(pi / 4) in three ways.

First.
This method is most often used when solving trigonometric equations in school. It consists in using , which contains the values ​​of four trigonometric functions from the most common arguments.

Such tables exist in several versions. They differ in that the values ​​of the angles are presented in degrees, in radians, or both in degrees and radians (which is most convenient).
In the table we find the angle (in this case pi / 4) and the desired function (we need the cosine function) and at the intersection of these values ​​we get the root of 2 / 2.
Mathematically it is written like this:

Second.
Also a common way that can always be used if there is no table. It consists in using (or a trigonometric circle).


On such a trigonometric circle, the cosine values ​​​​are located on the horizontal axis - the abscissa axis, and the arguments - on the curve of the circle itself.
In our case, the argument of the cosine is pi / 4. Let's determine where this value is located on the circle. Next, we lower the perpendicular to the x-axis. The value in which the end of this perpendicular will be will be the value of the given cosine. Therefore, the cosine of pi / 4 is the square root of 2 / 2.

The third.
It is also convenient to use the graph of the corresponding function - . It's easy to remember what it looks like.


When using a graph, some knowledge is needed to determine the value of the cosine pi / 4, which is . In this case, you need to understand that the value of the fraction is greater than 0.5 and less than 1.
There are, of course, several other ways. For example, calculating the value of cosine using a calculator. But for this you first need to convert the angle pi / 4 to degrees. Bradis tables may also be useful.

Degree measure of an angle. The radian measure of an angle. Convert degrees to radians and vice versa.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

In the previous lesson, we mastered the counting of angles on a trigonometric circle. Learned how to count positive and negative angles. Realized how to draw an angle greater than 360 degrees. It's time to deal with the measurement of angles. Especially with the number "Pi", which strives to confuse us in tricky tasks, yes ...

Standard tasks in trigonometry with the number "Pi" are solved quite well. Visual memory helps. But any deviation from the template - knocks down on the spot! In order not to fall - understand necessary. What we will successfully do now. In a sense - we understand everything!

So, what do angles count? In the school course of trigonometry, two measures are used: degree measure of an angle and radian measure of an angle. Let's take a look at these measures. Without this, in trigonometry - nowhere.

Degree measure of an angle.

We are somehow used to degrees. Geometry, at the very least, went through ... Yes, and in life we ​​often meet with the phrase "turned 180 degrees", for example. Degree, in short, a simple thing ...

Yes? Answer me then what is a degree? What doesn't work right off the bat? Something...

Degrees were invented in ancient Babylon. It was a long time ago ... 40 centuries ago ... And they just came up with it. They took and broke the circle into 360 equal parts. 1 degree is 1/360 of a circle. And that's it. Could be broken into 100 pieces. Or by 1000. But they broke it into 360. By the way, why exactly by 360? Why is 360 better than 100? 100 seems to be somehow more even... Try to answer this question. Or weak against Ancient Babylon?

Somewhere at the same time, in ancient Egypt, they were tormented by another issue. How many times greater is the circumference of a circle than the length of its diameter? And so they measured, and that way ... Everything turned out a little more than three. But somehow it turned out shaggy, uneven ... But they, the Egyptians, are not to blame. After them, they suffered for another 35 centuries. Until they finally proved that no matter how finely cut the circle into equal pieces, from such pieces to make smooth the length of the diameter is impossible ... In principle, it is impossible. Well, how many times the circumference is larger than the diameter, of course. About. 3.1415926... times.

This is the number "Pi". That's shaggy, so shaggy. After the decimal point - an infinite number of digits without any order ... Such numbers are called irrational. This, by the way, means that from equal pieces of a circle, the diameter smooth do not fold. Never.

For practical use, it is customary to remember only two digits after the decimal point. Remember:

Since we have understood that the circumference of a circle is greater than the diameter by "Pi" times, it makes sense to remember the formula for the circumference of a circle:

Where L is the circumference, and d is its diameter.

Useful in geometry.

For general education, I will add that the number "Pi" sits not only in geometry ... In various sections of mathematics, and especially in probability theory, this number appears constantly! By itself. Beyond our desires. Like this.

But back to degrees. Have you figured out why in ancient Babylon the circle was divided into 360 equal parts? But not 100, for example? Not? OK. I'll give you a version. You can't ask the ancient Babylonians... For construction, or, say, astronomy, it is convenient to divide a circle into equal parts. Now figure out what numbers are divisible by completely 100, and which ones - 360? And in what version of these dividers completely- more? This division is very convenient for people. But...

As it turned out much later than Ancient Babylon, not everyone likes degrees. Higher mathematics does not like them... Higher mathematics is a serious lady, arranged according to the laws of nature. And this lady declares: “Today you broke the circle into 360 parts, tomorrow you will break it into 100 parts, the day after tomorrow into 245 ... And what should I do? No really ...” I had to obey. You can't fool nature...

I had to introduce a measure of the angle that does not depend on human notions. Meet - radian!

The radian measure of an angle.

What is a radian? The definition of a radian is based on a circle anyway. An angle of 1 radian is the angle that cuts an arc from a circle whose length is ( L) is equal to the length of the radius ( R). We look at the pictures.

Such a small angle, there is almost none of it ... We move the cursor over the picture (or touch the picture on the tablet) and we see about one radian. L=R

Feel the difference?

One radian is much larger than one degree. How many times?

Let's look at the next picture. On which I drew a semicircle. The expanded angle is, of course, 180 ° in size.

And now I will cut this semicircle into radians! We hover over the picture and see that 3 radians with a tail fit into 180 °.

Who can guess what this ponytail is!?

Yes! This tail is 0.1415926.... Hello Pi, we haven't forgotten you yet!

Indeed, there are 3.1415926 ... radians in 180 degrees. As you can imagine, writing 3.1415926 all the time... is inconvenient. Therefore, instead of this infinite number, they always write simply:

And here is the number on the Internet

it is inconvenient to write ... Therefore, in the text I write it by name - "Pi". Don't get confused...

Now, it is quite meaningful to write an approximate equality:

Or exact equality:

Determine how many degrees are in one radian. How? Easily! If there are 180 degrees in 3.14 radians, then 1 radian is 3.14 times less! That is, we divide the first equation (the formula is also an equation!) By 3.14:

This ratio is useful to remember. There are approximately 60° in one radian. In trigonometry, you often have to figure out, evaluate the situation. This is where knowledge helps a lot.

But the main skill of this topic is converting degrees to radians and vice versa.

If the angle is given in radians with the number "pi", everything is very simple. We know that "pi" radians = 180°. So we substitute instead of "Pi" radians - 180 °. We get the angle in degrees. We reduce what is reduced, and the answer is ready. For example, we need to find out how much degrees in the corner "Pi"/2 radian? Here we write:

Or, more exotic expression:

Easy, right?

The reverse translation is a little more complicated. But not much. If the angle is given in degrees, we must figure out what one degree is in radians and multiply that number by the number of degrees. What is 1° in radians?

We look at the formula and realize that if 180° = "Pi" radians, then 1° is 180 times smaller. Or, in other words, we divide the equation (the formula is also an equation!) By 180. There is no need to represent "Pi" as 3.14, it is always written with a letter anyway. We get that one degree is equal to:

That's all. Multiply the number of degrees by this value to get the angle in radians. For example:

Or, similarly:

As you can see, in a leisurely conversation with lyrical digressions, it turned out that radians are very simple. Yes, and the translation is without problems ... And "Pi" is a completely tolerable thing ... So where is the confusion from !?

I'll reveal the secret. The fact is that in trigonometric functions the degrees icon is written. Always. For example, sin35°. This is sine 35 degrees . And the radians icon ( glad) is not written! He is implied. Either the laziness of mathematicians seized, or something else ... But they decided not to write. If there are no icons inside the sine - cotangent, then the angle - in radians ! For example, cos3 is the cosine of three radians .

This leads to misunderstandings ... A person sees "Pi" and believes that it is 180 °. Anytime and anywhere. By the way, this works. For the time being, while the examples are standard. But Pi is a number! The number 3.14 is not degrees! That's "Pi" radians = 180°!

Once again: "Pi" is a number! 3.14. Irrational, but a number. Same as 5 or 8. You can, for example, take about "Pi" steps. Three steps and a little more. Or buy "Pi" kilograms of sweets. If an educated salesman gets caught...

"Pi" is a number! What, I got you with this phrase? Have you already understood everything? OK. Let's check. Can you tell me which number is greater?

Or what is less?

This is from a series of slightly non-standard questions that can drive into a stupor ...

If you also fell into a stupor, remember the spell: "Pi" is a number! 3.14. In the very first sine, it is clearly indicated that the angle - in degrees! Therefore, it is impossible to replace "Pi" by 180 °! "Pi" degrees is about 3.14 degrees. Therefore, we can write:

There are no symbols in the second sine. So there - radians! Here, replacing "Pi" with 180 ° will work quite well. Converting radians to degrees, as written above, we get:

It remains to compare these two sines. What. forgot how? With the help of a trigonometric circle, of course! We draw a circle, draw approximate angles of 60° and 1.05°. We look at the sines of these angles. In short, everything, as at the end of the topic about the trigonometric circle, is painted. On a circle (even the crooked one!) it will be clearly seen that sin60° significantly more than sin1.05°.

We will do exactly the same with cosines. On the circle we draw angles of about 4 degrees and 4 radian(remember, what is approximately 1 radian?). The circle will say everything! Of course, cos4 is less than cos4°.

Let's practice handling angle measures.

Convert these angles from degrees to radians:

360°; 30°; 90°; 270°; 45°; 0°; 180°; 60°

You should end up with these values ​​in radians (in a different order!)

0

By the way, I have specially marked out the answers in two lines. Well, let's figure out what the corners are in the first line? Whether in degrees or radians?

Yes! These are the axes of the coordinate system! If you look at the trigonometric circle, then the moving side of the angle at these values fits right on the axle. These values ​​need to be known ironically. And I noted the angle of 0 degrees (0 radians) not in vain. And then some cannot find this angle on the circle in any way ... And, accordingly, they get confused in the trigonometric functions of zero ... Another thing is that the position of the moving side at zero degrees coincides with the position at 360 °, so coincidences on the circle are all the time near.

In the second line there are also special angles... These are 30°, 45° and 60°. And what is so special about them? Nothing special. The only difference between these corners and all the others is that you should know about these corners. all. And where are they located, and what are the trigonometric functions of these angles. Let's say the value sin100° you don't have to know. BUT sin45°- please be kind! This is mandatory knowledge, without which there is nothing to do in trigonometry ... But more on this in the next lesson.

Until then, let's keep practicing. Convert these angles from radians to degrees:

You should get results like this (in a mess):

210°; 150°; 135°; 120°; 330°; 315°; 300°; 240°; 225°.

Happened? Then we can assume that converting degrees to radians and vice versa- not your problem anymore.) But translating angles is the first step to understanding trigonometry. In the same place, you still need to work with sines-cosines. Yes, and with tangents, cotangents too ...

The second powerful step is the ability to determine the position of any angle on a trigonometric circle. Both in degrees and radians. About this very skill, I will boringly hint to you in all trigonometry, yes ...) If you know everything (or think you know everything) about the trigonometric circle, and the counting of angles on the trigonometric circle, you can check it out. Solve these simple tasks:

1. What quarter do the corners fall into:

45°, 175°, 355°, 91°, 355° ?

Easily? We continue:

2. In which quarter do the corners fall:

402°, 535°, 3000°, -45°, -325°, -3000°?

Also no problem? Well, look...)

3. You can place corners in quarters:

Were you able? Well, you give ..)

4. What axes will the corner fall on:

and corner:

Is it easy too? Hm...)

5. What quarter do the corners fall into:

And it worked!? Well, then I really don't know...)

6. Determine which quarter the corners fall into:

1, 2, 3 and 20 radians.

I will give the answer only to the last question (it is slightly tricky) of the last task. An angle of 20 radians will fall into the first quarter.

I won’t give the rest of the answers out of greed.) Just if you didn't decide something doubt as a result, or spent on task No. 4 more than 10 seconds you are poorly oriented in a circle. This will be your problem in all trigonometry. It is better to get rid of it (a problem, not trigonometry!) right away. This can be done in the topic: Practical work with a trigonometric circle in section 555.

It tells how to solve such tasks simply and correctly. Well, these tasks are solved, of course. And the fourth task was solved in 10 seconds. Yes, so decided that anyone can!

If you are absolutely sure of your answers and you are not interested in simple and trouble-free ways to work with radians, you can not visit 555. I do not insist.)

A good understanding is a good enough reason to move on!)

If you like this site...

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. Learning - with interest!)

you can get acquainted with functions and derivatives.

Table of values ​​of trigonometric functions

Note. This table of values ​​for trigonometric functions uses the √ sign to denote the 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