigonometry monsters. The first is the tangent of 1.5 one and a half degrees 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 values of the functions sine and cosine visually represents the signs of sine and cosine depending on the magnitude of the angle. Especially for blondes, the cosine values are underlined with a green dash to reduce confusion. The conversion of degrees to radians is also very clearly presented when radians are expressed in terms of pi.
This trigonometric table presents 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 should 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 the trigonometric functions at the bottom of the trigonometric table are different from the names at the top 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. Sine has positive values from 0 to 180 degrees, or 0 to pi. Sine has negative values 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 pi to 3/2 pi. Negative values of tangent and cotangent are from 90 to 180 degrees and from 270 to 360 degrees, or from 1/2 pi to pi and from 3/2 pi to 2 pi. When determining the signs of trigonometric functions for angles greater than 360 degrees or 2 pi, you should use the periodicity properties of these functions.
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 a negative angle will be positive. Sign rules must be followed when multiplying and dividing trigonometric functions.
The table of values for the trigonometric sine function shows the values for the following angles
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