Counting corners on a trigonometric circle. Positive and negative angles. The distribution of corners in quarters. Corner measurement can dedresses be negative

Angle: ° π RAD \u003d

Convert to: radians degrees 0 - 360 ° 0 - 2π positive negative calculation

When direct intersects, then four different areas are obtained relative to the intersection point.
These new areas are called corners.

The picture shows 4 different angle formed by the intersection of direct AB and CD

Typically, the angles are measured in degrees, which is indicated as °. When the object performs a full circle, that is, moving from point d via B, C, A, and then back to D, then they say that it turned 360 degrees (360 °). Thus, the degree is $ \\ FRAC (1) (360) $ circle.

Angles More than 360 degrees

We talked about when the object makes a full circle around the point, then it passes 360 °, however, when the object makes more than one circle, it makes an angle of more than 360 degrees. This is a common phenomenon in everyday life. The wheel passes many circles when the car moves, that is, it forms an angle greater than 360 °.

In order to find out the number of cycles (circles covered) when the object is rotated, we consider the number of times that you need to add 360 to yourself to get a number equal to or smaller than this angle. Similarly, we find the number we multiply on 360 to get a number of smaller, but the closest to this corner.

Example 2.
1. Find the number of circles described by the angle
a) 380 °
b) 770 °
c) 1000 °
Decision
a) 380 \u003d (1 × 360) + 20
The object described one circle and 20 °
Since $ 20 ^ (\\ CIRC) \u003d \\ FRAC (20) (360) \u003d \\ FRAC (1) (18) $ circle
The object described $ 1 \\ FRAC (1) (18) $ circles.

B) 2 × 360 \u003d 720
770 \u003d (2 × 360) + 50
The object described two circles and 50 °
$ 50 ^ (\\ CIRC) \u003d \\ FRAC (50) (360) \u003d \\ FRAC (5) (36) $ Circle
The object described $ 2 \\ FRAC (5) (36) $ Circle
c) 2 × 360 \u003d 720
1000 \u003d (2 × 360) + 280
$ 280 ^ (\\ CIRC) \u003d \\ FRAC (260) (360) \u003d \\ FRAC (7) (9) $ circles
The object described $ 2 \\ FRAC (7) (9) $ circles

When the object rotates clockwise, it forms a negative angle of rotation, and when it rotates counterclockwise - a positive angle. Up to this point, we considered only positive angles.

In the form of a diagram, a negative angle can be depicted as shown below.

The figure below shows the sign of the angle, which is measured from the total line, 0 axis (abscissal axis axis)

This means that in the presence of a negative angle, we can get the corresponding positive angle.
For example, the lower part of the vertical direct is 270 °. When measured in a negative direction, then we get -90 °. We simply subtract 270 out of 360. Having a negative angle, we add 360 in order to get a corresponding positive angle.
When an angle is -360 °, this means that the object made more than one circle clockwise.

Example 3.
1. Find the appropriate positive angle
a) -35 °
b) -60 °
C) -180 °
d) - 670 °

2. Find the corresponding negative angle of 80 °, 167 °, 330 ° and 1300 °.
Decision
1. In order to find the appropriate positive angle, we add 360 to the corner value.
a) -35 ° \u003d 360 + (-35) \u003d 360 - 35 \u003d 325 °
b) -60 ° \u003d 360 + (-60) \u003d 360 - 60 \u003d 300 °
c) -180 ° \u003d 360 + (-180) \u003d 360 - 180 \u003d 180 °
d) -670 ° \u003d 360 + (-670) \u003d -310
This means one circle clockwise (360)
360 + (-310) \u003d 50 °
The angle is 360 + 50 \u003d 410 °

2. In order to obtain an appropriate negative angle, we subtract 360 from the angle value.
80 ° \u003d 80 - 360 \u003d - 280 °
167 ° \u003d 167 - 360 \u003d -193 °
330 ° \u003d 330 - 360 \u003d -30 °
1300 ° \u003d 1300 - 360 \u003d 940 (one circle passed)
940 - 360 \u003d 580 (the second round passed)
580 - 360 \u003d 220 (the third circle passed)
220 - 360 \u003d -140 °
The angle is -360 - 360 - 360 - 140 \u003d -1220 °
Thus, 1300 ° \u003d -1220 °

Radian

Radine is an angle of the center of the circle, into which the arc is concluded, the length of which is equal to the radius of this circle. This is a unit of measurement of an angular value. Such an angle is approximately 57.3 °.
In most cases, it is indicated as glad.
Thus $ 1 Rad \\ APPROX 57.3 ^ (\\ CIRC) $

Radius \u003d R \u003d OA \u003d OB \u003d AB
BOA angle is equal to one radia

Since the circumference length is set as $ 2 \\ pi R $, then in the circumference of $ 2 \\ pi $ radii, which means in general the circle $ 2 \\ pi $ radian.

Radians are usually expressed by $ \\ pi $ to avoid decimal parts in the calculations. In most books, abbreviation rAD (RAD) Not found, but the reader should know that when it comes to an angle, then it is set through $ \\ pi $, and the units of measurement automatically become radians.

$ 360 ^ (\\ CIRC) \u003d 2 \\ PI \\ RAD $
$ 180 ^ (\\ CIRC) \u003d \\ PI \\ RAD $
$ 90 ^ (\\ CIRC) \u003d \\ FRAC (\\ PI) (2) RAD $
$ 30 ^ (\\ CIRC) \u003d \\ FRAC (30) (180) \\ pi \u003d \\ FRAC (\\ PI) (6) RAD $
$ 45 ^ (\\ CIRC) \u003d \\ FRAC (45) (180) \\ pi \u003d \\ FRAC (\\ PI) (4) RAD $
$ 60 ^ (\\ CIRC) \u003d \\ FRAC (60) (180) \\ pi \u003d \\ FRAC (\\ PI) (3) RAD $
$ 270 ^ (\\ CIRC) \u003d \\ FRAC (270) (180) \\ pi \u003d \\ FRAC (27) (18) \\ pi \u003d 1 \\ FRAC (1) (2) \\ PI \\ RAD $

Example 4.
1. Convert 240 °, 45 °, 270 °, 750 ° and 390 ° to radians through $ \\ pi $.
Decision
I multiply the corners on $ \\ FRAC (\\ PI) (180) $.
$ 240 ^ (\\ CIRC) \u003d 240 \\ Times \\ FRAC (\\ PI) (180) \u003d \\ FRAC (4) (3) \\ pi \u003d 1 \\ FRAC (1) (3) \\ pi $
$ 120 ^ (\\ CIRC) \u003d 120 \\ Times \\ FRAC (\\ pi) (180) \u003d \\ FRAC (2 \\ PI) (3) $
$ 270 ^ (\\ CIRC) \u003d 270 \\ Times \\ FRAC (1) (180) \\ pi \u003d \\ FRAC (3) (2) \\ pi \u003d 1 \\ FRAC (1) (2) \\ pi $
$ 750 ^ (\\ CIRC) \u003d 750 \\ Times \\ FRAC (1) (180) \\ pi \u003d \\ FRAC (25) (6) \\ pi \u003d 4 \\ FRAC (1) (6) \\ pi $
$ 390 ^ (\\ CIRC) \u003d 390 \\ Times \\ FRAC (1) (180) \\ pi \u003d \\ FRAC (13) (6) \\ pi \u003d 2 \\ FRAC (1) (6) \\ pi $

2. Convert the following angles to degrees.
a) $ \\ FRAC (5) (4) \\ pi $
b) $ 3.12 \\ pi $
c) 2.4 radians
Decision
$ 180 ^ (\\ CIRC) \u003d \\ pi $
a) $ \\ FRAC (5) (4) \\ pi \u003d \\ FRAC (5) (4) \\ Times 180 \u003d 225 ^ (\\ CIRC) $
b) $ 3.12 \\ pi \u003d 3.12 \\ Times 180 \u003d 561.6 ^ (\\ CIRC) $
c) 1 please \u003d 57.3 °
$ 2.4 \u003d \\ FRAC (2.4 \\ TIMES 57.3) (1) \u003d 137.52 $

Negative angles and angles more than $ 2 \\ pi $ radian

In order to convert a negative angle into a positive, we fold it with $ 2 \\ pi $.
In order to convert a positive angle to negative, we will deduct $ 2 \\ pi $ from it.

Example 5.
1. Convert $ - \\ FRAC (3) (4) \\ pi $ and $ - \\ FRAC (5) (7) \\ pi $ into positive angles in radians.

Decision
Add to the corner of $ 2 \\ pi $
$ - \\ FRAC (3) (4) \\ pi \u003d - \\ FRAC (3) (4) \\ pi + 2 \\ pi \u003d \\ frac (5) (4) \\ pi \u003d 1 \\ FRAC (1) (4) \\ $ - \\ FRAC (5) (7) \\ pi \u003d - \\ FRAC (5) (7) \\ pi + 2 \\ pi \u003d \\ frac (9) (7) \\ pi \u003d 1 \\ FRAC (2) (7) \\ When the object rotates an angle greater than $ 2 \\ pi $;, then it makes more than one circle.

In order to determine the number of revolutions (circles or cycles) in such angle, we find such a number, which is multiplying which $ 2 \\ pi $ is equal to or less, but as close as possible to this number.

Example 6.
1. Find the number of circles covered by the object in these corners

a) $ -10 \\ pi $
b) $ 9 \\ pi $
c) $ \\ FRAC (7) (2) \\ pi $
a) $ -10 \\ pi \u003d 5 (-2 \\ pi) $;
$ -2 \\ pi $ implies one cycle towards clockwise direction, then this means that

Decision
The object made 5 cycles clockwise.
b) $ 9 \\ pi \u003d 4 (2 \\ pi) + \\ pi $, $ \\ pi \u003d $ floor cycle
The object made four and a half cycle counterclockwise

c) $ \\ FRAC (7) (2) \\ pi \u003d 3.5 \\ pi \u003d 2 \\ pi + 1.5 \\ pi $, $ 1.5 \\ pi $ is three quarters of a $ cycle (\\ FRAC (1.5 \\ PI) (2 \\ PI) \u003d \\ FRAC (3) (4)) $
The object passed one and three quarters of the cycle counterclockwise

trigonometry, like science, originated in the ancient East. The first trigonometric ratios were derived by astronomers to create an accurate calendar and focus on the stars. These calculations belonged to spherical trigonometry, while in the school course, the ratios of the parties and an angle of a flat triangle are studied.
Trigonometry is a section of mathematics engaged in the properties of trigonometric functions and dependence between the parties and the corners of the triangles.

During the heyday of the culture and science of the first millennium, our era of knowledge has spread from the ancient East to Greece. But the main discoveries of trigonometry is the merit of husbands of the Arab Caliphate. In particular, the Turkmen scientist al-Marazvi entered the functions such as Tangent and Kotangent, compiled first tables for sinus values, tangents and catangents. The concept of sine and cosine is introduced by Indian scientists. Trigonometry is devoted to a lot of attention in the writings of such great leaders of antiquity, like Euclidea, Archimedes and Eratosthene.

The main values \u200b\u200bof trigonometry

The main trigonometric functions of the numerical argument are sinus, cosine, tangent and catangent. Each of them has its own schedule: sinusoid, cosineida, tangensoid and catangensoid.

The basis of the formulas for calculating the values \u200b\u200bof the specified quantities is the Pythagoreo theorem. Schoolchildren are more known in the wording: "Pythagoras pants, in all directions are equal," since the proof is given on an example of an equally sized rectangular triangle.

Sinus, cosine and other dependences establish a link between sharp corners and sides of any rectangular triangle. We give formulas for calculating these values \u200b\u200bfor angle A and trace the relationship of trigonometric functions:

As can be seen, TG and CTG are inverse functions. If you submit catat A as a piece of SIN A and hypotenuses with, and roll b in the form of COS A * C, we will receive the following formulas for Tangent and Kotangent:

Trigonometric circle

The graphically, the ratio of said values \u200b\u200bcan be represented as follows:

Circle, in this case, is all possible angle α - from 0 ° to 360 °. As can be seen from the figure, each function takes a negative or positive value depending on the corner value. For example, SIN α will be with the "+" sign, if α belongs to the I and II of the quarter of the circle, that is, it is between 0 ° to 180 °. With α from 180 ° to 360 ° (III and IV quarters), SIN α can only be a negative value.

Let's try to build trigonometric tables for specific angles and find out the value of values.

The values \u200b\u200bα are 30 °, 45 °, 60 °, 90 °, 180 °, and so on - are called special cases. The values \u200b\u200bof trigonometric functions are calculated for them and are presented in the form of special tables.

These angles are not chosen by any accident. The designation π in the tables stands for radians. Rad is an angle at which the length of the circumference arc corresponds to its radius. This value was introduced in order to establish a universal dependence, when calculating radians, the actual radius length in cm does not matter.

Corners in tables for trigonometric functions correspond to radian values:

So, it is not difficult to guess that 2π is a complete circle or 360 °.

Properties of trigonometric functions: sinus and cosine

In order to consider and compare the main properties of sinus and cosine, tangent and catangens, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.

Consider a comparative table of properties for sinusoids and cosineids:

Determine whether the function is even or not very simple. It is enough to present a trigonometric circle with signs of trigonometric values \u200b\u200band mentally "folded" the schedule relative to the OX axis. If the signs coincide, the function is even, otherwise - an odd.

Introduction of radians and transfer of the main properties of sinusoids and cosineids allow you to bring the following regularity:

Make sure the formula is very simple. For example, for x \u003d π / 2 sinus is 1, as well as cosine x \u003d 0. You can check for the tables or tracing the functions of functions for the specified values.

Properties of Tangensoids and Kotangensoids

The graphs of the functions of Tangent and Kotangent differ significantly from sinusoids and cosineids. The values \u200b\u200bof TG and CTG are back to each other.

1. Y \u003d TG x.
2. TangentSoid tends to values \u200b\u200by at x \u003d π / 2 + πk, but never reaches them.
3. The lowest positive period of tangensoid is equal to π.
4. TG (- x) \u003d - TG x, i.e., the function is odd.
5. TG x \u003d 0, at x \u003d πk.
6. The function is increasing.
7. TG x\u003e 0, at x ε (πk, π / 2 + πk).
8. TG x \u003c0, at x ε (- π / 2 + πk, πk).
9. Derivative (TG x) '\u003d 1 / COS 2 \u2061X.

Consider the graphic image of the catangensoids below the text.

The main properties of Kotangensoids:

1. Y \u003d CTG x.
2. Unlike the functions of sinus and cosine, in TangentSoid Y, it can take the values \u200b\u200bof many valid numbers.
3. Kothangensoid tends to values \u200b\u200by at x \u003d πk, but never reaches them.
4. The smallest positive period of the catangensoid is equal to π.
5. CTG (- x) \u003d - CTG X, i.e. function is odd.
6. CTG x \u003d 0, at x \u003d π / 2 + πk.
7. The function is descending.
8. CTG x\u003e 0, at x ε (πk, π / 2 + πk).
9. CTG x \u003c0, at x ε (π / 2 + πk, πk).
10. Derivative (CTG x) '\u003d - 1 / sin 2 \u2061x fix

A pair of different beams of OA and OB, emerging from one point O, is called an angle and is indicated by the symbol (A, B). The point O is called the peak of the angle, and the rays of the UB UB - the sides of the angle. If a and b - two points of the OA and OB rays, then (a, b) is also indicated by the symbol of AOs (Fig. 1.1).

The angle (a, b) is called expanded if the rays of OA and OB, emerging from one point lie on one straight and do not coincide (that is the oppositely directed).

Fig.1.1

Two angle are considered equal if one angle can be applied to another so that the side of the corners coincide. The bisector angle is called a beam with the beginning at the top of the angle, dividing the angle into two equal corners.

It is said that a ray of OS, coming from the top of the angle ao, lies between its parties if it crosses the segment of AV (Fig. 1.2). It is said that the point C lies between the sides of the angle, if a ray can be held through this point with the beginning at the top of the angle, lying between the corner sides. The set of all points of the plane lying between the sides of the angle forms an inner area of \u200b\u200bthe angle (Fig. 1.3). The set of points of the plane that do not belong to the inner region and the sides of the angle forms an outer area of \u200b\u200bthe angle.

An angle (A, B) is considered more angle (C, D) if the angle (C, D) can be imposed on an angle (A, B) so that after the combination of one pair of sides, the second side of the angle (C, D) will lie between The sides of the angle (A, B). In fig. 1.4 AOs More AOS.

Let the beam with lies between the sides of the angle (A, B) (Fig. 1.5). Couples of rays A, C and C, B form two corners. About the angle (A, B) they say that it is the sum of two angles (A, C) and (C, B), and write: (A, B) \u003d (A, C) + (C, B).

Fig.1.3

Usually in geometry they deal with angles smaller than the deployed. However, as a result of the addition of two angles, an angle can be turned out to be more deployed. In this case, the part of the plane, which is considered an internal area of \u200b\u200bthe angle, is marked with arc. In fig. 1.6 The inside of the angle aos, obtained as a result of the addition of the angles of AOC and OV and more deployed, is marked by arc.

Fig.1.5.

There are also angles of large 360 \u200b\u200b°. Such angles are formed, for example, by the rotation of the plane propeller, the rotation of the drum, on which the rope is wound, and so on.

In the future, when considering each angle, we agree to consider one of the sides of this angle from its initial party, and the other is the ultimate party.

Any angle, for example, the angle of AOs (Fig. 1.7), can be obtained as a result of rotation of the movable beam around the vertex of the initial side of the angle (OA) to its end side (s). We will measure this angle, considering the complete number of revolutions made around the point O, as well as the direction in which rotation happened.

Positive and negative angles.

Let we have an angle formed by the Rays of OA and OI (Fig.1.8). The movable ray, rotating around the point O from its initial position (OA), may take the final position (s) at two different directions of rotation. These directions are shown in Figure 1.8 with the corresponding arrows.

Fig.1.7

Just as on a numeric axis, one of two directions is considered positive, and the other is negative, two different directions of rotation of the movable beam are distinguished. It was agreed to be considered a positive direction of rotation that direction that is opposite to the direction of rotation of the clockwise. The direction of rotation coincides with the direction of rotation of the clockwise, is considered negative.

In accordance with these definitions, the angles are also divided into positive and negative.

A positive angle is called an angle formed by the rotation of the movable beam around the starting point in the positive direction.

Figure 1.9 has some positive angles. (The direction of rotation of the movable beam is shown in the drawings of the arrows.)

A negative angle is called an angle formed by the rotation of the movable beam around the starting point in the negative direction.

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

But the two coinciding rays can also form and the angles of + 360 ° P and -360 ° C (n \u003d 0,1,2,3, ...). Denote by used the smallest possible non-negative angle of rotation, which has been transferring the OA ray to the OS position. If now the ray of ov is made additionally a full turn around the point O, then we get another value of the angle, namely: avo \u003d b + 360 °.

Measurement of corners of circle arcs. Units of measurement of arcs and corners

In some cases, it turns out to be convenient to measure the angles using the circle arc. The possibility of such a measurement the basis on the well-known proposal of the planimery that the central angles and the corresponding arcs are in direct proportional dependence on them are in one circle.

Let some arc of this circle adopted per unit of measurement of the arc. The central angle corresponding to this arc will take the unit of measurement of the corners. With this condition, any circumference arc and the corresponding central angle corresponding to this arc will contain the same number of measurement units. Therefore, measuring circle arcs, it is possible to determine the magnitude of the central corners corresponding to these arcs.

Consider the two most common systems for measuring arcs and angles.

Degree Measuring Corners

During a degreesal measurement of the angles as the main unit of measurement of angles (the reference angle, with which various angles are compared) an angle is taken into one degree (designated 1?). The angle of one degree is an angle equal to 1/180 of the expanded angle. An angle equal to 1/60 of an angle of 1 ° is an angle of one minute (designated 1 "). Angle equal to 1/60 part of an angle in one minute is an angle in one second (designated 1").

Radian Corner Measurement Measurement

Along with the degree measurement measurement of angles in geometry and trigonometry, the other measurement measurement is also used, called radian. Consider the circle of the radius R with the center of O. We will carry out two radius about A and OB so that the length of the AV arc is equal to the circle radius (Fig. 1.12). The central angle of AOs obtained at the same time will be an angle of one radian. The angle of 1 radian is adopted per unit of measurement of the radicular measurement measurement measurement. When the angles are radic, the detailed angle is equal to R radians.

The degree and radian units of measurement of the angles are associated with equalities:

1 radian \u003d 180? / P57 ° 17 "45"; 1? \u003d P / 180 radian0.017453Radian;

1 "\u003d p / 180 * 60 radian0.000291 radian;

1 "" \u003d R / 180 * 60 * 60 radian0.000005 radian.

The degree (or radian) measure of the angle is also called the corner. The angle value of AOs is sometimes denoted /

Classification of corners

An angle equal to 90 °, or in the radicular measure of P / 2, is called a direct angle; It is often denoted by the letter d. An angle less than 90 ° is called sharp; The angle greater than 90 °, but the smaller 180 ° is called stupid.

Two angles having one common side and in the amount of 180 ° components are called adjacent angles. Two angles having one common side and in the amount of 90 ° components are called additional corners.

Counting corners on a trigonometric circle.

Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")

He is almost like in the previous lesson. There are axes, circle, angle, all rank chinar. Added quarters (in the corners of a large square) - from the first to fourth. And then what if someone does not know? As you can see, a quarter (they are also called the beautiful word "quadrants") numbers against the course of the clockwise. Added angle values \u200b\u200bon the axes. Everything is clear, no problems.

And the green arrow has been added. With a plus. What does it mean? Let me remind you that the stationary side of the corner always It is nailed to the positive semi-axis oh. So, if we will turn the moving side of the angle on the arrow with a plus. Ascending quarta numbers, the angle will be considered positive. For example, the picture shows a positive angle + 60 °.

If we hold off the corners in the opposite direction, along the clockwise arrow, the angle will be considered negative. Mouse over the picture (or tap pictures on the tablet), see the blue arrow with a minus. This is the direction of the negative reference of the corners. For example, a negative angle is shown (60 °). And you will see how the diquses have changed on the axes ... I also transferred them to negative angles. Numbering quadrants does not change.

Here, usually, the first misunderstandings begin. How so!? And if the negative angle on the circle coincides with positive!? And in general, it turns out that, the same position of the movable side (or point on the numerical circle) can be called as a negative angle and positive!?

Yes. Exactly. Let's say a positive angle of 90 degrees occupies on a circle exactly the same The situation is as a negative angle in minus 270 degrees. Positive angle, for example, + 110 ° degrees occupies exactly the same position as the negative angle -250 °.

No problem. Above correctly.) The choice of a positive or negative calculus of the angle depends on the condition of the task. If nothing is said in the condition open text about the corner sign, (type "to determine the smallest positive Corner ", etc.), we work with comfortable values.

Except (and how without them?!) Are trigonometric inequalities, but there we will master this chip.

And now the question you. How did I recognize that the position of the angle of 110 ° coincides with the position of the angle -250 °?
Nickname that this is due to the full turn. At 360 ° ... not clear? Then draw a circle. We draw, on paper. We mark the corner about 110 °. AND considerhow much it remains to full turnover. It will remain just 250 ° ...

Caught? And now - attention! If the angles are 110 ° and -250 ° occupy in a circle same position, what? Yes that the angles are 110 ° and -250 ° completely identical Sinus, Kosinus, Tangent and Cotangent!
Those. sin110 ° \u003d sin (-250 °), CTG110 ° \u003d CTG (-250 °) and so on. This is already really important! And in itself - there is a lot of tasks, where it is necessary to simplify expressions, and as a base for the subsequent development of the formulas of bringing and other wisdom of trigonometry.

Clear case, 110 ° and -250 ° I took the Namaum, purely for example. All these equalities work for any corners that occupy one position in the circle. 60 ° and -300 °, -75 ° and 285 °, and so on. I note immediately that the angles in these couples - different. And here are trigonometric functions of them - the same.

I think that such negative angles you understand. It is quite simple. Against the clockwise progress - a positive countdown. In the course - negative. Read the angle positive, or negative depends on us. From our desire. Well, and from the task, of course ... I hope you understand and how to move in trigonometric functions from negative angles to positive and back. Draw a circle, an approximate angle, but see how much lacks up to full turnover, i.e. up to 360 °.

Corners are greater than 360 °.

Corners that are more than 360 °. Are there any such? There are, of course. How to draw them in a circle? Yes, not a problem! Suppose we need to understand which quarter will get an angle of 1000 °? Easily! We make one full turn against the time of the clockwise (the angle was given positive!). Moved 360 °. Well, and wind on! Another turn - already it turned out 720 °. How much is left? 280 °. There is not enough for a complete turn ... But angle is greater than 270 ° - and this is the border between the third and fourth quarter. It was our angle in 1000 ° enters the fourth quarter. Everything.

As you can see, it's quite easy. Once again I remind you that the angle is 1000 ° and an angle of 280 °, which we got through the discarding "unnecessary" full revolutions - it is, strictly speaking, different Corners. But trigonometric functions of these corners completely identical! Those. SIN1000 ° \u003d SIN280 °, COS1000 ° \u003d COS280 °, etc. If I were sinus, I would not notice the difference between these two corners ...

Why do you need all this? Why do we need to translate corners from one to another? Yes, everything is the same.) In order to simplify expressions. Simplification of expressions, actually, the main task of school mathematics. Well, in the way, the head is training.)

Well, practice?)

Answer questions. First simple.

1. What quarter does the corner -325 ° fall?

2. What quarter does the angle of 3000 ° fall?

3. What quarter does the angle -3000 ° fall?

There is a problem? Or uncertainty? We go to section 555, practical work with a trigonometric circle. There, in the first lesson this very "practical work ..." everything is detailed ... in such Insecurity issues to be not!

4. What sign does sin555 ° have?

5. What sign is TG555 °?

Defined? Excellent! Doubt? It is necessary to section 555 ... By the way, they will learn how to draw Tangent and Cotangent on a trigonometric circle. Very useful thing.

And now questions in the root.

6. Certify the expression SIN777 ° to the sinus of the smallest positive angle.

7. Create an expression COS777 ° to the cosine of the greatest negative angle.

8. Provide the COS expression (-777 °) to the cacinus of the smallest positive angle.

9. Certify the expression SIN777 ° to the sinus of the highest negative angle.

What, questions 6-9 puzzled? Get used to, on the exam and not such wording meet ... So be, I will translate. Only for you!

The words "bring an expression to ..." mean to convert an expression so that its value not changed And the appearance has changed in accordance with the task. So, in the task 6 and 9 we must get sinus, within which it costs mute positive corner. Everything else - it does not matter.

Answers will be issued in order (in violation of our rules). And what to do, the sign is only two, and the quarter is only four ... you will not run in the options.

6. SIN57 °.

7. COS (-57 °).

8. COS57 °.

9. -sin (-57 °)

I assume that the answers to questions 6 -9 someone confused. Special -sin (-57 °)Is it true?) Indeed, in the elementary rules of reference corners there is a place for errors ... that is why I had to do a lesson: "How to identify the signs of functions and bring the corners on a trigonometric circle?" In section 555. There are 4 - 9 tasks disassembled. Well disassembled, with all underwater stones. And they are here.)

In the next lesson, we will deal with mysterious radians and the number "PI". We will learn to easily and correctly translate degrees into radians and back. And with surprise you will find that this elementary information on the site already grabs To solve some non-standard trigonometry tasks!

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In trigonometry, an important concept is angle of rotation. Below we will consistently give an idea of \u200b\u200bthe rotation, and enter all the concomitant concepts. Let's start with the general presentation of the turn, let's say about the full turn. Next, we proceed to the concept of an angle of rotation and consider its main characteristics, such as the direction and the magnitude of the rotation. Finally, we will give the definition of the shape of the figure around the point. All theory on the text will be supplied with explanatory examples and graphic illustrations.

Navigating page.

What is called turning the point around the point?

Immediately, we note that along with the phrase "turn around the point" we will also use the phrase "turn around the point" and "turn relative to the point", which means the same.

We introduce the concept of rotation of the point around the point.

First we give the definition of the center of rotation.

Definition.

Point relative to which turns the turn is called turning Center.

Now let's say what is obtained as a result of the rotation of the point.

As a result of the rotation of some point A relative to the center of turning O, the point A 1 is obtained (which in the case of a certain amount can coincide with A), and the point A 1 lies on the circle with the center at the OA radius point. In other words, when turning relative to the point O point A processes to the point A 1, lying on the circle with the center at the OA radius point.

It is believed that the point O when turned around himself goes into itself. That is, as a result of turning around the center of rotation, the point O goes into itself.

It is also worth noting that the rotation of the point A around the point o should be considered as moving as a result of the movement of the point A around the circle with the center at the OA radius point.

For clarity, we present an illustration of the rotation of the point and around the point O, in the figures below, move the point A to the point A 1, we show with the help of an arrow.

Full turn

You can perform such a rotation of the point A relative to the center of the turn O, which point A, having passed all the points of the circle, will turn out to be at the same place. At the same time they say that the point A accomplished around the point o.

Let's give a graphic illustration of a full turnover.

If you do not stop on one turn, but to continue the movement of the point around the circumference, then you can perform two, three and so on complete revolutions. In the drawing below the right shows how two complete turns can be produced, and the left is three turns.

The concept of rotation angle

From the point introduced in the first paragraph, the point of rotation is clear that there is an infinite set of points of rotation of the point and around the point O. Indeed, any point of the circumference with the center at the OA radius point can be considered as a point A 1 obtained as a result of the rotation of the point a. Therefore, to distinguish one turn from the other, introduced the concept of rotation angle.

One of the characteristics of the angle of rotation is turn direction. In the direction of rotation, judge how the rotation of the point is carried out - clockwise or counterclockwise.

Another characteristic of the angle of rotation is its value. The angles of rotation are measured in the same units as: the most common degrees and radians. It is worth noting here that the angle of rotation can be expressed in degrees by any real number from the interval of the minus of infinity to the plus of infinity, unlike the corner in geometry, the value of which is positive in degrees and does not exceed 180.

To refer to the angles of rotation, the lowercase letters of the Greek alphabet are commonly used: etc. To refer to a large number of corners of turning, one letter with lower indexes often use, for example, .

Now let's talk about the characteristics of the angle of rotation more and in order.

Turn direction

Let the circle with the center at the point o marked points a and a 1. At point A 1, you can get from point a by turning around the center O either clockwise or - counterclockwise. These turns are logically considered different.

We illustrate the turns in the positive and negative direction. In the drawing below, the turn is shown in the positive direction, and on the right - in the negative.

The magnitude of the angle of rotation, an angle of arbitrary

The angle of rotation of the point other than the center of rotation is fully determined by the indication of its value, on the other hand, the value of the corner of the turn can be judged on how this turn was carried out.

As we have already mentioned above, the magnitude of the angle of rotation in degrees is expressed by the number from -∞ to + ∞. In this case, the plus sign corresponds to turn clockwise, and the minus sign is rotating counterclockwise.

Now it remains to establish a correspondence between the value of the angle of rotation and the fact that it corresponds to.

Let's start with an angle of rotation equal to zero degrees. This corner turns the movement of the point and in itself. In other words, when turning 0 degrees around the point O point A remains in place.

Go to the rotation of the point and around the point O, in which the rotation occurs within half of the turnover. We will assume that the point A goes to point A 1. In this case, the absolute angle of AOA 1 in degrees does not exceed 180. If the rotation occurred in the positive direction, the magnitude of the angle of rotation is considered to be equal to the angle of AOA 1, and if the rotation occurred in the negative direction, its value is considered to be equal to the angle of AOA 1 with a minus sign. For example, we present a drawing showing the angles of rotation of 30, 180 and -150 degrees.

The angles of rotation are large 180 degrees and smaller -180 degrees are determined on the basis of the following sufficiently obvious properties of successive turns: Several serial turns of the point A around the center o are equivalent to one turn, the value of which is equal to the sum of the values \u200b\u200bof these turns.

Let us give an example illustrating this property. We will rotate the point A relative to the point O 45 degrees, and then turn this point by 60 degrees, after which we turn this point on -35 degrees. Denote intermediate points at these turns as a 1, a 2 and a 3. In the same point and 3, we could get, by performing one turn of the point A to the angle of 45 + 60 + (- 35) \u003d 70 degrees.

So, the angles of rotation, large 180 degrees, we will represent as a few consecutive turns on the corners, the sum of the values \u200b\u200bof which gives the value of the initial angle of rotation. For example, an angle of rotation of 279 degrees corresponds to sequential turns of 180 and 99 degrees, or 90, 90, 90 and 9 degrees, or 180, 180 and -81 degrees, or 279 consecutive turns of 1 degree.

The angles of rotation are defined in the same way, smaller -180 degrees. For example, an angle of rotation -520 degrees can be interpreted as consistent turns of the point to -180, -180 and -160 degrees.

Summarize. We determined the angle of rotation, the magnitude of which in degrees is expressed by some valid number from the gap from -∞ to + ∞. In trigonometry, we will work with turning angles, although the word "turn" is often lowered, and they say simply "angle." Thus, in trigonometry we will operate with angle angles, under which we will understand the angles of turn.

In conclusion of this paragraph, we note that the total turnover in the positive direction corresponds to the angle of rotation of 360 degrees (or 2 · π radians), and in the negative - the angle of rotation in -360 degrees (or -2 · π is glad). At the same time, it is convenient for large angles of turning to represent as a certain amount of complete revolutions and another turn by an angle of magnitude from -180 to 180 degrees. For example, take an angle of rotation of 1,340 degrees. It is easy to represent 1 340 as 360 · 4 + (- 100). That is, the initial angle of rotation corresponds 4 full turns in the positive direction and the next turn on -100 degrees. Another example: an angle of rotation -745 degrees can be interpreted as two turns against a clockwise arrow and a subsequent rotation of -25 degrees, since -745 \u003d (- 360) · 2 + (- 25).

Rotate the shape around the point at the angle

The concept of rotation of the point is easily expanding on rotate any shape around the point at the angle (This is about such a turn as the point relative to which the turn is carried out, and the figure that turns, lie in the same plane).

Under the turn of the figure we will understand the rotation of all the points of the figure around the specified point at the given angle.

As an example, we give an illustration of the following action: perform a rotation of the cut AB to the angle relative to the point O, this segment when turning turns into a segment A 1 B 1.

Bibliography.

• Algebra: Studies. For 9 cl. environments Shk. /u. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov; Ed. S. A. Telikovsky. - M.: Education, 1990.- 272 C.: Il.- ISBN 5-09-002727-7
• Bashmakov M. I. Algebra and start analysis: studies. For 10-11 cl. environments shk. - 3rd ed. - M.: Enlightenment, 1993. - 351 C.: Il. - ISBN 5-09-004617-4.
• Algebra and starting analysis: studies. For 10-11 cl. general education. institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn, etc.; Ed. A. N. Kolmogorova.- 14th ed. - M.: Enlightenment, 2004.- 384 C.: Il.- ISBN 5-09-013651-3.
• Gusev V. A., Mordkovich A. G. Mathematics (benefit for applicants in technical schools): studies. benefit. - m.; Higher. Shk., 1984.-351 p., Il.