The angle is positive and negative. Negative angle

Counting angles on a trigonometric circle.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

It is almost the same as in the previous lesson. There are axes, a circle, an angle, everything is in order. Added quarter numbers (in the corners of the large square) - from the first to the fourth. What if someone doesn’t know? As you can see, the quarters (they are also called the beautiful word “quadrants”) are numbered counterclockwise. Added angle values ​​on axes. Everything is clear, no problems.

And a green arrow is added. With a plus. What does it mean? Let me remind you that the fixed side of the angle Always nailed to the positive semi-axis OX. So, if we rotate the movable side of the angle along the arrow with a plus, i.e. in ascending order of quarter numbers, the angle will be considered positive. As an example, the picture shows a positive angle of +60°.

If we put aside the corners in the opposite direction, clockwise, the angle will be considered negative. Hover your cursor over the picture (or touch the picture on your tablet), you will see a blue arrow with a minus sign. This is the direction of negative angle reading. For example, a negative angle (- 60°) is shown. And you will also see how the numbers on the axes have changed... I also converted them to negative angles. The numbering of the quadrants does not change.

This is where the first misunderstandings usually begin. How so!? What if a negative angle on a circle coincides with a positive one!? And in general, it turns out that the same position of the moving side (or point on the number circle) can be called both a negative angle and a positive one!?

Yes. Exactly. Let's say a positive angle of 90 degrees takes on a circle exactly the same position as a negative angle of minus 270 degrees. A positive angle, for example, +110° degrees takes exactly the same position as negative angle -250°.

No problem. Anything is correct.) The choice of positive or negative angle calculation depends on the conditions of the task. If the condition says nothing in clear text about the sign of the angle, (like "determine the smallest positive angle", etc.), then we work with values ​​that are convenient for us.

The exception (how could we live without them?!) are trigonometric inequalities, but there we will master this trick.

And now a question for you. How did I know that the position of the 110° angle is the same as the position of the -250° angle?
Let me hint that this is connected with a complete revolution. In 360°... Not clear? Then we draw a circle. We draw it ourselves, on paper. Marking the corner approximately 110°. AND we think, how much time remains until a full revolution. Just 250° will remain...

Got it? And now - attention! If angles 110° and -250° occupy a circle same situation, then what? Yes, the angles are 110° and -250° exactly the same sine, cosine, tangent and cotangent!
Those. sin110° = sin(-250°), ctg110° = ctg(-250°) and so on. Now this is really important! And in itself, there are a lot of tasks where you need to simplify expressions, and as a basis for the subsequent mastery of reduction formulas and other intricacies of trigonometry.

Of course, I took 110° and -250° at random, purely as an example. All these equalities work for any angles occupying the same position on the circle. 60° and -300°, -75° and 285°, and so on. Let me note right away that the angles in these pairs are different. But they have trigonometric functions - the same.

I think you understand what negative angles are. It's quite simple. Counterclockwise - positive counting. Along the way - negative. Consider the angle positive or negative depends on us. From our desire. Well, and also from the task, of course... I hope you understand how to move in trigonometric functions from negative angles to positive ones and back. Draw a circle, an approximate angle, and see how much is missing to complete a full revolution, i.e. up to 360°.

Angles greater than 360°.

Let's deal with angles that are greater than 360°. Are there such things? There are, of course. How to draw them on a circle? No problem! Let's say we need to understand which quarter an angle of 1000° will fall into? Easily! We make one full turn counterclockwise (the angle we were given is positive!). We rewinded 360°. Well, let's move on! One more turn - it’s already 720°. How much is left? 280°. It’s not enough for a full turn... But the angle is more than 270° - and this is the border between the third and fourth quarter. Therefore, our angle of 1000° falls into the fourth quarter. All.

As you can see, it's quite simple. Let me remind you once again that the angle of 1000° and the angle of 280°, which we obtained by discarding the “extra” full revolutions, are, strictly speaking, different corners. But the trigonometric functions of these angles exactly the same! Those. sin1000° = sin280°, cos1000° = cos280°, etc. If I were a sine, I wouldn't notice the difference between these two angles...

Why is all this needed? Why do we need to convert angles from one to another? Yes, all for the same thing.) In order to simplify expressions. Simplifying expressions is, in fact, the main task of school mathematics. Well, and, along the way, the head is trained.)

Well, let's practice?)

We answer questions. Simple ones first.

1. Which quarter does the -325° angle fall into?

2. Which quarter does the 3000° angle fall into?

3. Which quarter does the angle -3000° fall into?

There is a problem? Or uncertainty? Go to Section 555, Trigonometric Circle Practice. There, in the first lesson of this very “Practical work...” everything is detailed... In such questions of uncertainty to be shouldn't!

4. What sign does sin555° have?

5. What sign does tg555° have?

Have you determined? Great! Do you have any doubts? You need to go to Section 555... By the way, there you will learn to draw tangent and cotangent on a trigonometric circle. A very useful thing.

And now the questions are more sophisticated.

6. Reduce the expression sin777° to the sine of the smallest positive angle.

7. Reduce the expression cos777° to the cosine of the largest negative angle.

8. Reduce the expression cos(-777°) to the cosine of the smallest positive angle.

9. Reduce the expression sin777° to the sine of the largest negative angle.

What, questions 6-9 puzzled you? Get used to it, on the Unified State Exam you don’t find such formulations... So be it, I’ll translate it. Only for you!

The words "bring an expression to..." mean to transform the expression so that its meaning hasn't changed and the appearance changed in accordance with the task. So, in tasks 6 and 9 we must get a sine, inside of which there is smallest positive angle. Everything else doesn't matter.

I will give out the answers in order (in violation of our rules). But what to do, there are only two signs, and there are only four quarters... You won’t be spoiled for choice.

6. sin57°.

7. cos(-57°).

8. cos57°.

9. -sin(-57°)

I assume that the answers to questions 6-9 confused some people. Especially -sin(-57°), really?) Indeed, in the elementary rules for calculating angles there is room for errors... That is why I had to do a lesson: “How to determine the signs of functions and give angles on a trigonometric circle?” In Section 555. Tasks 4 - 9 are covered there. Well sorted, with all the pitfalls. And they are here.)

In the next lesson we will deal with the mysterious radians and the number "Pi". Let's learn how to easily and correctly convert degrees to radians and vice versa. And we will be surprised to discover that this basic information on the site enough already to solve some custom trigonometry problems!

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.

In the last lesson, we successfully mastered (or repeated, depending on who) the key concepts of all trigonometry. This trigonometric circle , angle on a circle , sine and cosine of this angle , and also mastered signs of trigonometric functions by quarters . We mastered it in detail. On the fingers, one might say.

But this is not enough yet. To successfully apply all these simple concepts in practice, we need one more useful skill. Namely - correct working with corners in trigonometry. Without this skill in trigonometry, there is no way. Even in the most primitive examples. Why? Yes, because the angle is the key operating figure in all trigonometry! No, not trigonometric functions, not sine and cosine, not tangent and cotangent, namely the corner itself. No angle means no trigonometric functions, yes...

How to work with angles on a circle? To do this, we need to firmly grasp two points.

1) How Are angles measured on a circle?

2) What are they counted (measured)?

The answer to the first question is the topic of today's lesson. We will deal with the first question in detail right here and now. I will not give the answer to the second question here. Because it is quite developed. Just like the second question itself is very slippery, yes.) I won’t go into details yet. This is the topic of the next separate lesson.

Shall we get started?

How are angles measured on a circle? Positive and negative angles.

Those who read the title of the paragraph may already have their hair standing on end. How so?! Negative angles? Is this even possible?

To negative numbers We've already gotten used to it. We can depict them on the number axis: to the right of zero are positive, to the left of zero are negative. Yes, and we periodically look at the thermometer outside the window. Especially in winter, in the cold.) And the money on the phone is in the minus (i.e. duty) sometimes they leave. This is all familiar.

What about the corners? It turns out that negative angles in mathematics there are too! It all depends on how to measure this very angle... no, not on the number line, but on the number circle! That is, on a circle. The circle - here it is, an analogue of the number line in trigonometry!

So, How are angles measured on a circle? There’s nothing we can do, we’ll have to draw this very circle first.

I'll draw this beautiful picture:

It is very similar to the pictures from the last lesson. There are axes, there is a circle, there is an angle. But there is also new information.

I also added 0°, 90°, 180°, 270° and 360° numbers on the axes. Now this is more interesting.) What kind of numbers are these? Right! These are the angle values ​​measured from our fixed side that fall to the coordinate axes. We remember that the fixed side of the angle is always tightly tied to the positive semi-axis OX. And any angle in trigonometry is measured precisely from this semi-axis. This basic starting point for angles must be kept firmly in mind. And the axes – they intersect at right angles, right? So we add 90° in each quarter.

And more added red arrow. With a plus. Red is on purpose so that it catches the eye. And it is well etched in my memory. Because this must be remembered reliably.) What does this arrow mean?

So it turns out that if we twist our corner along the arrow with a plus(counterclockwise, according to the numbering of quarters), then the angle will be considered positive! As an example, the figure shows an angle of +45°. By the way, please note that the axial angles 0°, 90°, 180°, 270° and 360° are also rewound in the positive direction! Follow the red arrow.

Now let's look at another picture:

Almost everything is the same here. Only the angles on the axes are numbered reversed. Clockwise. And they have a minus sign.) Still drawn blue arrow. Also with a minus. This arrow is the direction of the negative angles on the circle. She shows us that if we postpone our corner clockwise, That the angle will be considered negative. For example, I showed an angle of -45°.

By the way, please note that the numbering of quarters never changes! It doesn’t matter whether we move the angles to plus or minus. Always strictly counterclockwise.)

Remember:

1. The starting point for angles is from the positive semi-axis OX. By the clock – “minus”, against the clock – “plus”.

2. The numbering of quarters is always counterclockwise, regardless of the direction in which the angles are calculated.

By the way, labeling angles on the axes 0°, 90°, 180°, 270°, 360°, each time drawing a circle, is not at all mandatory. This is done purely for the sake of understanding the point. But these numbers must be present in your head when solving any trigonometry problem. Why? Yes, because this basic knowledge provides answers to so many other questions in all of trigonometry! The most important question is Which quarter does the angle we are interested in fall into? Believe it or not, answering this question correctly solves the lion's share of all other trigonometry problems. We will deal with this important task (distributing angles into quarters) in the same lesson, but a little later.

The values ​​of the angles lying on the coordinate axes (0°, 90°, 180°, 270° and 360°) must be remembered! Remember it firmly, until it becomes automatic. And both a plus and a minus.

But from this moment the first surprises begin. And along with them, tricky questions addressed to me, yes...) What happens if there is a negative angle on a circle coincides with the positive? It turns out that the same point on a circle can be denoted by both a positive and a negative angle???

Absolutely right! This is true.) For example, a positive angle of +270° occupies a circle same situation , the same as a negative angle of -90°. Or, for example, a positive angle of +45° on a circle will take same situation , the same as the negative angle -315°.

We look at the next drawing and see everything:

In the same way, a positive angle of +150° will fall in the same place as a negative angle of -210°, a positive angle of +230° will fall in the same place as a negative angle of -130°. And so on…

And now what i can do? How exactly to count angles, if you can do it this way and that? Which is correct?

Answer: in every way correct! Mathematics does not prohibit either of the two directions for counting angles. And the choice of a specific direction depends solely on the task. If the assignment does not say anything in plain text about the sign of the angle (such as "define the largest negative corner" etc.), then we work with the angles that are most convenient for us.

Of course, for example, in such cool topics as trigonometric equations and inequalities, the direction of angle calculation can have a huge impact on the answer. And in the relevant topics we will consider these pitfalls.

Remember:

Any point on a circle can be designated by either a positive or a negative angle. Anyone! Whatever we want.

Now let's think about this. We found out that an angle of 45° is exactly the same as an angle of -315°? How did I find out about these same 315° ? Can't you guess? Yes! Through a full rotation.) In 360°. We have an angle of 45°. How long does it take to complete a full rotation? Subtract 45° from 360° - so we get 315° . Move in the negative direction and we get an angle of -315°. Still not clear? Then look at the picture above again.

And this should always be done when converting positive angles to negative (and vice versa) - draw a circle, mark approximately a given angle, we calculate how many degrees are missing to complete a full revolution, and move the resulting difference in the opposite direction. That's all.)

What else is interesting about angles that occupy the same position on a circle, do you think? And the fact that at such corners exactly the same sine, cosine, tangent and cotangent! Always!

For example:

Sin45° = sin(-315°)

Cos120° = cos(-240°)

Tg249° = tg(-111°)

Ctg333° = ctg(-27°)

But this is extremely important! For what? Yes, all for the same thing!) To simplify expressions. Because simplifying expressions is a key procedure for a successful solution any math assignments. And in trigonometry as well.

So, we figured out the general rule for counting angles on a circle. Well, if we started talking about full turns, about quarter turns, then it’s time to twist and draw these very corners. Shall we draw?)

Let's start with positive corners They will be easier to draw.

We draw angles within one revolution (between 0° and 360°).

Let's draw, for example, an angle of 60°. Everything is simple here, no hassles. We draw coordinate axes and a circle. You can do it directly by hand, without any compass or ruler. Let's draw schematically: We are not drawing with you. You don’t need to comply with any GOSTs, you won’t be punished.)

You can (for yourself) mark the angle values ​​on the axes and point the arrow in the direction against the clock. After all, we are going to save as a plus?) You don’t have to do this, but you need to keep everything in your head.

And now we draw the second (moving) side of the corner. In what quarter? In the first, of course! Because 60 degrees is strictly between 0° and 90°. So we draw in the first quarter. At an angle approximately 60 degrees to the fixed side. How to count approximately 60 degrees without a protractor? Easily! 60° is two thirds of a right angle! We mentally divide the first devil of the circle into three parts, taking two thirds for ourselves. And we draw... How much we actually get there (if you attach a protractor and measure) - 55 degrees or 64 - it doesn’t matter! It’s important that it’s still somewhere about 60°.

We get the picture:

That's all. And no tools were needed. Let's develop our eye! It will come in handy in geometry problems.) This unsightly drawing is indispensable when you need to quickly scribble a circle and an angle, without really thinking about beauty. But at the same time scribble Right, without errors, with all the necessary information. For example, as an aid in solving trigonometric equations and inequalities.

Let's now draw an angle, for example, 265°. Let's figure out where it might be located? Well, it’s clear that not in the first quarter and not even in the second: they end at 90 and 180 degrees. You can figure out that 265° is 180° plus another 85°. That is, to the negative semi-axis OX (where 180°) you need to add approximately 85°. Or, even simpler, guess that 265° does not reach the negative semi-axis OY (where 270° is) some unfortunate 5°. In short, in the third quarter there will be this angle. Very close to the negative semi-axis OY, to 270 degrees, but still in the third!

Let's draw:

Again, absolute precision is not required here. Let in reality this angle turn out to be, say, 263 degrees. But to the most important question (what quarter?) we answered correctly. Why is this the most important question? Yes, because any work with an angle in trigonometry (it doesn’t matter whether we draw this angle or not) begins with the answer to exactly this question! Always. If you ignore this question or try to answer it mentally, then mistakes are almost inevitable, yes... Do you need it?

Remember:

Any work with an angle (including drawing this very angle on a circle) always begins with determining the quarter in which this angle falls.

Now, I hope you can accurately depict angles, for example, 182°, 88°, 280°. IN correct quarters. In the third, first and fourth, if that...)

The fourth quarter ends with an angle of 360°. This is one full revolution. It is clear that this angle occupies the same position on the circle as 0° (i.e., the origin). But the angles don't end there, yeah...

What to do with angles greater than 360°?

“Are there really such things?”- you ask. They do happen! There is, for example, an angle of 444°. And sometimes, say, an angle of 1000°. There are all kinds of angles.) It’s just that visually such exotic angles are perceived a little more difficult than the angles we are used to within one revolution. But you also need to be able to draw and calculate such angles, yes.

To correctly draw such angles on a circle, you need to do the same thing - find out Which quarter does the angle we are interested in fall into? Here, the ability to accurately determine the quarter is much more important than for angles from 0° to 360°! The procedure for determining the quarter itself is complicated by just one step. You'll see what it is soon.

So, for example, we need to figure out which quadrant the 444° angle falls into. Let's start spinning. Where? A plus, of course! They gave us a positive angle! +444°. We twist, we twist... We twisted it one turn - we reached 360°.

How long is there left until 444°?We count the remaining tail:

444°-360° = 84°.

So, 444° is one full rotation (360°) plus another 84°. Obviously this is the first quarter. So, the angle 444° falls in the first quarter. Half the battle is done.

Now all that remains is to depict this angle. How? Very simple! We make one full turn along the red (plus) arrow and add another 84°.

Like this:

Here I didn’t bother cluttering the drawing - labeling the quarters, drawing angles on the axes. All this good stuff should have been in my head for a long time.)

But I used a “snail” or a spiral to show exactly how an angle of 444° is formed from angles of 360° and 84°. The dotted red line is one full revolution. To which 84° (solid line) are additionally screwed. By the way, please note that if this very full revolution is discarded, this will not affect the position of our angle in any way!

But this is important! Angle position 444° completely coincides with an angle position of 84°. There are no miracles, that’s just how it turns out.)

Is it possible to discard not one full revolution, but two or more?

Why not? If the angle is hefty, then it’s not only possible, but even necessary! The angle won't change! More precisely, the angle itself will, of course, change in magnitude. But his position on the circle is absolutely not!) That’s why they full revolutions, that no matter how many copies you add, no matter how many you subtract, you will still end up at the same point. Nice, isn't it?

Remember:

If you add (subtract) any angle to an angle whole the number of full revolutions, the position of the original angle on the circle will NOT change!

For example:

Which quarter does the 1000° angle fall into?

No problem! We count how many full revolutions sit in a thousand degrees. One revolution is 360°, another is already 720°, the third is 1080°... Stop! Too much! This means that it sits at an angle of 1000° two full turns. We throw them out of 1000° and calculate the remainder:

1000° - 2 360° = 280°

So, the position of the angle is 1000° on the circle the same, as at an angle of 280°. Which is much more pleasant to work with.) And where does this corner fall? It falls into the fourth quarter: 270° (negative semi-axis OY) plus another ten.

Let's draw:

Here I no longer drew two full turns with a dotted spiral: it turns out to be too long. I just drew the remaining tail from zero, discarding All extra turns. It’s as if they didn’t exist at all.)

Once again. In a good way, the angles 444° and 84°, as well as 1000° and 280°, are different. But for sine, cosine, tangent and cotangent these angles are - the same!

As you can see, in order to work with angles greater than 360°, you need to determine how many full revolutions sits in a given large angle. This is the very additional step that must be done first when working with such angles. Nothing complicated, right?

Rejecting full revolutions is, of course, a pleasant experience.) But in practice, when working with absolutely terrible angles, difficulties arise.

For example:

Which quarter does the angle 31240° fall into?

So what, are we going to add 360 degrees many, many times? It's possible, if it doesn't burn too much. But we can not only add.) We can also divide!

So let’s divide our huge angle into 360 degrees!

With this action we will find out exactly how many full revolutions are hidden in our 31240 degrees. You can divide it into a corner, you can (whisper in your ear:)) on a calculator.)

We get 31240:360 = 86.777777….

The fact that the number turned out to be fractional is not scary. Only us whole I'm interested in the revs! Therefore, there is no need to divide completely.)

So, in our shaggy coal sits as many as 86 full revolutions. Horror…

It will be in degrees86·360° = 30960°

Like this. This is exactly how many degrees can be painlessly thrown out of a given angle of 31240°. Remains:

31240° - 30960° = 280°

All! The position of the angle 31240° is fully identified! Same place as 280°. Those. fourth quarter.) I think we've already depicted this angle before? When was the 1000° angle drawn?) There we also went 280 degrees. Coincidence.)

So, the moral of this story is:

If we are given a scary hefty angle, then:

1. Determine how many full revolutions sit in this corner. To do this, divide the original angle by 360 and discard the fractional part.

2. We count how many degrees there are in the resulting number of revolutions. To do this, multiply the number of revolutions by 360.

3. We subtract these revolutions from the original angle and work with the usual angle ranging from 0° to 360°.

How to work with negative angles?

No problem! Exactly the same as with positive ones, only with one single difference. Which one? Yes! You need to turn the corners reverse side, minus! Going clockwise.)

Let's draw, for example, an angle of -200°. First, everything is as usual for positive angles - axes, circle. Let's also draw a blue arrow with a minus and sign the angles on the axes differently. Naturally, they will also have to be counted in a negative direction. These will be the same angles, stepping through 90°, but counted in the opposite direction, to the minus: 0°, -90°, -180°, -270°, -360°.

The picture will look like this:

When working with negative angles, there is often a feeling of slight bewilderment. How so?! It turns out that the same axis is, say, +90° and -270° at the same time? No, something is fishy here...

Yes, everything is clean and transparent! We already know that any point on a circle can be called either a positive or a negative angle! Absolutely any. Including on some of the coordinate axes. In our case we need negative angle calculus. So we snap all the corners to minus.)

Now drawing the angle -200° correctly is not difficult at all. This is -180° and minus another 20°. We begin to swing from zero to minus: we fly through the fourth quarter, we also miss the third, we reach -180°. Where should I spend the remaining twenty? Yes, everything is there! By the hour.) Total angle -200° falls within second quarter.

Now do you understand how important it is to firmly remember the angles on the coordinate axes?

The angles on the coordinate axes (0°, 90°, 180°, 270°, 360°) must be remembered precisely in order to accurately determine the quarter where the angle falls!

What if the angle is large, with several full turns? It's OK! What difference does it make whether these full revolutions are turned to positive or negative? A point on a circle will not change its position!

For example:

Which quarter does the -2000° angle fall into?

All the same! First, we count how many full revolutions sit in this evil corner. In order not to mess up the signs, let’s leave the minus alone for now and simply divide 2000 by 360. We’ll get 5 with a tail. We don’t care about the tail for now, we’ll count it a little later when we draw the corner. We count five full revolutions in degrees:

5 360° = 1800°

Wow. This is exactly how many extra degrees we can safely throw out of our corner without harming our health.

We count the remaining tail:

2000° – 1800° = 200°

But now we can remember about the minus.) Where will we wind the 200° tail? Minus, of course! We are given a negative angle.)

2000° = -1800° - 200°

So we draw an angle of -200°, only without any extra revolutions. I just drew it, but so be it, I’ll draw it one more time. By hand.

It is clear that the given angle -2000°, as well as -200°, falls within second quarter.

So, let’s go crazy... sorry... on our head:

If a very large negative angle is given, then the first part of working with it (finding the number of full revolutions and discarding them) is the same as when working with a positive angle. The minus sign does not play any role at this stage of the solution. The sign is taken into account only at the very end, when working with the angle remaining after removing full revolutions.

As you can see, drawing negative angles on a circle is no more difficult than positive ones.

Everything is the same, only in the other direction! By the hour!

Now comes the most interesting part! We looked at positive angles, negative angles, large angles, small angles - the full range. We also found out that any point on a circle can be called a positive and negative angle, we discarded full revolutions... Any thoughts? It must be postponed...

Yes! Whatever point on the circle you take, it will correspond to infinite number of angles! Big ones and not so big ones, positive ones and negative ones - all kinds! And the difference between these angles will be whole number of full revolutions. Always! That’s how the trigonometric circle works, yes...) That’s why reverse the task is to find the angle using the known sine/cosine/tangent/cotangent - solvable ambiguous. And much more difficult. In contrast to the direct problem - given an angle, find the entire set of its trigonometric functions. And in more serious topics of trigonometry ( arches, trigonometric equations And inequalities ) we will encounter this trick all the time. We're getting used to it.)

1. Which quarter does the -345° angle fall into?

2. Which quarter does the angle 666° fall into?

3. Which quarter does the angle 5555° fall into?

4. Which quarter does the -3700° angle fall into?

5. What sign doescos999°?

6. What sign doesctg999°?

And did it work? Wonderful! There is a problem? Then you.

Answers:

1. 1

2. 4

3. 2

4. 3

5. "+"

6. "-"

This time the answers were given in order, breaking with tradition. For there are only four quarters, and there are only two signs. You won’t run away too much...)

In the next lesson we will talk about radians, about the mysterious number "pi", we will learn how to easily and simply convert radians to degrees and vice versa. And we will be surprised to discover that even this simple knowledge and skills will be quite enough for us to successfully solve many non-trivial trigonometry problems!

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A pair of different rays Oa and Ob emanating from one point O is called an angle and is denoted by the symbol (a, b). Point O is called the vertex of the angle, and the rays Oa and Ob are called the sides of the angle. If A and B are two points of the rays Oa and Ob, then (a, b) is also denoted by the symbol AOB (Fig. 1.1).

An angle (a, b) is called unfolded if the rays Oa and Ob, coming out from the same point, lie on the same straight line and do not coincide (i.e., opposite directions).

Fig.1.1

Two angles are considered equal if one angle can be superimposed on the other so that the sides of the angles coincide. The bisector of an angle is a ray with its origin at the vertex of the angle, dividing the angle into two equal angles.

They say that ray OS emanating from the vertex of angle AOB lies between its sides if it intersects segment AB (Fig. 1.2). Point C is said to lie between the sides of an angle if through this point it is possible to draw a ray with its origin at the vertex of the angle and lying between the sides of the angle. The set of all points of the plane lying between the sides of the angle forms the internal region of the angle (Fig. 1.3). The set of points of the plane that do not belong to the internal region and sides of the angle forms the external region of the angle.

Angle (a, b) is considered greater than angle (c, d) if angle (c, d) can be superimposed on angle (a, b) so that after combining one pair of sides, the second side of angle (c, d) will lie between sides of the angle (a, b). In Fig. 1.4 AOB is greater than AOC.

Let the ray c lie between the sides of the angle (a, b) (Fig. 1.5). Pairs of rays a, c and c, b form two angles. The angle (a, b) is said to be the sum of two angles (a, c) and (c, b), and they write: (a, b) = (a, c) + (c, b).

Fig.1.3

Usually in geometry we deal with angles smaller than the unfolded angle. However, the addition of two angles can result in an angle larger than the unfolded one. In this case, that part of the plane that is considered the inner area of ​​the angle is marked with an arc. In Fig. 1.6, the inner part of the angle AOB, obtained by adding the angles AOS and COB and the larger unfolded one, is marked with an arc.

Fig.1.5

There are also angles greater than 360°. Such angles are formed, for example, by the rotation of an airplane propeller, the rotation of a drum on which a rope is wound, etc.

In the future, when considering each angle, we will agree to consider one of the sides of this angle as its initial side, and the other as its final side.

Any angle, for example angle AOB (Fig. 1.7), can be obtained by rotating a moving beam around vertex O from the initial side of the angle (OA) to its final side (OB). We will measure this angle, taking into account the total number of revolutions made around point O, as well as the direction in which the rotation occurred.

Positive and negative angles.

Let us have an angle formed by rays OA and OB (Fig. 1.8). The moving beam, rotating around point O from its initial position (OA), can take its final position (OB) in two different directions of rotation. These directions are shown in Figure 1.8 by the corresponding arrows.

Fig.1.7

Just as on the number axis one of the two directions is considered positive and the other negative, two different directions of rotation of the moving beam are also distinguished. We agreed to consider the positive direction of rotation to be the direction opposite to the direction of clockwise rotation. The direction of rotation coinciding with the direction of rotation clockwise is considered negative.

According to these definitions, angles are also classified into positive and negative.

A positive angle is the angle formed by rotating the moving beam around the starting point in a positive direction.

Figure 1.9 shows some positive angles. (The direction of rotation of the moving beam is shown in the drawings by arrows.)

A negative angle is the angle formed by rotating the moving beam around the starting point in a negative direction.

Figure 1.10 shows some negative angles. (The direction of rotation of the moving beam is shown in the drawings by arrows.)

But two coinciding rays can also form angles +360°n and -360°n (n = 0,1,2,3,...). Let us denote by b the smallest possible non-negative angle of rotation that transfers the beam OA to position OB. If now ray OB makes an additional full revolution around point O, then we obtain a different angle value, namely: ABO = b + 360°.

Measuring angles using circular arcs. Units for arcs and angles

In some cases it turns out to be convenient to measure angles using circular arcs. The possibility of such a measurement is based on the well-known proposal of planimetry that in one circle (or in equal circles) the central angles and the corresponding arcs are in direct proportion.

Let some arc of a given circle be taken as the unit of measurement of arcs. We take the central angle corresponding to this arc as the unit of measurement for angles. Under this condition, any arc of a circle and the central angle corresponding to this arc will contain the same number of units of measurement. Therefore, by measuring the arcs of a circle, it is possible to determine the value of the central angles corresponding to these arcs.

Let's look at the two most common systems for measuring arcs and angles.

Degree measure of angles

When measuring angles by degrees, an angle of one degree (denoted 1?) is taken as the basic unit of measurement of angles (the reference angle with which various angles are compared). An angle of one degree is an angle equal to 1/180 of the reversed angle. An angle equal to 1/60th of an angle of 1° is an angle of one minute (denoted 1"). An angle equal to 1/60th of an angle of one minute is an angle of one second (denoted 1").

Radian measure of angles

Along with the degree measure of angles, geometry and trigonometry also use another measure of angles, called the radian. Let's consider a circle of radius R with center O. Let's draw two radii O A and OB so that the length of the arc AB is equal to the radius of the circle (Fig. 1.12). The resulting central angle AOB will be an angle of one radian. An angle of 1 radian is taken as the radian unit of measurement for angles. When measuring angles in radians, the rotated angle is equal to p radians.

The degree and radian units of measurement of angles are related by the equalities:

1 radian =180?/р57° 17" 45"; 1?=p/180 radians0.017453 radians;

1"=p/180*60 radian0.000291 radian;

1""=p/180*60*60 radian0.000005 radian.

The degree (or radian) measure of an angle is also called the angle magnitude. The angle AOB is sometimes denoted /

Classification of angles

An angle equal to 90°, or in radian measure p/2, is called a right angle; it is often denoted by the letter d. An angle less than 90° is called acute; An angle greater than 90° but less than 180° is called obtuse.

Two angles that have one common side and add up to 180° are called adjacent angles. Two angles that have one common side and add up to 90° are called supplementary angles.