Degree measure of angles. Degree measure of an angle

How to find the degree measure of an angle?

For many in school, geometry is a real challenge. One of the basic geometric shapes is the angle. This concept means two rays that originate at one point. To measure the value (value) of an angle, degrees or radians are used. How to find the degree measure of an angle, you will learn from our article.

Types of corners

Let's say we have a corner. If we expand it into a straight line, then its value will be equal to 180 degrees. Such an angle is called deployed, and 1/180 of its part is considered one degree.

In addition to the developed angle, there are also sharp (less than 90 degrees), obtuse (greater than 90 degrees) and right (equal to 90 degrees) angles. These terms are used to characterize the degree measure of an angle.

Angle measurement

The angle is measured with a protractor. This is a special device on which the semicircle is already divided into 180 parts. Place the protractor against the corner so that one side of the corner lines up with the bottom of the protractor. The second beam must intersect the arc of the protractor. If this does not happen, remove the protractor and use a ruler to lengthen the beam. If the corner "opens" to the right of the top, read its value on the upper scale, if to the left - on the lower one.

In the SI system, it is customary to measure the magnitude of an angle in radians, not degrees. Only 3.14 radians fit in a full angle, so this value is inconvenient and almost never used in practice. That is why you need to know how to convert radians to degrees. There is a formula for this:

Degrees = radians/π x 180

For example, the angle value is 1.6 radians. Convert to degrees: 1.6 / 3.14 * 180 = 92

Corner Properties

Now you know how to measure and convert degree measures of angles. But to solve problems, you also need to know the properties of angles. To date, the following axioms have been formulated:

Any angle can be expressed in degrees greater than zero. The value of the expanded angle is 360.
If an angle consists of several angles, then its degree measure is equal to the sum of all angles.
In a given half-plane from any ray, it is possible to construct an angle of a given value less than 180 degrees, and only one.
The values of equal angles are the same.
To add two angles, you need to add their values.

Understanding these rules and being able to measure angles is the key to a successful study of geometry.

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.

An angle is a figure that consists of a point - the vertex of the angle and two different half-lines emanating from this point - the sides of the angle (Fig. 14). If the sides of an angle are complementary half-lines, then the angle is called a straight angle.

An angle is indicated either by indicating its vertex, or by indicating its sides, or by indicating three points: a vertex and two points on the sides of the angle. The word "angle" is sometimes replaced

Angle in Figure 14 can be represented in three ways:

It is said that a ray c passes between the sides of an angle if it comes from its vertex and intersects some segment with ends on the sides of the angle.

In figure 15, the ray c passes between the sides of the angle since it intersects the segment

In the case of a straight angle, any ray emanating from its vertex and different from its sides passes between the sides of the angle.

Angles are measured in degrees. If you take a straight angle and divide it into 180 equal angles, then the degree measure of each of these angles is called a degree.

The main properties of measuring angles are expressed in the following axiom:

Each angle has a certain degree measure greater than zero. The developed angle is 180°. The degree measure of an angle is equal to the sum of the degree measures of the angles into which it is divided by any ray passing between its sides.

This means that if the ray c passes between the sides of the angle, then the angle is equal to the sum of the angles

The degree measure of an angle is found using a protractor.

An angle equal to 90° is called a right angle. An angle less than 90° is called an acute angle. An angle greater than 90° and less than 180° is called an obtuse angle.

Let us formulate the main property of laying off corners.

From any half-line into a given half-plane, one can lay off an angle with a given degree measure less than 180 °, and only one.

Consider the half line a. We extend it beyond the starting point A. The resulting straight line divides the plane into two half-planes. Figure 16 shows how to use a protractor to set aside from the half-line a to the upper half-plane an angle with a given degree measure of 60 °.

T. 1. 2. If two angles are set aside from a given half-line in one half-plane, then the side of the smaller angle, which is different from the given half-line, passes between the sides of the larger angle.

Let be the angles from the given half-line a into one half-plane, and let the angle be less than the angle . Theorem 1.2 states that the ray passes between the sides of the angle (Fig. 17).

The bisector of an angle is a ray that comes from its vertex, passes between the sides and divides the angle in half. In figure 18, the ray is the bisector of the angle

In geometry, there is the concept of a plane angle. A plane angle is a part of a plane bounded by two different rays emanating from the same point. These rays are called sides of the angle. There are two flat corners with given sides. They are called extras. In Figure 19, one of the flat corners with sides a and

Mathematics, geometry - for many, these sciences, as well as most other exact sciences, are extremely difficult. It is difficult for people to understand formulas and strange terminology. What is hidden under this strange concept?

Definition

For starters, you just need to consider the measure of the angle. The image of a ray and a straight line will help with this. First you need to draw, for example, a horizontal straight line. Then, from its first point, a ray is drawn that is not parallel to the straight line. Thus, a certain distance, a small angle, appears between the straight line and the ray. The measure of an angle is the size of this very rotation of the beam.

This concept denotes a certain digital value that will be greater than zero. It is expressed in degrees, as well as its constituent parts, that is, minutes and seconds. The number of degrees that fits into the angle between the ray and the straight line will be the degree measure.

Corner Properties

Absolutely each angle will have a certain degree measure.
If it is fully deployed, then the number will be equal to 180 degrees.
To find the degree measure, the sum of all the angles that the beam has broken is considered.
With the help of any ray, you can create a half-plane in which it is realistic to make an angle. It will have a degree measure, the value of which will be less than 180, and there can be only one such angle.

How to find the measure of an angle?

As a rule, the minimum degree measure is 1 degree, which is 1/180 of a straightened angle. However, sometimes you can not get such a clear figure. In these cases, seconds and minutes are used.

When they are found, the value can be converted to degrees, thus getting a fraction of a degree. Sometimes fractional numbers are used, like 80.7 degrees.

It is also important to remember the key values. A right angle will always be 90 degrees. If the measure is greater, then it will be considered blunt, and if less, then sharp.

Angles are measured in different units. It can be degrees, radians. Most often, angles are measured in degrees. (This degree should not be confused with a measure of temperature, where the word "degree" is also used.)

1 degree is an angle that is equal to 1/180 of a straightened angle. In other words, if we take a developed angle and divide it into 180 equal parts-angles, then each such small angle will be equal to 1 degree. The size of all other angles is determined by how many of these small angles can be placed inside the measured angle.

The degree is denoted by the sign °. This is not zero and not the letter O. This is such a special symbol introduced to denote a degree.

Thus, a straight angle is 180°, a right angle is 90°, acute angles are smaller than 90°, and obtuse angles are larger than 90°.

The metric system uses a meter to measure distance. However, both larger and smaller units are used. For example, centimeter, millimeter, kilometer, decimeter. By analogy with this, minutes and seconds are also distinguished in the degree measure of angles.

One degree minute is equal to 1/60 of a degree. It is denoted by one sign ".

One degree second is equal to 1/60 of a minute or 1/3600 of a degree. The second is denoted by two signs ", that is, "".

In school geometry, degree minutes and seconds are rarely used, but one must be able to understand, for example, such a record: 35 ° 21 "45"". This means that the angle is 35 degrees + 21 minutes + 45 seconds.

On the other hand, if the angle cannot be measured exactly in whole degrees, then it is not necessary to enter minutes and seconds. It is enough to use fractional degrees. For example, 96.5°.

It is clear that minutes and seconds can be converted to degrees, expressing them in fractions of a degree. For example, 30" equals (30/60)° or 0.5°. And 0.3° equals (0.3 * 60)" or 18". So using minutes and seconds is just a matter of convenience.