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 developed angle.
An angle is designated either by indicating its vertex, or by indicating its sides, or by indicating three points: the vertex and two points on the sides of the angle. The word "angle" is sometimes replaced
The Angle symbol in Figure 14 can be designated in three ways:
A ray c is said to pass between the sides of an angle if it starts from its vertex and intersects some segment with ends on the sides of the angle.
In Figure 15, ray c passes between the sides of the angle as 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 by 180 equal angles then the degree measure of each of these angles is called a degree.
The basic properties of angle measurement are expressed in the following axiom:
Each angle has a certain degree measure greater than zero. The rotated angle is 180°. Degree measure 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 a ray c passes between the sides of an 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 acute angle. An angle greater than 90° and less than 180° is called obtuse.
Let us formulate the main property of setting aside corners.
From any half-line, into a given half-plane, you can put 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 divides the plane into two half-planes. Figure 16 shows how, using a protractor, to plot an angle with a given degree measure of 60° from a half-line to the upper half-plane.
T. 1. 2. If two angles from a given half-line are put 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 laid off 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 the 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 plane angle. A plane angle is a 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 plane angles with given sides. They are called additional. In Figure 19, one of the plane angles with sides a and is shaded.
Degree measure of 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 “very much…”)
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. 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 the very 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. We 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 weak 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 greater 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 does not like them... Higher mathematics is 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 and a half 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. Therefore, instead of this infinite number, they 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 the 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 one of a series of slightly non-standard questions that 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 are they located, and what are these angles? trigonometric functions. 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 into 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.)
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. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
How to find the degree measure of an angle?
For many people at school, geometry is a real test. One of the basic geometric shapes is an angle. This concept means two rays that originate at the same point. To measure the value (magnitude) of an angle, degrees or radians are used. You will learn how to find the degree measure of an angle in our article.
Types of angles
Let's say we have an angle. If we expand it into a straight line, then its value will be equal to 180 degrees. Such an angle is called a turned angle, and 1/180 of its part is considered one degree.
In addition to a straight angle, there are also acute (less than 90 degrees), obtuse (more than 90 degrees) and right angles (equal to 90 degrees). These terms are used to characterize 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. Attach the protractor to the corner so that one of the sides of the corner coincides 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 angle “opens” to the right of the vertex, its value is read 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, rather than in degrees. Only 3.14 radians fit in the unfolded angle, so this value is inconvenient and is almost never used in practice. This 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. Convert to degrees: 1.6/3.14 * 180 = 92
Properties of corners
Now you know how to measure and recalculate degrees 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 size of the rotated 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 knowing how to measure angles is the key to successfully learning geometry.
Mathematics, geometry - these sciences, as well as most other exact sciences, are extremely difficult for many. People find it difficult to understand formulas and strange terminology. What is hidden under this strange concept?
Definition
To begin with, you need to consider simply 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 a ray is drawn from its first point, 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 beam rotation.
This concept denotes a certain digital value that will be greater than zero. It is expressed in degrees, and also its components, 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.
Properties of corners
- Absolutely each angle will have a certain degree measure.
- If it is fully deployed, the number will be 180 degrees.
- To find the degree measure, the sum of all angles broken by the beam is considered.
- Using any ray, you can create a half-plane in which you can actually make an angle. It will have a degree measure, the value of which will be less than 180, and there can only be one such angle.
How to find out the measure of an angle?
As a rule, the minimum degree measure is 1 degree, which is 1/180 of the rotated angle. However, sometimes you cannot get such a clear figure. In these cases, seconds and minutes are used.
Once found, the value can be converted to degrees, thus getting a fraction of a degree. Sometimes used fractional numbers, like 80.7 degrees.
It is also important to remember key quantities. A right angle will always be 90 degrees. If the measure is greater, then it will be considered obtuse, and if less, then sharp.
Degree measure of angle is a positive number showing how many times a degree and its parts fit into the angle.
The word "angle" has different interpretations. In geometry, an angle is a part of a plane bounded by two rays that issue from one point, the so-called vertex. When right, acute and straight angles are considered, it is geometric angles that are meant.
Like any geometric shapes, angles can be compared. In the field of geometry, it is not difficult today to describe that one angle is larger or smaller compared to another.
The unit of measurement for angles is a degree - 1/180 of a rotated angle.
Every angle has a degree measure greater than zero. A straight angle corresponds to 180 degrees. The degree measure of an angle is equal to the sum of all degree measures of angles into which the original angle can be divided by rays.
From any beam to given plane You can set aside an angle with a degree measure of no more than 180 degrees. The measure of a plane angle, which 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 that contains the half-plane is denoted by the number 360 - ?, where? is the degree measure of a complementary plane angle.
A right angle is always equal to 90 degrees, an obtuse angle is less than 180 degrees, but more than 90, and an acute angle does not exceed 90 degrees.
In addition to the degree measure of angle, there is a radian measure. In planimetry, the length of a circular arc is denoted as L, the radius as r, and the corresponding central corner got the designation - ?.. The relationship between these parameters looks like this: ? = L/r.
