Cos 7 degrees. Argument and meaning

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 (in degrees)	0	90	180	270	360
Angle x (in radians)	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 (in degrees)	0	30	45	60	90
Angle x (in radians)	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.

Examples:

\(\cos(⁡30^°)=\)\(\frac(\sqrt(3))(2)\)
\(\cos⁡\)\(\frac(π)(3)\) \(=\)\(\frac(1)(2)\)
\(\cos⁡2=-0.416…\)

Argument and meaning

Cosine of an acute angle

Cosine of an acute angle can be determined using a right triangle - it is equal to the ratio of the adjacent leg to the hypotenuse.

Example :

1) Let an angle be given and we need to determine the cosine of this angle.

2) Let us complete any right triangle on this angle.

3) Having measured the required sides, we can calculate the cosine.

The cosine of an acute angle is greater than \(0\) and less than \(1\)

If, when solving a problem, the cosine of an acute angle turns out to be greater than 1 or negative, then there is an error somewhere in the solution.

Cosine of a number

The number circle allows you to determine the cosine of any number, but usually you find the cosine of numbers somehow related to: \(\frac(π)(2)\) , \(\frac(3π)(4)\) , \(-2π\ ).

For example, for the number \(\frac(π)(6)\) - the cosine will be equal to \(\frac(\sqrt(3))(2)\) . And for the number \(-\)\(\frac(3π)(4)\) it will be equal to \(-\)\(\frac(\sqrt(2))(2)\) (approximately \(-0 ,71\)).

For cosine for other numbers often encountered in practice, see.

The cosine value always lies in the range from \(-1\) to \(1\). In this case, the cosine can be calculated for absolutely any angle and number.

Cosine of any angle

Thanks to the number circle, you can determine the cosine of not only an acute angle, but also an obtuse, negative, and even greater than \(360°\) (full revolution). How to do this is easier to see once than to hear \(100\) times, so look at the picture.

Now an explanation: suppose we need to determine the cosine of the angle KOA with degree measure in \(150°\). Combining the point ABOUT with the center of the circle, and the side OK– with the \(x\) axis. After this, set aside \(150°\) counterclockwise. Then the ordinate of the point A will show us the cosine of this angle.

If we are interested in an angle with a degree measure, for example, in \(-60°\) (angle KOV), we do the same, but we set \(60°\) clockwise.

And finally, the angle is greater than \(360°\) (angle CBS) - everything is similar to the stupid one, only after going clockwise a full turn, we go to the second circle and “get the lack of degrees”. Specifically, in our case, the angle \(405°\) is plotted as \(360° + 45°\).

It’s easy to guess that to plot an angle, for example, in \(960°\), you need to make two turns (\(360°+360°+240°\)), and for an angle in \(2640°\) - whole seven.

It's worth remembering that:

The cosine of a right angle is zero. The cosine of an obtuse angle is negative.

Cosine signs by quarters

Using the cosine axis (that is, the abscissa axis, highlighted in red in the figure), it is easy to determine the signs of the cosines along the numerical (trigonometric) circle:

Where the values ​​on the axis are from \(0\) to \(1\), the cosine will have a plus sign (I and IV quarters - green area),
- where the values ​​on the axis are from \(0\) to \(-1\), the cosine will have a minus sign (II and III quarters - purple area).

Example. Determine the sign of \(\cos 1\).
Solution: Let's find \(1\) on the trigonometric circle. We will start from the fact that \(π=3.14\). This means that one is approximately three times closer to zero (the “start” point).

If you draw a perpendicular to the cosine axis, it becomes obvious that \(\cos⁡1\) is positive.
Answer: plus.

Relation to other trigonometric functions:

- the same angle (or number): the basic trigonometric identity \(\sin^2⁡x+\cos^2⁡x=1\)
- the same angle (or number): by the formula \(1+tg^2⁡x=\)\(\frac(1)(\cos^2⁡x)\)
- and the sine of the same angle (or number): the formula \(ctgx=\)\(\frac(\cos(x))(\sin⁡x)\)
For other most commonly used formulas, see.

Function \(y=\cos(x)\)

If we plot the angles in radians along the \(x\) axis, and the cosine values ​​corresponding to these angles along the \(y\) axis, we get the following graph:

This graph is called and has the following properties:

The domain of definition is any value of x: \(D(\cos(⁡x))=R\)
- range of values ​​– from \(-1\) to \(1\) inclusive: \(E(\cos(x))=[-1;1]\)
- even: \(\cos⁡(-x)=\cos(x)\)
- periodic with period \(2π\): \(\cos⁡(x+2π)=\cos(x)\)
- points of intersection with coordinate axes:
abscissa axis: \((\)\(\frac(π)(2)\) \(+πn\),\(;0)\), where \(n ϵ Z\)
Y axis: \((0;1)\)
- intervals of constancy of sign:
the function is positive on the intervals: \((-\)\(\frac(π)(2)\) \(+2πn;\) \(\frac(π)(2)\) \(+2πn)\), where \(n ϵ Z\)
the function is negative on the intervals: \((\)\(\frac(π)(2)\) \(+2πn;\)\(\frac(3π)(2)\) \(+2πn)\), where \(n ϵ Z\)
- intervals of increase and decrease:
the function increases on the intervals: \((π+2πn;2π+2πn)\), where \(n ϵ Z\)
the function decreases on the intervals: \((2πn;π+2πn)\), where \(n ϵ Z\)
- maximums and minimums of the function:
the function has a maximum value \(y=1\) at points \(x=2πn\), where \(n ϵ Z\)
the function has a minimum value \(y=-1\) at points \(x=π+2πn\), where \(n ϵ Z\).