The degree measure of the angle is equal to the sum. Definition
Degree measure of angle- this is a positive number showing how many times a degree and its parts fit into an angle.
The word "corner" has different meanings. In geometry, an angle is called a part of a plane bounded by two rays that come out of one point, the so-called vertex. When straight, sharp and unfolded angles are considered, it is geometrical angles that are meant.
As with any geometric shape, angles can be compared. In the field of geometry, it is not difficult today to describe that one corner is larger or smaller than another.
A degree is taken as a unit of measurement of angles - 1/180 of the unfolded angle.
Each angle has a degree measure that is greater than zero. The unfolded angle corresponds to 180 degrees. The degree measure of the angle is equal to the sum of all the degree measures of the angles into which the original angle can be divided by rays.
From any ray to a given plane, you can postpone an angle with a degree measure of no more than 180 degrees. A plane angle measure that is part of a half-plane is a degree measure of an angle that has similar sides. The measure of the plane of the angle, which contains the half-plane, is denoted by the number 360 -?, Where? is the degree measure of the complementary plane angle.
Right angle is always equal to 90 degrees, obtuse - less than 180 degrees, but more than 90, acute - does not exceed 90 degrees.
In addition to the degree measure of the angle, there is a radian. In planimetry, the length of the arc of a circle is denoted as L, the radius is r, and the corresponding central angle is designated as? .. The ratio of these parameters looks like this:? = L / r.
The degree measure of the angle. Radian measure of angle. Converting degrees to radians and vice versa.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very ..."
And for those who are "very even ...")
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 figure out how to measure angles. Especially with the number "Pi", which strives to confuse us in tricky tasks, yes ...
Standard tasks on trigonometry with the number "Pi" are solved well. Visual memory helps out. But any deviation from the template - knocks on the spot! In order not to fall - understand necessary. What we will do now with success. In the sense - we will understand everything!
So, what is are angles counted? The school trigonometry course uses two measures: degree measure of an angle and radian measure of angle... Let's analyze these measures. Without this, in trigonometry - nowhere.
The degree measure of the angle.
We are somehow used to degrees. At the very least, we passed geometry ... Yes, and in life we often come across the phrase "turned 180 degrees," for example. Degree, in short, a simple thing ...
Yes? Answer me then, what is degree? What, it doesn’t work right off the bat? 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. Took and broke a circle 360 equal parts... 1 degree is 1/360 of a circle. And that's all. Could be broken into 100 parts. Or 1000. But we broke it into 360. By the way, why exactly 360? Why is 360 better than 100? 100, it seems, somehow smoother ... 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 is the circumference of a circle longer than 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, for another 35 centuries, they suffered. 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 cannot be ... In principle, it is impossible. Well, of course, how many times the circumference is larger than the diameter. About. 3.1415926 ... times.
This is the number "Pi". So shaggy, so shaggy. After the decimal point - an infinite number of digits without any order ... Such numbers are called irrational. By the way, this means that out of equal pieces of a circle, the diameter smooth do not fold. Never.
For practical application it is customary to memorize only two digits after the decimal point. Remember:
Since we realized that the circumference is greater than the diameter in "pi" times, it makes sense to remember the formula for the circumference:
Where L is the circumference, and d- its diameter.
It will come in handy in geometry.
For general education I will add that the number "Pi" sits not only in geometry ... In various branches of mathematics, and especially in the theory of probability, 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? And not 100, for example? Not? OK. I'll give you a version. You cannot ask the ancient Babylonians ... For construction, or, say, astronomy, it is convenient to divide the circle into equal parts. Now figure out what numbers are divisible by entirely 100, and what 360? And in which version of these dividers entirely- 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, arranged according to the laws of nature. And this lady declares: "Today you have broken a circle into 360 parts, tomorrow you will break it by 100, the day after tomorrow by 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!
Radian measure of 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 ( L) is equal to the length of the radius ( R). We look at the pictures.
Such a small angle, there is almost no one ... Hover the cursor over the picture (or touch the picture on the tablet) and see about one radian. L = R
Do you feel the difference?
One radian is much more than one degree. How many times?
See the next picture. On which I drew a semicircle. The developed angle is, of course, 180 °.
Now I'm going to cut this semicircle into radians! Hover the cursor over the picture and see that 180 ° fits 3 with a tail of radians.
Who can guess what this ponytail is equal to !?
Yes! This ponytail is 0.1415926 .... Hello, Pi, we haven't forgotten you yet!
Indeed, in 180 ° degrees 3.1415926 ... radians fit. As you can imagine, writing 3.1415926 all the time ... is inconvenient. Therefore, instead of this infinite number, they always write simply:
But on the Internet, the number
it is inconvenient to write ... Therefore, in the text I write it by the name - "Pi". Do not get confused, go? ...
Now you can write down the approximate equality in a completely meaningful way:
Or exact equality:
Let's determine how many degrees are in one radian. How? Easy! If 3.14 radians are 180 ° degrees, then 1 radian is 3.14 times less! That is, we divide the first equation (the formula is also an equation!) By 3.14:
It is useful to remember this ratio. In one radian, about 60 °. In trigonometry, very often you have to figure out, evaluate 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 pi, it's very simple. We know Pi is radian = 180 °. So we substitute radians for "Pi" - 180 °. We get the angle in degrees. We shorten what is shortened, and the answer is ready. For example, we need to figure out how much degrees in the corner "Pi" / 2 radian? So we write:
Or, a more exotic expression:
Easy, right?
The reverse translation is a little more difficult. But not much. If the angle is given in degrees, we have to 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 less. 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 and get the angle in radians. For instance:
Or, similarly:
As you can see, in a leisurely conversation with lyrical digressions, it turned out that radians are very simple. And translation without any problems ... And "Pi" is quite a tolerable thing ... So where does the confusion come from !?
I will reveal the secret. The fact is that in trigonometric functions the degrees icon is written. Is always. For example, sin35 °. This is sine 35 degrees ... And the radians icon ( glad) - not written! It is implied. Either the mathematicians were overwhelmed by laziness, or something else ... But they decided not to write. If there are no signs inside the sine - cotangent, then the angle is 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. This, by the way, works. For the time being, the examples are standard. But Pi is a number! The number is 3.14, 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, take about Pi steps. Three steps and a little more. Or buy "Pi" kilograms of candy. If an educated seller comes across ...
Pi is a number! What, did I get you with this phrase? Have you understood everything for a long time? OK. Let's check. Tell me, which number is higher?
Or what is less?
This is from a series of slightly non-standard questions that can drive you into a stupor ...
If you also fell into a stupor, remember the spell: "Pi" is a number! 3.14. The very first sine clearly states that the angle is in degrees! Therefore, it is impossible to replace "Pi" by 180 °! Pi degrees are roughly 3.14 degrees. Therefore, we can write:
There is no designation in the second sine. So, there - radians! Here, replacing "Pi" by 180 ° is quite good enough. We convert radians to degrees, as written above, we get:
It remains to compare these two sines. What. forgot how? Using the trigonometric circle, of course! Draw a circle, draw rough angles of 60 ° and 1.05 °. We look at the sinuses of these angles. In short, everything is described as at the end of the topic about the trigonometric circle. On the circle (even the most crooked!) It will be clearly seen that sin60 ° substantially more than sin1.05 °.
We will do exactly the same with cosines. On the circle we will draw corners of about 4 degrees and 4 radians(remember what is roughly 1 radian?). 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 have specially 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 along the trigonometric circle, then the movable side of the angle at these values fits exactly on the axes... These values need to be known ironically. And I noted the angle of 0 degrees (0 radians) for a reason. And then some of this angle cannot be found on the circle ... And, accordingly, in trigonometric functions they get confused ... 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 are they located, and what are these corners trigonometric functions... Let's say the value sin100 ° you don't have to know. A sin45 °- be so kind! This is an obligatory knowledge, without which there is nothing to do in trigonometry ... But more on this in the next lesson.
In the meantime, let's continue training. Convert these angles from radian 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 comprehending trigonometry. In the same place, it is also necessary to work with sine-cosines. And with tangents, cotangents too ...
The second powerful step is the ability to determine the position of any angle on the trigonometric circle. Both in degrees and in radians. About this very skill, I'll be boringly hinting to you in all trigonometry, yes ...) If you know everything (or think that you know everything) about the trigonometric circle, and the counting of angles on the trigonometric circle, you can check. Solve these simple tasks:
1. In which quarter do the corners fall:
45 °, 175 °, 355 °, 91 °, 355 °?
Easy? We continue:
2. In which quarter do the corners fall:
402 °, 535 °, 3000 °, -45 °, -325 °, -3000 °?
No problem too? Well, look ...)
3. You can place corners in quarters:
Could you? Well, you give ..)
4. On which axles the corner will fall:
and corner:
Easy too? HM...)
5. In which quarter do the corners fall:
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.
The rest of the answers will not be given out of greed.) Just if you didn't decide something doubt as a result, or spent on task # 4 more than 10 seconds, you are poorly guided in a circle. This will be your problem in all trigonometry. Better to get rid of it (problems, not trigonometry!)) Right away. This can be done in the topic: Practical work with the trigonometric circle in section 555.
It tells how to easily and correctly solve such tasks. Well, these tasks have been solved, of course. And the fourth task was solved in 10 seconds. Yes, it is so decided that anyone can!
If you are absolutely sure of your answers and you are not interested in simple and trouble-free ways of working with radians, you can skip visiting 555. I don’t insist.)
Good understanding- 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. Instant validation testing. Learning - with interest!)
you can get acquainted with functions and derivatives.
How to find the degree measure of an angle?
For many in school, geometry is a real test. One of the basic geometric shapes is the corner. This concept means two rays that originate at the same point. To measure the value (magnitude) of the angle, degrees or radians are used. You will learn how to find the degree measure of an angle from our article.
Types of angles
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 unfolded, and 1/180 of its part is considered one degree.
In addition to the extended angle, there are still sharp (less than 90 degrees), obtuse (more than 90 degrees) and straight (equal to 90 degrees) angles. These terms are used to characterize the magnitude of the degree measure of an angle.
Angle measurement
The angle is measured using 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 is aligned with the bottom of the protractor. The second ray should cross the protractor arc. If not, remove the protractor and lengthen the beam using a ruler. If the angle "opens" to the right of the top, read its value on the upper scale, if to the left - on the bottom.
In the SI system, it is customary to measure the value of the angle in radians, and not in degrees. The unfolded angle fits only 3.14 radians, so this value is inconvenient and almost never applied 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 is 1.6 radians. We translate into degrees: 1.6 / 3.14 * 180 = 92
Corner properties
Now you know how to measure and recalculate the 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 unfolded 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, you can build an angle of a given value, less than 180 degrees, and only one.
- The values of equal angles are the same.
- To add two corners, add their values.
Understanding these rules and knowing how to measure angles is the key to successfully learning geometry.
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 the corner are additional half-lines, then the angle is called unfolded.
An angle is indicated either by specifying its vertex, or by specifying its sides, or by specifying three points: the vertex and two points on the sides of the corner. The word "corner" is sometimes replaced by
The angle symbol in Figure 14 can be denoted in three ways:
They say that a ray c passes between the sides of an angle if it emanates from its vertex and crosses some segment with ends on the sides of the angle.
In Figure 15, ray c passes between the sides of the corner as it crosses the segment
In the case of a flat corner, any ray emanating from its vertex and other than its sides passes between the sides of the corner.
Angles are measured in degrees. If you take an extended angle and divide it into 180 equal angles, then the degree measure of each of these angles is called a degree.
The basic properties of measuring angles are expressed in the following axiom:
Each angle has a certain degree measure, greater than zero. The flattened angle is 180 °. The degree measure of the 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 the 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 obtuse.
Let us formulate the main property of the deposition of corners.
From any half-line to a given half-plane, you can postpone an angle with a given degree measure less than 180 °, and only one.
Consider the half-line a. Let us extend it beyond the starting point A. The resulting straight line splits the plane into two half-planes. Figure 16 shows how, using a protractor, to set aside an angle with a given degree measure of 60 ° from the half-line a to the upper half-plane.
T. 1. 2. If two corners are laid off from a given half-line into one half-plane, then the side of the smaller angle, different from the given half-line, passes between the sides of the larger angle.
Let be the angles plotted from a 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 emanates 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 flat angle. A plane angle is the part of a plane bounded by two different rays emanating from one point. These rays are called the sides of the angle. There are two planar corners with these sides. They are called complementary. Figure 19 shaded one of the flat corners with sides a and
Angles are measured in different units. It can be degrees, radians. Most often, angles are measured in degrees. (Do not confuse this degree with a measure of temperature measurement, which also uses the word "degree").
1 degree is an angle that is 1/180 of the flattened angle. In other words, if you take an expanded 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 corners is determined by how many of these small corners can be placed within the measured corner.
The degree is indicated by the ° sign. This is not a zero and not the letter O. It is such a special symbol introduced to denote a degree.
Thus, the unfolded angle is 180 °, the 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 of angles.
One degree minute is equal to 1/60 of a degree. It is designated 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 characters ", that is," ".
In school geometry, degree minutes and seconds are rarely used, but one must be able to understand, for example, the following notation: 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 only, then it is not necessary to enter minutes and seconds. It is sufficient to use fractional degrees. For example, 96.5 °.
It is clear that minutes and seconds can be converted into degrees, expressing them in fractions of a degree. For example, 30 "is equal to (30/60) ° or 0.5 °. And 0.3 ° is equal to (0.3 * 60)" or 18 ". So the use of minutes and seconds is just a matter of convenience.
