Table 30-45-60 degrees. Sine (sin x) and cosine (cos x) – powers, graphs, formulas
We'll figure out exactly what it looks like from the statistics table of trigonometric values, sine, cosine, tangent and cotangent. Let's look at the basic meanings of trigonometric functions, such as 0,30,45,60,90,...,360 degrees. I wonder how to use these tables to calculate the values of trigonometric functions.
Let's take a look first table of cosine, sine, tangent and cotangent view at 0, 30, 45, 60, 90, .. degrees. The values of these values are given by the values of the functions of the cutoffs 0 and 90 degrees:
sin 0 0 =0, cos 0 0 = 1. tg 00 = 0, cotangent 00 will be insignificant
sin 90 0 = 1, cos 90 0 =0, ctg90 0 = 0, tangent 90 0 will be insignificant
If you take straight-cut knitted ones from 30 to 90 degrees. We reject:
sin 30 0 = 1/2, cos 30 0 = √3/2, tan 30 0 = √3/3, cos 30 0 = √3
sin 45 0 = √2/2, cos 45 0 = √2/2, tan 45 0 = 1, cos 45 0 = 1
sin 60 0 = √3/2, cos 60 0 = 1/2, tg 60 0 =√3, cot 60 0 = √3/3
Imaginably all the meanings have been taken away trigonometric table:
Table of sines, cosines, tangents and cotangents!
If you use the reduction formula, our table will increase, adding values for cutoffs up to 360 degrees. Viglyadatime wona yak:
Also, based on the power of periodicity, the table can be expanded by replacing the values with 0 0 +360 0 *z .... 330 0 +360 0 *z, in which z is the whole number. In this table you can determine the values of all the cut-off points in a single count.
Let's sort out the table from the solution.
Everything is very simple. The fragments of necessary importance lie at the point where the crossbars we need are crossed. For example, if we take cos at 60 degrees, the table will look like this:
The table of the basic meanings of trigonometric functions is the same. Also in this table you can find out how many tangents are in stock at 1020 degrees, vin = -√3 Verifiable 1020 0 = 300 0 +360 0 *2. We know at the table.
Bradis table. For sine, cosine, tangent and cotangent.
The Bradys table is divided into several parts, consisting of a table of cosine and sine, tangent and cotangent - which is divided into two parts (tg up to 90 degrees and ctg of small ones).
Sine and cosine
tg from 00 to 760, ctg from 140 to 900.
tg up to 900 and ctg of small kuti.
Let's figure out how to use the Bradis tables at the top level.
We know the designation of sin (the designation is in the column on the left edge) 42 hvilini (the designation is on the top row). Let's figure out the value, vono = 0.3040.
The size of the hulls is indicated with a gap of six hulls, as we need to put more value into this gap. Let’s take 44 values, and in the table there are only 42. We take 42 as a basis and accelerate them with additional clauses on the right side, take 2nd amendment and add to 0.3040 + 0.0006, subtract 0.3046.
When sin 47 xv we take 48 xv as a basis and add 1 correction from it, then 0.3057 - 0.0003 = 0.3054
When calculating cos, we proceed in the same way as sin, but we take the bottom row of the table as a basis. For example, cos 20 0 = 0.9397
The tg values are up to 90 0 and the cot is small, but they have no corrections. For example, tg 78 0 37хв = 4.967 
a ctg 20 0 13хв = 25.83 
Well, we also looked at the basic trigonometric tables. We are confident that this information was extremely confusing for you. Please write your own food table in the comments as soon as the stinks have appeared!
Wall tappers - a tapping board for protecting walls. Go to the instructions for frameless wall embossers (http://www.spi-polymer.ru/otboyniki/) and read the report.
The study of trigonometry is different from the recticutaneous tricutaneous. It is significant that both sine and cosine, as well as tangent and cotangent of the sharp cut. These are the basics of trigonometry.
Guess what straight cut- it’s like 90 degrees. In other words, half of a flared kut.
Gostrii kut- less than 90 degrees.
Dumb cut- the larger one is 90 degrees. This is so much “stupid” - not an image, but a mathematical term :-)
We paint a straight-cut tricutnik. Directly kut zazvichay appears. I really appreciate that the side that lies opposite the corner is represented by the same letter, albeit a small one. So, the side that lies opposite to cut A is designated .
Kut is designated by a typical Greek letter.
Hypotenuse straight cut tricutule - this is the side that lies opposite the straight cut.
Kateti- The sides should lie opposite the sharp cuts.
The katet, which lies opposite the kut, is called pre-ulcers(According to the current date). The other leg, which lies on one of the sides of the cut, is called let's lie down.
Sine acute cut in the rectum tricucut - this is the extension of the protilage leg to the hypotenusus:
Cosine of the acute cut in the rectum tricut - the extension of the adjoining leg to the hypotenusus:
Tangent a tight cut in a straight cut - the development of the protidal leg to the adjacent leg:
Another (equal) value: the tangent of the sharp cut is called the ratio of the sine of the cut to the second cosine:
Cotangent a sharp cut in a rectal tricut - the extension of the adjacent leg to the prostrate leg (or, at the same time, the extension of the cosine to the sine):
Return to the basic relationships for sine, cosine, tangent and cotangent, which are indicated below. The stench will become a stench for us at the height of the task.
Let's get to the bottom of them.
Good, we gave the appointment and wrote down the formulas. What do you need for sine, cosine, tangent and cotangent?
We know what the sum of kuti be-any trikutnik is as old as.
Known relationship between parties straight-cut tricut. This is the Pythagorean theorem: .
To go out, if you know two corners of the trikutnik, you can find the third. Knowing the two sides of the recticutaneous tricut, you can know the third. This means that for the lovers there is their own relationship, for the parties - their own. What can you do if a straight-cut tricutnik has only one cut (except the straight one) and one side, but you need to know the other sides?
Here people stood in the past, forming maps of the locality and the dawning sky. Once again you can completely destroy all sides of the tricutaneous.
Sine, cosine and tangent - what are they called? trigonometric functions- give communication between partiesі kutami Tricutnik. If you know this, you can find out all the trigonometric functions using special tables. And if you know the sines, cosines and tangents of the tricutaneous and one of its sides, you can know others.
We also draw up a table of the values of sine, cosine, tangent and cotangent for “good” cutoffs up to .
Change the value by two digits in the table. With these values, the tangent and cotangent do not exist.
Let's sort out the task from trigonometry from the Bank of the FID task.
1. The trikutnik has a lot of money. Find.
The mystery disappears in just a few seconds.
Oskolki, .
2. The trikutnik has a dear , , . Find.
We know the Pythagorean theorem.
The story is over.
Often in problems there is a mix of tricutniks with kutas or with kutas i. The main information for them, remember!
For a trikutnik with kutas and a leg, which lies opposite the kuta, it is ancient half hypotenuse.
Trikutnik with kuts and equal thighs. In this case, the hypotenuse is several times larger than the leg.
We looked at the hidden secrets of the straight-cut tights - then the discovery of unknown sides of the kuts. That's not all! In the options ЄДІ from mathematics there is no specific requirement, which is represented by sine, cosine, tangent and cotangent of the external cut of the tricutaneous. About this – in the current statistics.
Trigonometry, as a science, originated at the Ancient Gathering. The first trigonometric relationships were developed by astronomers to create an accurate calendar and orientation behind the mirrors. These calculations were based on spherical trigonometry, just as the school course teaches the relationship between the sides and sides of a flat tricuput.
Trigonometry is a branch of mathematics that deals with the powers of trigonometric functions and the relationships between the sides and cuts of tricutaneous elements.
During the development of culture and science in the first millennium of our era, knowledge expanded from the Ancient Gathering to Greece. All the main developments of trigonometry are the merit of the people of the Arab Caliphate. Zokrema, the Turkmen teachings of al-Marazwi in the centuries such functions as tangent and cotangent, including the first table of values for sines, tangents and cotangents. The concept of sine and cosine was introduced by Indians. Trigonometry was held in high esteem by such great figures of old as Euclid, Archimedes and Eratosthenes.
Basic quantities of trigonometry
The main trigonometric functions of a numerical argument are sine, cosine, tangent and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.
The Pythagorean theorem is the basis for the formulas for understanding the values of quantities. Schoolchildren are more familiar with the formula: “Pythagorean trousers, equal on all sides,” since the proof is based on the butt of the equifemoral tricucutineum.
Sine, cosine and other conditions establish connections between the edges and sides of any rectilinear tricut. Let us introduce formulas for the breakdown of these quantities for cut A and the simple interconnection of trigonometric functions:
As you can see, tg and ctg are gate functions. If we see leg a as an addition to sin A and hypotenuse c, and leg b as cos A * c, then we can derive the following formulas for tangent and cotangent:
Trigonometric colo
A graphical representation of the predicted quantities can be done as follows:
The circle has all possible values of α - from 0° to 360°. As can be seen from the baby, skin function acquires a negative or positive value depending on the size of the skin. For example, sin α will be a “+” sign, since α is placed on the 1st and 2nd quarters of the stake, so that it lies between 0° and 180°. When viewing from 180° to 360° (III and IV quarters), sin α can only have negative values.
Let's try to create trigonometric tables for specific values and determine the values of quantities.
The values of α levels 30°, 45°, 60°, 90°, 180° are also called close drops. The values of trigonometric functions are analyzed and presented in a special table.
This image is not at all casual. The π values in the tables are for radians. Radium is the same, when the end of the arc of the stake indicates radius. This value of the boule was introduced in order to establish a universal value; with expansions in radians, the effective doubling of the radius, see.
The values in the tables for trigonometric functions are based on radian values:
Well, it’s not important to guess that 2π is outside the circle or 360 °.
Power of trigonometric functions: sine and cosine
In order to look at and equalize the main powers of sine and cosine, tangent and cotangent, it is necessary to name their functions. This can be done by looking at a curve drawn from a two-dimensional coordinate system.
Take a look at the table of powers for sine and cosine:
| Sinusoid | 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 at x = π/2 + πk, de k ϵ Z |
| sin x = 1, for x = π/2 + 2πk, de k ϵ Z | cos x = 1 at x = 2πk, where k ϵ Z |
| sin x = - 1 at x = 3π/2 + 2πk, de k ϵ Z | cos x = - 1 at x = π + 2πk, de k ϵ Z |
| sin (-x) = - sin x, then the function is unpaired | cos (-x) = cos x, which is the parn function |
| the function is periodic, the shortest period is 2π | |
| sin x › 0, at x place I and II quarters or from 0° to 180° (2πk, π + 2πk) | cos x › 0, at x place the I and IV quarters or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk) |
| sin x ‹ 0, at x lie on the III and IV quarters or from 180° to 360° (π + 2πk, 2π + 2πk) | cos x ‹ 0, at x lie in the II and III quarters or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk) |
| grows by interval [- π/2 + 2πk, π/2 + 2πk] | grows by interval [-π + 2πk, 2πk] |
| changes at intervals [π/2 + 2πk, 3π/2 + 2πk] | changes between intervals |
| pokhіdna (sin x)’ = cos x | Pokhidna (cos x)' = - sin x |
This means that the function of the steam room is not very simple. It is enough to identify the trigonometric circle with the signs of trigonometric quantities and think about “folding” the graph along the OX axis. If the signs are matched, the function is paired, otherwise it is unpaired.
The introduction of radians and the re-interpretation of the main powers of sine and cosine allow us to establish a regularity:
It is very easy to change the correctness of the formula. For example, for x = π/2, the sine is equal to 1, as is the cosine of x = 0. The verification can be done in the table or by running the curve functions for the given values.
The power of tangents and cotangents
The graphs of the tangent and cotangent functions significantly differ from sine and cosine. The values tg and ctg are identical.
- Y = tan x.
- The tangent is equal to the value of y at x = π/2 + πk, but does not reach їх.
- The smallest positive period of the tangent is equal to π.
- Tg (-x) = - tg x, then the function is unpaired.
- Tg x = 0 at x = πk.
- The function is growing.
- Tg x › 0, for x ϵ (πk, π/2 + πk).
- Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
- Pokhidna (tg x)' = 1/cos 2 x .
Let's take a look at the graphic image of the cotangent below the text.
Main power cotangents:
- Y = cot x.
- By substituting the sine and cosine functions, the tangent Y can display the values of all real numbers.
- The cotangent is equal to the value of y at x = πk, but does not reach ich.
- The smallest positive period of the cotangent is equal to π.
- Ctg (-x) = - ctg x, then the function is unpaired.
- Ctg x = 0, for x = π/2 + πk.
- The function is inactive.
- Ctg x › 0, for x ϵ (πk, π/2 + πk).
- Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
- Pokhіdna (ctg x)’ = — 1/sin 2 x Vipravity
Respect!
Up to this point and additional information
materials from section 555.
For those who are very “uneasy...”
And for those who “already tell…”)
Beforehand, I’ll tell you a simple, but very simple, symbol from the lesson “What is sine and cosine? What is tangent and cotangent?”
The axis of this visor:
Sine, cosine, tangent and cotangent are closely related to their terms. We know one thing, which means we know something else.
In other words, the skin is subject to its own constant sine and cosine. And the skin has its own tangent and cotangent. Chomu Mayzhe? About the price below.
This knowledge miraculously helps in learning! There is a lot of instructions when it is necessary to move from the sinuses to the corners and back again. For whom is it necessary? tables of sines. Similarly, for a factor with a cosine - tables of cosines. And, as you already guessed, it turns out tangent tableі table of cotangents.)
Tables vary. For a long time, you can marvel at how ancient, say, sin37°6' is. Opening the Bradys table, we find that there are thirty seven degrees six degrees and most importantly the value is 0.6032. Obviously, there is no need to remember this number (and thousands of other table values).
In fact, in our time there is no need for long tables of cosines of sines of tangents and cotangents. One hell of a calculator replaces them completely. Alas, the foundation of such a table does not matter. For zagal erudition.)
What's the next lesson? - Power up the vi.
And the axis is here. Among the countless number of cuties there are especially, about your responsibilities to know Mustache. These courses covered all school geometry and trigonometry. This is, in a way, a “multiplication table” of trigonometry. If you don’t know what is comparable to sin50°, no one can sue you.) If you don’t know what is similar to sin30°, be prepared to take away a well-deserved deuce...
Such especially Kutiv can also gain decent weight. School assistants call to kindly sing until you remember table of sines and table of cosines for seventeen kutіv. Well, I understand, table of tangents and table of cotangents for these seventeen revelers themselves... Tobto. 68 values are stored. Signs that are similar to each other are repeated and changed over and over again. For a person without perfect visual memory, it’s a lost cause...)
We will take a different route. Replace mechanical memory with logic and intelligence. Then we will have to memorize 3 (three!) values for the table of sines and the table of cosines. І 3 (three!) Values for the table of tangents and the table of cotangents. And that's all. Six values are easier to remember, lower than 68, less difficult...)
All other necessary values are taken from these six using the following simple legal cheat sheet - Trigonometric stake. If you haven’t learned this topic, go and get it, don’t be lazy. This lesson needs more than this. Vin is irreplaceable for all trigonometry at a time. It’s simply a sin not to use such a tool! Don't you want to? On the right is yours. Get started tables of sines. Table of cosines. Table of tangents. Table of cotangents. Usi 68 value for different styles.)
So let's see. For the cob, we separate all the special cutlets into three groups.
First group of kouts.
Let's take a look at the group Kutiv from seventeen especially. There are 5 cuts: 0°, 90°, 180°, 270°, 360°.
The axis looks like this in the table of sines, cosines, tangents, cotangents for these lines:
Kut x
|
0 |
90 |
180 |
270 |
360 |
Kut 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 |
If you want to remember, remember. Let me just say right away that all the ones and zeros are already getting lost in my head. Much more strongly, but I don’t want to.) So we turn on logic and trigonometric colo.
The color is indicated on the same scale: 0 °, 90 °, 180 °, 270 °, 360 °. I mean this in red specks:
It is immediately obvious why these cuisines are so special. So! This is what we eat exactly on the coordinate axis! Vlasna, that’s why people get lost... But let’s not get lost. Let's figure out how to figure out trigonometric functions of these parts without special memorization.
Before speech, the position of the corner is 0 degrees completely avoided The position is 360 degrees. This means that the sines, cosines, and tangents of these terms are absolutely the same. Around 360 degrees, I mean to close the circle.
Let’s say, in a difficult stressful situation, you somehow began to doubt... Why is the sine of 0 degrees so important? Otherwise zero... And then one?! Mechanical memory is such a thing. Doubts begin to arise in the minds of such people...)
Calm down, just calm down!) I will show you a practical technique, which looks like a hundred hundred times correct answer and will clear up all your doubts.
Let’s figure out how to clearly and reliably calculate, say, the sine of 0 degrees. And at the same time, and cosine 0. It’s not surprising that people often get confused in these values.
For whom on earth do we paint satisfying kut X. At the first quarter of the day it was not far from 0 degrees. It is significant on the sine and cosine axes X, everything is in order. The axis is like this:
And now - wow! Zmenshimo kut X, close to the rukhomy b_k to the axis OH. Hover your cursor over the images (or click on the images on your tablet) and you’ll be done.
Now let's turn on elementary logic! It’s amazing and dimensional: How to deal with sinx when the cut x is changed? Is it close to zero? It's changing! And cosx is getting bigger! If you don’t get sick, what will happen to your sinus if you don’t know anything at all? What if the side of the kut (point A) settles down to all OX and the kut becomes equal to zero? Obviously, the sine goes to zero. And the cosine will increase to... to... Why is the dovzhina of a ruffled side of a kut (the radius of a trigonometric stake) more expensive? One!
Axis and axis. Sine of 0 degrees is equal to 0. Cosine of 0 degrees is equal to 1. Completely interesting and without any doubts!) Just something different buti can't.
In exactly the same way, you can determine (or clarify) the sine of 270 degrees, for example. Abo cosine 180. Namalyuvati colo, satisfying where in the fourth order from all the coordinates, which is to tell us, we are thinking about sponging around the corner and catching what the sine and cosine will become if the side of the corner fits all together. That's all.
As you see, for this group of kuts there is no need to learn anything. Not needed here table of sines... That's it cosine table- same.) Before speaking, after several times of standing up the trigonometric stake, all the meanings are forgotten by themselves. And if you forget - having painted the stake in 5 seconds and clarified it. It’s easier to call others from the toilet with a smock for a certificate, right?)
What is the difference between tangent and cotangent - all the same. We paint the tangent (cotangent) on the line - and everything is clearly visible. In some cases, the stink is as good as zero, but in other words, it doesn’t exist. What, you don’t know about lines of tangent and cotangent? This is complicated, but it can be corrected.) Published Section 555 Tangent and cotangent on a trigonometric scale - and there are no problems!
As soon as you understand how to clearly determine the sine, cosine, tangent and cotangent for these five factors - I will tell you! In case of any problem, I would like to inform you that you can now define functions be any kind of kuti that will be wasted on the axle. And this is 450 °, 540 °, 1800 °, and even more infinite capacity...) After washing (correctly!) Cut on the stake - and there are no problems with the functions.
All the same problems and solutions are solved with the help of the rest of the world... How to overcome them, it is written in the lesson: How to paint (detail) any kind of situation on a trigonometric scale in degrees. It’s elementary, but it really helps in the fight against pardons.)
And the main lesson: How to paint (drawing) whatever it is on a trigonometric scale in radians - it will be cooler. Have a sense of possibility. Let's say, it means that due to the fact that four birds are consumed by the
you can do it in a few seconds. I'm not kidding! In just a few seconds. Well, obviously, not just 345 “pi”...) I 121, I 16, I -1345. Any participant is suitable for the meeting.
What about Kut?
Big deal! Make sure to go out in 10 seconds. For any shot value of radians from the double at the banner.
Vlasne, cym and good trigonometric colo. Tim, what should you do with active kutami win automatically expands to endless impersonality Kutiv.
Well, five out of seventeen have separated.
Another group of revelers.
A group of kuti is advancing - the kuti are 30°, 45° and 60°. Why ci itself, and not, for example, 20, 50 or 80? So it seems to have turned out this way... Historically.) Later it will be seen what garni and kuti are.
The table of sines, cosines, tangents, cotangents for these cuts looks like this:
Kut x
|
0 |
30 |
45 |
60 |
90 |
Kut x
|
0 |
||||
sin x |
0 |
1 |
|||
cos x |
1 |
0 |
|||
tg x |
0 |
1 |
noun |
||
ctg x |
noun |
1 |
0 |
I have deleted the values for 0° and 90° from the front table to complete the picture.) So that it can be seen what lies in the first quarter and grows. From 0 to 90. This will give us a lot.
Table values for cutoffs 30°, 45° and 60° are stored. Memorize it however you want. And here, too, is the ability to make life easier for yourself. Return respect to sine table values a lot of fun. I equalize values of the cosine table...
So! Stinks the same! Only a few are taken out in order. Kuti grows (0, 30, 45, 60, 90) - that value of the sine grow from 0 to 1. You can go to the calculator. And the value of the cosine is subside from 1 to zero. Moreover, the meaning itself the same thing. For kids 20, 50, 80 it wouldn’t work out that way...
Star of the brown crown. Enough vivchiti three values for cutoffs are 30, 45, 60 degrees. And remember that the sine grows, and the cosine changes. Next to the sine.) At halfway (45 °) the sounds become sharper, so that the sine of 45 degrees is the same as the cosine of 45 degrees. And then we diverge again... You can count three values, right?
With tangents and cotangents the picture is exactly the same. One to one. Others are less important. These values (three more!) also need to be read.
Well, practically all the memorization is over. You understand (I believe) how to calculate the values for five corners on the axis and calculate the values for corners of 30, 45, 60 degrees. Usyogo 8.
I lost contact with the remaining group of 9 kuts.
Axis:
120°; 135°; 150°; 210°; 225°; 240°; 300°; 315°; 330°. For these purposes, it is important to know the table of sines, the table of cosines, etc.
Nightmare, right?)
How about adding something here, like: 405 °, 600 °, or 3000 ° and so many other beautiful things?)
What do you mean by radians? For example, about kuti:
and a lot of others, you are guilty of nobility Mustache.
What do you know? Mustache - impossible in principle. How to improve mechanical memory.
And it’s very easy, in fact elementary – just like vikorystuvati trigonometrically. Once you master the practical work with a trigonometric stake, all these thirsty questions in degrees will be easily and elegantly reduced to the good old ones:
Before I speak, I still have a few more great sites for you.)
You can practice with advanced tools and learn your skill. Testing with mitta verification. Check it out - with interest!)
You can learn about the functions and related ones.
The concepts of sine (), cosine (), tangent (), cotangent () are inextricably linked with the concepts of kuta. In order to get a good understanding of these, at first glance, simple concepts (as the rich schoolchildren cry out), and realize that “the devil is not as terrible as they paint him,” let’s finally get to the bottom of the concept.
Meaning of kuta: radian, degree
Let's marvel at the little one. The vector has “turned” around the point by a small amount. So the axis of the world of this rotation is in a cob position and is visible kut.
What else do you need to know about the concept of kuta? Well, of course, only a few will die!
Both geometry and trigonometry can be expressed in degrees and radians.
Kutom (one degree) is the name given to the central corner in the stake, which spirals into a circular arc equal to part of the stake. In this way, everything is made up of “pieces” of circular arcs, and the place that is described by the stake is ancient.
So that the little one is more like a cut, rive, then that whole cut spirals into a circular arc the size of a dove stake.
Kutom among the Radians is the name given to the central kutum in the stake, which spirals into a circular arc, the length of which is equal to the radius of the stake. Well, have you gotten over it? If not, then let’s take it easy on the little one.
Well, the little image is equal to the radian, so that this cut spirals into a circular arc, the length of which is equal to the radius of the stake (the length is the same as the radius or the radius is the same as the arc). Thus, the dovzhina of the arc is calculated using the following formula:
De is the central corner of the radians.
Well, can you, if you know, tell us how many radians there are in the circle that the circle describes? So, for this purpose, guess the formula of Dovzhini Kola. Axis:
Well, now it’s clear that there are two formulas and we can see what is being described in the same way. Then, based on the magnitude in degrees and radians, we can deduce that. Apparently, . As you can see, in the submenu of “degrees”, the word “radian” is omitted, so that only one word can be understood in context.
How many radians add up? Everything is true!
Having caught it? Then fasten it forward:
Vinikly problems? Then marvel types:
Rectocutant: sine, cosine, tangent, cotangent cotangent
Well, we've got the hang of things. What is sine, cosine, tangent, cotangent? Let's find out. For this purpose, a straight-cut tricutnik will help us.
What are the names of the sides of the straight-cut tricutaneous? Everything is correct, the hypotenuse and the legs: the hypotenuse is the side that lies opposite the straight cut (the butt has the same side); legs - there are two sides that have lost i (those that fit up to the straight edge), and, if you look at the legs straight from the edge, then the leg is a snug leg, and the leg is a flat one. Well, now let’s give you an answer to the question: what is sine, cosine, tangent and cotangent?
Sinus kuta- Placement of the protilage (distant) leg to the hypotenusus.
For our trikutnik.
Cosine of Kuta- The aim is to place an adjacent (close) leg to the hypotenusus.
For our trikutnik.
Tangent of cut- Place the protilage (distant) leg to the adjacent (close) leg.
For our trikutnik.
Cotangent kuta- The aim is to place the adjacent (close) leg to the protimal (far) leg.
For our trikutnik.
These are essential remember! To make it easier to remember which leg to divide, it is necessary to clearly understand what is in tangentі cotangent sit only side by side, and the hypotenuse appears only in sinusesі cosine. And then you can come up with a little association. For example, the axis is like this:
Cosine → stick → stick → diligent;
Cotangent → sticking → touching → diligent.
First of all, it is necessary to remember that the sine, cosine, tangent and cotangent of the three sides of the triangle do not lie under the same sides (with one vugilla). Don't you believe it? Then look over, marveling at the little ones:
Let's look at, for example, the cosine of the cut. For the meanings, from the trikuputnik: , but we can calculate the cosine of the kut i from the trikuputnik: . Bachish, there is a lot of strife on both sides, and the value of the cosine of one is the same. Thus, the values of sine, cosine, tangent and cotangent lie inclusive of the value of kut.
Once you have sorted out the assignments, then fasten them forward!
For the tricutnik, depicted below the baby, we know.
Well, did you catch it? Then try it yourself: fuck those same ones for the kick.
Single (trigonometric) colo
Understanding the concepts of degrees and radians, we looked at them with equal radii. That's what it's called single. You really need some trigonometry. That's why we'll spend a little time on it.
As you may note, this was created in a Cartesian coordinate system. The radius of the stake is an ancient unit, in which the center of the stake lies on the cob coordinates, the cob position of the radius vector is fixed along the positive direction of the axis (in our example, the radius).
The skin point of the stake is indicated by two numbers: axial coordinate and axial coordinate. What are these coordinate numbers? And how can the stench linger to this day? For this purpose, it is necessary to make a fortune for the straight-cut tricutnik. On a small fish, if you look at it, you can spot as many as two straight cutlets. Let's take a look at the trikutnik. It is straight, the fragments are perpendicular to the axis.
What is similar to trikutnik? Everything is true. In addition, we know that this is the radius of a single stake, which means . Let's substitute the value into our formula for cosine. Axis to go:
And why is it similar to the tricutanea? Well, obviously! Let us substitute the radius value into this formula and remove it:
So, can you tell me what coordinates the point has to put the stake on? Well, so what? How can you figure out that they are not just numbers? Which coordinates does it indicate? Well, okay, coordinates! What coordinates does it indicate? Everything is correct, coordinates! In this manner, period.
Why are we jealous? Everything is correct, it is accelerated by the corresponding tangent and cotangent values and is taken away, as well.
What if it will be bigger? The axis, for example, is like this picture:
What has changed in whose butt? Let's find out. For this reason I am going wild to the straight-cut tricutaneous. Let's take a look at the straight-cut trikutnik: kut (yak lay down to the kut). What is the difference between the meanings of sine, cosine, tangent and cotangent? Everything is correct, we can calculate the following values of trigonometric functions:
Well, as a matter of fact, the value of the sine indicates the coordinates; cosine values - coordinates; and the meaning of tangent and cotangent in similar relationships. In this way, the relationship can be frozen until any rotation of the radius vector.
I had already guessed that the cob position of the radius vector is the reinstatement of the positive straightening of the axis. We have already wrapped this vector against the anniversary arrow, but what happens if we turn it behind the anniversary arrow? Nothing extraordinary, it will just come out of a mere magnitude, but it will be negative. In this manner, with the radius vector wrapped around the year arrow, go out positive vibes, and when wrapped behind the anniversary arrow - negative.
Well, we know that the whole circle of the radius vector along the stake can be set either. How can you rotate the radius vector by an hour? Well, of course, it’s possible! At the first drop, in this manner, the radius vector makes one additional turn and settles into position.
In the other case, the radius vector should be rotated three times and return to its original position.
In this way, from pointing the butts, we can create a pattern such that the parts that intersect on either (if it is an integer number) correspond to one position of the radius vector.
Below on the baby there is a picture of a cut. This image confirms the situation. This list can be continued ad infinitum. All qi can be written using the halal formula abo (de – be a whole number)
Now, knowing the meanings of the basic trigonometric functions and the vikorist ones, try to guess why the values are equal:
I will help you one more time:
Vinikly problems? Todi, let's find out. Well, we know what:
The stars indicate the coordinates of the points that correspond to the first approaches to the kut. Well, let's start in order: the corner is indicated by a point with coordinates, then:
Not sleeping;
Further, following the same logic, it is clear that points with coordinates represent points, obviously. As you know, it is easy to calculate the values of trigonometric functions at certain points. Try it yourself for the first time, and then see the testimonials.
Types:
In this manner, we can fold the following sign:
There is no need to remember all the values. It is sufficient to remember the type of coordinates of points on a single column and the value of trigonometric functions:
And the axis of significance of the trigonometric functions of cutives i, indicated below in the table, must be remembered:
No need to shout, we’ll show you one of the butts now To achieve simple memorization of the corresponding values:
To use this method in life, it is necessary to memorize the values of the sine for all three entries of the cut (), as well as the value of the tangent of the cut. Knowing these values, you can simply update the entire table - the cosine values are transferred consistently to the arrows, then:
Knowingly, it is possible to update the meaning. The numeral " " will be confirmed, and the sign " " will be confirmed. The cotangent values are transferred consistently to the arrows indicated on the panel. Once you know and remember the diagram with the arrows, it is enough to remember all the values in the table.
Coordinates of the point on the stake
How can you know the point (its coordinates) on the stake, know the coordinates to the center of the stake, its radius and turn?
Well, of course, it’s possible! Let's get to know each other Here's the formula for finding the coordinates of a point.
The axis, for example, is in front of us like this:
We are given that the point is the center of the stake. The radius of the stake is ancient. It is necessary to know the coordinates of the point reached by rotating the point by degrees.
As is obvious from the little one, the coordinates of the point are confirmed by the end of the cut. The last cut indicates the coordinates of the center of the stake, then it is aligned. Dovzhin can be expressed using the value of the cosine:
Then we can say what for the coordinate point.
Using the same logic, we can find the coordinate values for the point. In such a manner
Well, in a formal way, the coordinates of the points are indicated by the formulas:
Coordinate to the center of the stake,
Radius stake,
Let's turn the radius vector.
As you can note, for a single stake, which we can see, the formulas are significantly shortened, the fragments of the coordinate center become zero, and the radius becomes equal to one:
Well, let's try these formulas for relish, getting better from the well-known spots on the stake?
1. Find out the coordinates of a point on a single stake, determined by rotating the point on.
2. Find out the coordinates of a point on a single stake, determined by rotating the point on.
3. Find out the coordinates of a point on a single stake, determined by rotating the point on.
4. Speck - the center of the stake. The radius of the stake is ancient. It is necessary to know the coordinates of the point traced by rotating the cob radius vector.
5. Speck - the center of the stake. The radius of the stake is ancient. It is necessary to know the coordinates of the point traced by rotating the cob radius vector.
Are there problems with the known coordinates of the point on the stake?
Unravel the five butts (or take a good look at the solution) and you will learn to know them!
SHORT VIKLAD AND BASIC FORMULAS
The sinus of the cutus is the extension of the protilage (distant) leg to the hypotenusus.
The cosine of the cut is the value of the adjacent (close) leg to the hypotenusus.
The tangent of the cut is the extension of the protidal (far) leg to the adjacent (close) leg.
The cotangent of the cut is the extension of the adjacent (close) leg to the prostrate (far) leg.
Well, that's it, the topic is over. If you read a whole series, that means you are even cooler.
Because only 5% of people master this on their own. And once you’ve read to the end, you’ve lost 5%!
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