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

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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. Approximately. 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 constantly appears! 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? No? 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 move the cursor 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 (a 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°. 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!)

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. A 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.

Trigonometric functions are elementary functions whose argument is corner. Trigonometric functions describe the relationship between sides and acute angles 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.

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

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

sinus corner α y points M(x,y) to the radius r: sin α = y/r. Because the r= 1, then the sine is equal to the ordinate of the point M(x,y).

cosine corner α x points M(x,y) to the radius r: cos α = x/r = x

tangent corner α is called the ratio of the ordinate y points M(x,y) to its abscissa x:tan α = y/x, x ≠ 0

Cotangent corner α called the ratio of the abscissa x points M(x,y) to its ordinate y: cat α = x/y, y ≠ 0

Secant corner α is the radius ratio r to the abscissa x points M(x,y):sec α = r/x = 1/x, x ≠ 0

Cosecant corner α is the radius ratio r to the ordinate y points M(x,y): cosec α = r/y = 1/y, y ≠ 0

In a single projection circle x, y points M(x,y) and radius r form a right triangle in which x, y are legs, and r− hypotenuse. Therefore, the above definitions of trigonometric functions as applied to a right triangle are formulated as follows: sinus corner α is the ratio of the opposite leg to the hypotenuse. cosine corner α is the ratio of the adjacent leg to the hypotenuse. tangent corner α called the opposite leg to the adjacent. Cotangent corner α called the adjacent leg to the opposite.

sine function graph y= sin x, domain: x ∈ ℜ , range: −1 ≤ sin x ≤ 1

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

tangent function graph y= ttg x, domain: x ∈ ℜ , x ≠ (2k + 1)π /2, range: −∞< tg x < ∞

Graph of the cotangent function y=ctg x, domain: x ∈ ℜ , x ≠ kπ, range: −∞< ctg x < ∞

Let's look at the picture. The vector \(AB \) "turned" relative to the point \(A \) by a certain amount. So the measure of this rotation relative to the initial position will be angle \(\alpha \).

What else do you need to know about the concept of angle? Well, units of angle, of course!

Angle, both in geometry and trigonometry, can be measured in degrees and radians.

An angle in \(1()^\circ \) (one degree) is a central angle in a circle based on a circular arc equal to the \(\dfrac(1)(360) \) part of the circle.

So the whole circle is made up of \(360 \) "pieces" of circular arcs, or the angle described by the circle is \(360()^\circ \) .

That is, the figure above shows the angle \(\beta \) equal to \(50()^\circ \) , that is, this angle is based on a circular arc of size \(\dfrac(50)(360) \) of the circumference.

An angle in \(1 \) radians is a central angle in a circle, based on a circular arc, the length of which is equal to the radius of the circle.

So, the figure shows the angle \(\gamma \) equal to \(1 \) radian, that is, this angle is based on a circular arc, the length of which is equal to the radius of the circle (the length \(AB \) is equal to the length \(BB"\) or the radius \(r \) is equal to the length of the arc \(l \) ) Thus, the length of the arc is calculated by the formula:

\(l=\theta \cdot r \) , where \(\theta \) is the central angle in radians.

Well, knowing this, can you answer how many radians contains an angle described by a circle? Yes, for this you need to remember the formula for the circumference of a circle. Here she is:

\(L=2\pi \cdot r\)

Well, now let's correlate these two formulas and get that the angle described by the circle is \(2\pi \) . That is, correlating the value in degrees and radians, we get that \(2\pi =360()^\circ \) . Accordingly, \(\pi =180()^\circ \) . As you can see, unlike "degrees", the word "radian" is omitted, since the unit of measurement is usually clear from the context.

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1 radian [rad] = 57.2957795130823 degree [°]

Initial value

Converted value

degree radian deg gon minute second zodiac sector thousandth revolution circumference revolution quadrant right angle sextant

More about corners

General information

Flat angle - a geometric figure formed by two intersecting lines. A flat angle consists of two rays with a common origin, and this point is called the vertex of the ray. The rays are called the sides of the angle. Angles have many interesting properties, for example, the sum of all angles in a parallelogram is 360°, and in a triangle it is 180°.

Types of corners

Direct angles are 90°, sharp- less than 90°, and stupid- on the contrary, more than 90 °. Angles equal to 180° are called deployed, 360° angles are called complete, and angles greater than expanded but less than full are called non-convex. When the sum of two angles is 90°, that is, one angle complements the other up to 90°, they are called additional related, and if up to 360 ° - then conjugated

When the sum of two angles is 90°, that is, one angle complements the other up to 90°, they are called additional. If they complement each other up to 180°, they are called related, and if up to 360 ° - then conjugated. In polygons, the angles inside the polygon are called internal, and those conjugated to them are called external.

Two angles formed by the intersection of two lines that are not adjacent are called vertical. They are equal.

Angle measurement

Angles are measured using a protractor or calculated by a formula by measuring the sides of the angle from the vertex to the arc, and the length of the arc that limits these sides. Angles are usually measured in radians and degrees, although other units exist.

You can measure both the angles formed between two straight lines and between curved lines. To measure between curves, tangents are used at the point of intersection of the curves, that is, at the vertex of the corner.

Protractor

A protractor is a tool for measuring angles. Most protractors are shaped like a semicircle or a circle and can measure angles up to 180° and 360° respectively. Some protractors have an additional rotating ruler built in for ease of measurement. Scales on protractors are usually applied in degrees, although sometimes they are also in radians. Protractors are most often used at school in geometry lessons, but they are also used in architecture and engineering, in particular in tool making.

The use of angles in architecture and art

Artists, designers, craftsmen and architects have long used angles to create illusions, accents and other effects. The alternation of acute and obtuse angles or geometric patterns of acute angles are often used in architecture, mosaics and stained glass, for example in the construction of Gothic cathedrals and in Islamic mosaics.

One of the well-known forms of Islamic fine art is decoration with the help of geometric girih ornament. This pattern is used in mosaics, metal and wood carving, paper and fabric. The pattern is created by alternating geometric shapes. Traditionally, five figures are used with strictly defined angles from combinations of 72°, 108°, 144° and 216°. All these angles are divisible by 36°. Each shape is divided by lines into several smaller, symmetrical shapes to create a more subtle pattern. Initially, these figures themselves or pieces for mosaics were called girih, hence the name of the whole style came from. In Morocco, there is a similar geometric style of mosaic, the zellige or zilidj. The shape of the terracotta tiles that make up this mosaic is not as strictly observed as in girikha, and the tiles are often more bizarre in shape than the strict geometric figures in girikha. Despite this, zellige artists also use angles to create contrasting and whimsical designs.

In Islamic visual arts and architecture, the rub al-hizb is often used - a symbol in the form of one square superimposed on another at an angle of 45 °, as in the illustrations. It can be depicted as a solid figure, or in the form of lines - in this case, this symbol is called the star of Al-Quds (al quds). The rub al-hizb is sometimes decorated with small circles at the intersection of squares. This symbol is used in the coats of arms and on the flags of Muslim countries, for example, on the coat of arms of Uzbekistan and on the flag of Azerbaijan. The bases of the world's tallest twin towers at the time of writing (spring 2013), the Petronas Towers, are built in the form of a rub al-hizb. These towers are located in Kuala Lumpur in Malaysia and the Prime Minister of the country participated in their design.

Sharp corners are often used in architecture as decorative elements. They give the building an understated elegance. Obtuse corners, on the contrary, give buildings a cozy look. So, for example, we admire Gothic cathedrals and castles, but they look a little sad and even intimidating. But we will most likely choose a house for ourselves with a roof with obtuse angles between the slopes. Corners in architecture are also used to reinforce different parts of a building. Architects design the shape, size and angle of inclination depending on the load on the walls in need of reinforcement. This principle of strengthening with the help of a slope has been used since ancient times. For example, ancient builders learned to build arches without cement or other binding materials, laying stones at a certain angle.

Usually buildings are built vertically, but sometimes there are exceptions. Some buildings are deliberately built on a slope, and some are tilted due to errors. One example of leaning buildings is the Taj Mahal in India. The four minarets that surround the main building are built with an inclination from the center, so that in the event of an earthquake they would fall not inwards, onto the mausoleum, but in the other direction, and not damage the main building. Sometimes buildings are built at an angle to the ground for decorative purposes. For example, Abu Dhabi's Leaning Tower or Capital Gate is tilted 18° to the west. And one of the buildings in Stuart Landsborough's Puzzle World in Wanka, New Zealand leans 53° to the ground. This building is called "The Leaning Tower".

Sometimes the slope of a building is the result of a design error, such as the slope of the Leaning Tower of Pisa. The builders did not take into account the structure and quality of the soil on which it was built. The tower was supposed to stand straight, but the poor foundation could not support its weight and the building sagged, loping to one side. The tower has been restored many times; the most recent restoration in the 20th century stopped its gradual subsidence and increasing slope. It was possible to level it from 5.5° to 4°. The tower of the SuurHussen church in Germany is also tilted because its wooden foundation rotted on one side after the marshy soil on which it was built drained. At the moment, this tower is tilted more than the Leaning Tower of Pisa - about 5 °.

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