Degree measure of angle. Radian measure of angle. Converting degrees to radians and vice versa.

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In the previous lesson we learned how to measure angles on a trigonometric circle. Learned how to count positive and negative angles. We learned how to draw an angle greater than 360 degrees. It's time to figure out how to measure angles. Especially with the number "Pi", which strives to confuse us in tricky tasks, yes...

Standard problems in trigonometry with the number "Pi" are solved well. Visual memory helps out. But any deviation from the template is a disaster! To avoid falling - understand necessary. Which is what we will do now with success. I mean, we’ll understand everything!

So, what do angles count? IN school course trigonometry uses two measures: degree measure of angle And radian angle measure. Let's look at these measures. Without this, there is nowhere in trigonometry.

Degree measure of angle.

We somehow got used to degrees. At least we passed geometry... And in life we often come across the phrase “turned 180 degrees,” for example. A degree, in short, is a simple thing...

Yes? Answer me then what is a degree? What, it doesn’t work out right away? That's it...

Degrees were invented in Ancient Babylon. It was a long time ago... 40 centuries ago... And they came up with a simple idea. They took and divided the circle into 360 equal parts. 1 degree is 1/360 of a circle. That's all. They could have broken it into 100 pieces. Or 1000. But they divided it into 360. By the way, why exactly 360? How is 360 better than 100? 100 seems to be somehow smoother... Try to answer this question. Or weakly against Ancient Babylon?

Somewhere at the same time, in Ancient Egypt were tormented by another question. How many times is the length of a circle greater than the length of its diameter? And they measured it this way, and that way... Everything turned out to be 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 you cut a circle into equal pieces, from such pieces you can 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 was established, of course. Approximately. 3.1415926... times.

This is the number "Pi". So shaggy, so shaggy. After the decimal point there is an infinite number of numbers without any order... Such numbers are called irrational. This, by the way, means that from equal pieces of a circle the diameter smooth don't fold. Never.

For practical application It is customary to remember only two digits after the decimal point. Remember:

Since we understand that the circumference of a circle is greater than its diameter by “Pi” times, it makes sense to remember the formula for the circumference of a circle:

Where L- circumference, and d- its diameter.

Useful in geometry.

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

But let's return to degrees. Have you figured out why in Ancient Babylon the circle was divided into 360 equal parts? And not by 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 the circle into equal parts. Now figure out what numbers it is divisible by completely 100, and which ones - 360? And in what version of these divisors 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 doesn't like them... Higher mathematics- a serious lady, organized 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, the day after tomorrow into 245... And what should I do? No, really...” I had to listen. You can't fool nature...

We had to introduce a measure of angle that did not depend on human inventions. Meet - radian!

Radian measure of angle.

What is a radian? The definition of a radian is still based on a circle. 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). Let's look at the pictures.

Such a small angle, it’s almost non-existent... We move the cursor over the picture (or touch the picture on the tablet) and we see about one radian. L = R

Do you feel the difference?

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

Let's look at the next picture. On which I drew a semicircle. The unfolded angle is, naturally, 180°.

Now I'll cut this semicircle into radians! We hover the cursor over the picture and see that 180° fits 3 plus radians.

Who can guess what this tail is equal to!?

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

Indeed, 180° degrees contains 3.1415926... radians. As you yourself understand, writing 3.1415926 all the time... is inconvenient. So instead infinite number always write simply:

But on the Internet the number

It’s inconvenient to write... That’s why I write his name in the text - “Pi”. Don't get confused, okay?...

Now we can write down an approximate equality in a completely meaningful way:

Or exact equality:

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

This ratio is useful to remember. One radian is approximately 60°. In trigonometry, you often have to estimate and assess the situation. This is where this 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 radians for “Pi” - 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 many degrees in angle "Pi"/2 radian? So we write:

Or, a 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 equal to in radians and multiply that number by the number of degrees. What is 1° equal to 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 find that one degree is equal to:

That's it. We multiply the number of degrees by this value and 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. And the translation is no problem... And “Pi” is a completely tolerable thing... So where does the confusion come from!?

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

This leads to confusion... A person sees “Pi” and believes that it is 180°. Always and everywhere. By the way, this works. For the time being, the examples are standard. But "Pi" is a number! The number is 3.14, but not degrees! This is "Pi" radians = 180°!

Once again: “Pi” is a number! 3.14. Irrational, but a number. Same as 5 or 8. You can, for example, do about "Pi" steps. Three steps and a little more. Or buy "Pi" kilograms of sweets. If an educated seller comes across...

"Pi" is a number! What, did I annoy you with this phrase? Have you already understood everything long ago? OK. Let's check. Tell me, which number is greater?

Or what is less?

This is a bit of a series non-standard questions, which can drive you into a stupor...

If you, too, have fallen into a stupor, remember the spell: “Pi” is a number! 3.14. In the very first sine it is clearly stated that the angle is in degrees! Therefore, it is impossible to replace “Pi” by 180°! "Pi" degrees is approximately 3.14°. Therefore, we can write:

There are no notations in the second sine. So, there - radians! This is where replacing “Pi” by 180° will work just fine. Converting radians to degrees, as written above, we get:

It remains to compare these two sines. What. forgot how? Using a trigonometric circle, of course! Draw a circle, draw approximate angles of 60° and 1.05°. Let's see what sines these angles have. In short, everything is described as at the end of the topic about the trigonometric circle. On a circle (even the crooked one!) it will be clearly visible that sin60° significantly more than sin1.05°.

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

Let's practice using angle measures.

Convert these angles from degrees to radians:

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

You should get these values in radians (in a different order!)

0

By the way, I specifically highlighted the answers in two lines. Well, let's figure out what the corners are in the first line? At least in degrees, at least in radians?

Yes! These are the axes of the coordinate system! If you look at the trigonometric circle, then the moving side of the angle with these values fits exactly on the axes. These values need to be known. And I noted the angle of 0 degrees (0 radians) for good reason. And then some people just can’t find this angle on a circle... 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 there are always coincidences on the circle near.

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

In the meantime, let's continue training. Convert these angles from radian to degree:

You should get results like this (in disarray):

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

Did it work? Then we can assume that converting degrees to radians and back- no longer your problem.) But translating angles is the first step to understanding trigonometry. There you also need to work with sines and cosines. And with tangents and cotangents too...

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

1. Which quarter do the angles fall into:

45°, 175°, 355°, 91°, 355° ?

Easily? Let's continue:

2. Which quarter do the corners fall into:

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

No problem too? Well, look...)

3. You can place the corners in quarters:

Could you? Well, you give..)

4. Which axes will the corner fall on:

and corner:

Is it easy too? Hm...)

5. Which 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 an answer only to the last question (it’s a little tricky) of the last task. An angle of 20 radians will fall in the first quarter.

I won’t give the rest of the answers, not out of greed.) Simply, if you haven't decided something you doubt it 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 of trigonometry. It’s better to get rid of it (the problem, not trigonometry!) immediately. This can be done in the topic: Practical work with the trigonometric circle in section 555.

It tells how to solve such tasks simply and correctly. Well, these tasks have been solved, of course. And the fourth task was solved in 10 seconds. Yes, it’s been decided that anyone can do it!

If you are absolutely confident in your answers and you are not interested in simple and trouble-free ways of working with radians, you don’t have to visit 555. I don’t insist.)

Good understanding is enough good 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. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Trigonometric functions are elementary functions whose argument is corner. By using trigonometric functions describes the relationships between sides and acute angles in a right triangle. The areas of application of trigonometric functions are extremely diverse. 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 specified functions there is an inverse trigonometric function.

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

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

Cosine angle α x points M(x,y) to radius r:cos α = x/r = x

Tangent angle α called the ordinate ratio y points M(x,y) to its abscissa x:tan α = y/x, x ≠ 0

Cotangent angle α called abscissa ratio x points M(x,y) to its ordinate y:cot α = x/y, y ≠ 0

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

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

In the unit circle of projection 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 angle α is called the ratio of the opposite side to the hypotenuse. Cosine angle α called the ratio of the adjacent leg to the hypotenuse. Tangent angle α called the opposite side to the adjacent. Cotangent angle α called adjacent leg to the opposite one.

Graph of the sine function y= sin x, domain of definition: x ∈ ℜ , range: −1 ≤ sin x ≤ 1

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

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

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

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

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

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

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

Thus, the entire circle consists of \(360\) "pieces" of circular arcs, or the angle described by the circle is \(360()^\circ \) .

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

An angle in \(1\) radians is the central angle in a circle subtended by a circular arc whose length is equal to the radius of the circle.

So, the figure shows an angle \(\gamma \) equal to \(1 \) radians, that is, this angle rests 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 radius \(r\) equal to length arcs \(l\) ). Thus, the arc length 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 are contained in the angle described by the circle? Yes, for this you need to remember the formula for circumference. Here it is:

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

Well, now let’s correlate these two formulas and find that the angle described by the circle is equal to \(2\pi \) . That is, by correlating the value in degrees and radians, we find 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 degrees [°]

Initial value

Converted value

degree radian grad gon minute second zodiacal sector thousandth revolution circle revolution quadrant right angle sextant

More about angles

General information

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

Types of angles

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

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

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

Measuring angles

Angles are measured using a protractor or calculated using 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 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 angle.

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 into them for ease of measurement. Scales on protractors are often written in degrees, although sometimes they are also in radians. Protractors are most often used in geometry lessons at school, but they are also used in architecture and engineering, in particular in tool making.

Use of angles in architecture and art

Artists, designers, craftsmen and architects have long used angles to create illusions, accents and other effects. Alternating acute and obtuse angles or geometric patterns from sharp corners often used in architecture, mosaics and stained glass, such as in Gothic cathedrals and Islamic mosaics.

One of the famous forms of Islamic fine art is decoration using geometric girih designs. This pattern is used in mosaics, metal and wood carvings, on paper and fabric. The drawing is created by alternating geometric shapes. Traditionally, five figures are used with strict certain angles from combinations of 72°, 108°, 144° and 216°. All these angles are divisible by 36°. Each figure is divided by lines into several smaller ones symmetrical figures to create a thinner pattern. Initially, these figures or mosaic pieces themselves were called girikh, hence the name of the entire style. In Morocco, there is a similar geometric style of mosaic, zullage or zilij. The shape of the terracotta tiles from which this mosaic is made is not observed as strictly as in girikha, and the tiles are often more bizarre in shape than the strict ones geometric shapes in Giriha. Despite this, zullyaj artists also use angles to create contrasting and intricate patterns.

In Islamic fine 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. He can be depicted as solid figure, or in the form of lines - in this case this symbol is called the Al-Quds star (al Quds). The Rub al-Hizb is sometimes decorated with small circles at the intersection of the 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 tallest twin towers in the world at the time of writing (spring 2013), the Petronas Towers, are built in the form of rub al-hizb. These towers are located in Kuala Lumpur in Malaysia and the country's Prime Minister was involved in their design.

Sharp corners are often used in architecture as decorative elements. They give the building a strict elegance. Obtuse angles, on the contrary, give buildings a cozy appearance. For example, we admire Gothic cathedrals and castles, but they look a little sad and even scary. But we will most likely choose a house with a roof obtuse angles between the slopes. Corners in architecture are also used to strengthen different parts buildings. Architects design the shape, size and angle of inclination depending on the load on the walls that need strengthening. This principle of strengthening by tilting 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 intentionally built to slope, and some lean because of mistakes. One example of leaning buildings is the Taj Mahal in India. The four minarets that surround the main building were built with an inclination from the center, so that in the event of an earthquake they would not fall inward, on the mausoleum, but in the other direction, and would not damage the main building. Sometimes buildings are built at an angle to the ground for decorative purposes. For example, the Leaning Tower of Abu Dhabi or Capital Gate is tilted 18° to the west. And one of the buildings in Stuart Landsborough's Puzzle World in Wanka, New Zealand, tilts 53° to the ground. This building is called the “Leaning Tower”.

Sometimes the leaning of a building is the result of a design error, such as the leaning 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 sank, leaning 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. We managed to level it from 5.5° to 4°. The tower of the SuurHusen Church in Germany is also leaning because its wooden foundation rotted on one side after the marshy soil on which it was built drained. On at the moment this tower is tilted more than the Leaning Tower of Pisa - by about 5°.

Do you find it difficult to translate units of measurement from one language to another? Colleagues are ready to help you. Post a question in TCTerms and within a few minutes you will receive an answer.

Angles are measured in degrees or radians. It is important to understand the relationship between these units of measurement. Understanding this relationship allows you to operate with angles and make the transition from degrees to radians and back. In this article, we will derive a formula for converting degrees to radians and radians to degrees, and also look at several practical examples.

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Relationship between degrees and radians

To establish the connection between degrees and radians, it is necessary to know the degree and radian measure of an angle. For example, take the central angle, which is based on the diameter of a circle of radius r. To calculate the radian measure of this angle, it is necessary to divide the length of the arc by the length of the radius of the circle. The angle in question corresponds to the length of the arc, equal to half circumference π · r. Divide the length of the arc by the radius and get the radian measure of the angle: π · r r = π rad.

So, the angle in question is π radians. On the other hand, it is a reversed angle equal to 180°. Therefore 180° = π rad.

Relationship between degrees and radians

The relationship between radians and degrees is expressed by the formula

π radian = 180°

Formulas for converting radians to degrees and vice versa

From the formula obtained above, you can derive other formulas for converting angles from radians to degrees and from degrees to radians.

Let's express one radian in degrees. To do this, divide the left and right sides of the radius by pi.

1 r a d = 180 π ° - the degree measure of an angle of 1 radian is equal to 180 π.

You can also express one degree in radians.

1° = π 180 r a d

You can make approximate calculations of angle values in radians and vice versa. To do this, take the values of the number π with an accuracy of ten thousandths and substitute them into the resulting formulas.

1 r a d = 180 π ° = 180 3, 1416 ° = 57, 2956 °

So there are approximately 57 degrees in one radian

1° = π 180 r a d = 3.1416 180 r a d = 0.0175 r a d

One degree contains 0.0175 radians.

Formula for converting radians to degrees

x r a d = x 180 π °

To convert an angle from radians to degrees, you need to multiply the angle value in radians by 180 and divide by pi.