Trigonometry table. Well, let's try these formulas for a taste, practicing finding points on a circle? The proposed mathematical apparatus is a complete analog of the complex calculus for n-dimensional hypercomplex numbers with any number of degrees with

Attention!
There are additional
material in Special Section 555.
For those who strongly "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?"

Here is that output:

The sine, cosine, tangent, and cotangent are tightly connected to their angles. We know one thing, so we know something else.

In other words, each angle has its own fixed sine and cosine. And almost everyone has their own tangent and cotangent. Why almost? More on that below.

This knowledge will help you a lot! There are many tasks where you need to go from sines to angles and vice versa. For this there is sine table. Similarly, for jobs with cosine - cosine table. And, you guessed it, there is tangent table and cotangent table.)

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. Of course, memorizing this number (and thousands of others table values) is not required at all.

In fact, in our time, long tables of cosines, sines, tangents, and 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.)

Why then this lesson? - you ask.

But why. Among the infinite number of angles there are special, about which you should know 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, get ready to get a well-deserved deuce...

Such special corners are also decently typed. School textbooks are usually kindly offered for memorization. sine table and cosine table for seventeen corners. And, of course, tangent table and cotangent table for the same seventeen corners... That is. it is proposed to remember 68 values. Which, by the way, are very similar to each other, repeat and change signs every now and then. For a person without an ideal visual memory - that's another task ...)

We will go the other way. Let's replace mechanical memorization with logic and ingenuity. Then we 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. And that's it. Six values ​​are easier to remember than 68, I think...)

Other required values we'll get out of these six with a powerful legal cheat sheet - trigonometric circle. If you have not studied this topic, go to the link, do not be lazy. This circle is not only 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 sine table. cosine table. Tangent table. Cotangent table. All 68 values ​​for various angles.)

So, let's begin. To begin with, let's break all these special angles into three groups.

The first group of corners.

Consider the first group corners of seventeen special. These are 5 angles: 0°, 90°, 180°, 270°, 360°.

This is how the table of sines, cosines, tangents, and cotangents for these angles looks like:

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	not noun	0	not noun	0
ctg x	not noun	0	not noun	0	not noun

Those who want to remember - remember. But I must say right away that all these ones and zeros are very confused in my head. Much stronger than you want.) Therefore, we turn on the 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:

You can immediately see what the peculiarity of these corners is. Yes! These are the corners that fall exactly on the coordinate axis! Actually, that's why people get confused ... But we will not get confused. Let's figure out how to find the trigonometric functions of these angles without much memorization.

By the way, the position of the angle is 0 degrees completely coincides with a 360 degree angle. This means that the sines, cosines, tangents of these angles are exactly the same. I marked the 360 ​​degree angle to complete the circle.

Suppose, in a difficult stressful environment of the Unified State Examination, you somehow doubted ... What equals sine 0 degrees? It seems like zero ... What if it's a unit?! Mechanical memory is such a thing. In harsh conditions, doubts begin to gnaw ...)

Calm, only calm!) I'll tell you practical technique, which will give a 100% correct answer and completely remove all doubts.

As an example, let's figure out how to clearly and reliably determine, say, a 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 injection X. In the first quarter, so that it was not far from 0 degrees. Note on the axes the sine and cosine of this angle X, everything is chinar. Like this:

And now - attention! Decrease the angle X, bring the movable side to the axis OH. Hover over the picture (or touch the picture on the tablet) and see everything.

Now turn on the elementary logic!. Watch and think: How does sinx behave when 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 will the moving side of the angle (point A) settle down on the OX axis and the angle become equal to zero? Obviously, the sine of the angle will also 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)? Unity!

Here is the answer. The sine of 0 degrees is 0. The cosine of 0 degrees is 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 catch what the sine and cosine will become when the side of the angle settles 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 sine table... Yes and cosine table- too.) By the way, after several applications of the trigonometric circle, all these values ​​\u200b\u200bare remembered by themselves. And if they are forgotten, I drew a circle in 5 seconds and clarified it. Much easier than calling a friend from the toilet with the risk of a certificate, right?)

As for the tangent and cotangent, everything is the same. We draw a line of tangent (cotangent) on the circle - and everything is immediately visible. Where they are equal to zero, and where they do not exist. What, don't you know about the lines of tangent and cotangent? This is sad, but fixable.) Visited Section 555 Tangent and cotangent on a trigonometric circle - and no problem!

If you understand how to clearly define the sine, cosine, tangent and cotangent for these five angles - congratulations! Just in case, I inform you that you can now define functions any angles that fall on the axis. And this is 450°, and 540°, and 1800°, and even an infinite number ...) I counted (correctly!) The angle on the circle - and there are no problems with the functions.

But, just with the counting of angles, 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.)

And here is the lesson: How to draw (count) any angle on a trigonometric circle in radians - it will be more abrupt. In terms of possibilities. Let's say, determine which of the four semiaxes the angle falls on

you can 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 good for an instantaneous answer.

What if the angle

Think! The correct answer is obtained in 10 seconds. For any fractional value radians with a denominator of two.

Actually, this is good trigonometric circle. The fact that the ability to work with some corners it automatically expands to infinite set corners.

So, with five corners out of seventeen - figured it out.

The second group of angles.

Next group angles are 30°, 45° and 60°. Why these, and not, for example, 20, 50 and 80? Yes, it somehow happened like this ... Historically.) Further it will be seen how good these angles are.

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		not noun
ctg x	not noun		1		0

I left the values ​​for 0° and 90° from the previous table for completeness.) To make it clear that these angles lie in the first quarter and increase. From 0 to 90. This will be useful to us further.

The table values ​​for the angles 30°, 45° and 60° must be memorized. Scratch if you want. But here, too, there is an opportunity to make life easier for yourself.) Pay attention to sine table values these corners. And compare with cosine table values...

Yes! They are same! Only located in reverse order. The angles increase (0, 30, 45, 60, 90) - and the sine values increase from 0 to 1. You can verify with a calculator. And the cosine values ​​- decrease from 1 to zero. Moreover, the values ​​themselves same. For angles of 20, 50, 80 this would not have happened...

Hence a useful conclusion. Enough to learn three values ​​for angles 30, 45, 60 degrees. And remember that they increase in the sine, and decrease in the cosine. Towards the sine.) Half way (45°) they meet, i.e. 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 exclusively the same. One to one. Only the values ​​are different. These values ​​(three more!) also need to be learned.

Well, almost all memorization is over. You understood (hopefully) how to determine the values ​​for the five angles that fall 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 corners:
120°; 135°; 150°; 210°; 225°; 240°; 300°; 315°; 330°. For these angles, you need to know the iron table of sines, the table of cosines, etc.

Nightmare, right?)

And if you add angles here, like: 405 °, 600 °, or 3000 ° and many, many of the same beautiful?)

Or angles in radians? For example, about corners:

and many more you should know all.

The funniest thing is to know all - impossible in principle. If you use mechanical memory.

And it is very easy, actually elementary - if you use a trigonometric circle. If you get hands-on with the trigonometric circle, all those awful angles in degrees can be easily and elegantly reduced to the good old 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. Learning - with interest!)

you can get acquainted with functions and derivatives.

We begin our study of trigonometry with a right triangle. Let's define what the sine and cosine are, as well as the tangent and cotangent of an acute angle. These are the basics of trigonometry.

Recall that right angle is an angle equal to 90 degrees. In other words, half of the unfolded corner.

Sharp corner- less than 90 degrees.

Obtuse angle- greater than 90 degrees. In relation to such an angle, "blunt" is not an insult, but a mathematical term :-)

Let's draw a right triangle. A right angle is usually denoted . Note that the side opposite the corner is denoted by the same letter, only small. So, the side lying opposite the angle A is denoted.

An angle is denoted by the corresponding Greek letter.

Hypotenuse A right triangle is the side opposite the right angle.

Legs- sides opposite sharp corners.

The leg opposite the corner is called opposite(relative to angle). The other leg, which lies on one side of the corner, is called adjacent.

Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:

Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:

Tangent acute angle in a right triangle - the ratio of the opposite leg to the adjacent:

Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of an angle to its cosine:

Cotangent acute angle in a right triangle - the ratio of the adjacent leg to the opposite (or, equivalently, the ratio of cosine to sine):

Pay attention to the basic ratios for sine, cosine, tangent and cotangent, which are given below. They will be useful to us in solving problems.

Let's prove some of them.

Okay, we have given definitions and written formulas. But why do we need sine, cosine, tangent and cotangent?

We know that the sum of the angles of any triangle is.

We know the relationship between parties right triangle. This is the Pythagorean theorem: .

It turns out that knowing two angles in a triangle, you can find the third one. Knowing two sides in a right triangle, you can find the third. So, for angles - their ratio, for sides - their own. But what to do if in a right triangle one angle (except for a right one) and one side are known, but you need to find other sides?

This is what people faced in the past, making maps of the area and the starry sky. After all, it is not always possible to directly measure all the sides of a triangle.

Sine, cosine and tangent - they are also called trigonometric functions of the angle- give the ratio between parties and corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.

We will also draw a table of sine, cosine, tangent and cotangent values ​​for "good" angles from to.

Notice the two red dashes in the table. For the corresponding values ​​of the angles, the tangent and cotangent do not exist.

Let's analyze several problems in trigonometry from the Bank of FIPI tasks.

1. In a triangle, the angle is , . Find .

The problem is solved in four seconds.

Insofar as , .

2. In a triangle, the angle is , , . Find .

Let's find by the Pythagorean theorem.

Problem solved.

Often in problems there are triangles with angles and or with angles and . Memorize the basic ratios for them by heart!

For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.

A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.

We considered problems for solving right triangles - that is, for finding unknown parties or corners. But that's not all! AT USE options in mathematics, there are many problems where the sine, cosine, tangent or cotangent of the outer angle of the triangle appears. More on this in the next article.

Main table trigonometric functions for angles 0, 30, 45, 60, 90, … degrees

From the trigonometric definitions of the functions $\sin$, $\cos$, $\tan$, and $\cot$, one can find their values ​​for angles $0$ and $90$ degrees:

$\sin⁡0°=0$, $\cos0°=1$, $\tan 0°=0$, $\cot 0°$ not defined;

$\sin90°=1$, $\cos90°=0$, $\cot90°=0$, $\tan 90°$ is not defined.

AT school course geometries in the study of right triangles find the trigonometric functions of the angles $0°$, $30°$, $45°$, $60°$ and $90°$.

The found values ​​of trigonometric functions for the specified angles in degrees and radians respectively ($0$, $\frac(\pi)(6)$, $\frac(\pi)(4)$, $\frac(\pi)(3) $, $\frac(\pi)(2)$) for ease of memorization and use are entered in a table called trigonometric table, table of basic values ​​of trigonometric functions etc.

When using reduction formulas, trigonometric table can be expanded to $360°$ and $2\pi$ radians respectively:

Applying the periodicity properties of trigonometric functions, each angle that differs from the already known one by $360°$ can be calculated and recorded in a table. For example, the trigonometric function for the angle $0°$ will have the same value for the angle $0°+360°$, and for the angle $0°+2 \cdot 360°$, and for the angle $0°+3 \cdot 360°$ and etc.

Using a trigonometric table, you can determine the values ​​​​of all angles of a unit circle.

In the school course of geometry, it is supposed to memorize the basic values ​​​​of trigonometric functions collected in a trigonometric table, for the convenience of solving trigonometric problems.

Using a table

In the table, it is enough to find the necessary trigonometric function and the value of the angle or radian for which this function needs to be calculated. At the intersection of the row with the function and the column with the value, we get the desired value of the trigonometric function of the given argument.

In the figure you can see how to find the value $\cos⁡60°$ which is equal to $\frac(1)(2)$.

The extended trigonometric table is used similarly. The advantage of using it is, as already mentioned, the calculation of the trigonometric function of almost any angle. For example, you can easily find the value $\tan 1 380°=\tan (1 380°-360°)=\tan(1 020°-360°)=\tan(660°-360°)=\tan300°$:

Bradis tables of basic trigonometric functions

The ability to calculate the trigonometric function of absolutely any angle value for an integer value of degrees and an integer value of minutes gives the use of Bradis tables. For example, find the value $\cos⁡34°7"$. The tables are divided into 2 parts: the table of $\sin$ and $\cos$ values ​​and the table of $\tan$ and $\cot$ values.

Bradis tables make it possible to obtain an approximate value of trigonometric functions with an accuracy of up to 4 decimal places.

Using Bradis Tables

Using the tables of Bradys for sines, we find $\sin⁡17°42"$. To do this, in the column on the left of the table of sines and cosines we find the value of degrees - $17°$, and in the top line we find the value of minutes - $42"$. At their intersection, we get the desired value:

$\sin17°42"=0.304$.

To find the value of $\sin17°44"$, you need to use the correction on the right side of the table. In this case to the value of $42"$, which is in the table, you need to add a correction for $2"$, which is equal to $0.0006$. We get:

$\sin17°44"=0.304+0.0006=0.3046$.

To find the value of $\sin17°47"$, we also use the correction on the right side of the table, only in this case we take the value of $\sin17°48"$ as a basis and subtract the correction for $1"$:

$\sin17°47"=0.3057-0.0003=0.3054$.

When calculating the cosines, we perform similar actions, but we look at the degrees in the right column, and the minutes in the bottom column of the table. For example, $\cos20°=0.9397$.

There are no corrections for tangent values ​​up to $90°$ and small angle cotangent. For example, let's find $\tan 78°37"$, which according to the table is $4,967$.

1. Trigonometric functions represent elementary functions, whose argument is injection. With the help of trigonometric functions, the relationships between the sides and sharp corners in a right triangle. The areas of application of trigonometric functions are extremely diverse. So, for example, any periodic processes can be represented as a sum of trigonometric functions (Fourier series). These functions often appear when solving differential and functional equations.

2. Trigonometric functions include the following 6 functions: sinus, cosine, tangent,cotangent, secant and cosecant. For each of specified functions there is an inverse trigonometric function.

3. Geometric definition trigonometric functions are conveniently introduced using unit circle. The figure below shows a circle with radius r=1. The point M(x,y) is marked on the circle. The angle between the radius vector OM and the positive direction of the Ox axis is α.

4. sinus the angle α is the ratio of the ordinate y of the point M(x,y) to the radius r:
sinα=y/r.
Since r=1, then the sine is equal to the ordinate of the point M(x,y).

5. cosine the angle α is the ratio of the abscissa x of the point M(x,y) to the radius r:
cosα=x/r

6. tangent the angle α is the ratio of the ordinate y of the point M(x,y) to its abscissa x:
tanα=y/x,x≠0

7. Cotangent the angle α is the ratio of the abscissa x of the point M(x,y) to its ordinate y:
cotα=x/y,y≠0

8. Secant angle α is the ratio of the radius r to the abscissa x of the point M(x,y):
secα=r/x=1/x,x≠0

9. Cosecant angle α is the ratio of the radius r to the ordinate y of the point M(x,y):
cscα=r/y=1/y,y≠0

10. In unit circle the projections x, y of the point M(x,y) and the radius r form a right triangle in which x,y are the legs and r is the hypotenuse. Therefore, the above definitions of trigonometric functions as applied to right triangle are formulated in this way:
sinus angle α is the ratio of the opposite leg to the hypotenuse.
cosine angle α is the ratio of the adjacent leg to the hypotenuse.
tangent angle α is called the opposite leg to the adjacent one.
Cotangent angle α is called the adjacent leg to the opposite.
Secant angle α is the ratio of the hypotenuse to the adjacent leg.
Cosecant angle α is the ratio of the hypotenuse to the opposite leg.

11. sine function graph
y=sinx, domain: x∈R, domain: −1≤sinx≤1

12. Graph of the cosine function
y=cosx, domain: x∈R, range: −1≤cosx≤1

13. tangent function graph
y=tanx, domain: x∈R,x≠(2k+1)π/2, domain: −∞

14. Graph of the cotangent function
y=cotx, domain: x∈R,x≠kπ, domain: −∞

15. Graph of the secant function
y=secx, domain: x∈R,x≠(2k+1)π/2, domain: secx∈(−∞,−1]∪∪)