sinus numbers a called the ordinate of the point depicting this number on the number circle. The sine of the angle in a radian is called the sine of a number a.

Sinus- number function x. Her domain

Sine Range- segment from -1 before 1 , since any number of this segment on the y-axis is a projection of some point on the circle, but no point outside this segment is a projection of any of these points.

Sine period

Sine sign:

1. the sine is zero at , where n- any integer;

2. the sine is positive at , where n- any integer;

3. sine is negative at

Where n- any integer.

Sinus- function odd x and -x, then their ordinates - sines - will also be opposite. I.e for anyone x.

1. The sine increases on segments , where n- any integer.

2. The sine decreases on the segment , where n- any integer.

At ;

at .

Cosine

cosine numbers a is called the abscissa of the point depicting this number on the number circle. The cosine of the angle in a radian is called the cosine of a number a.

Cosine is a number function. Her domain- the set of all numbers, since for any number you can find the ordinate of the point representing it.

Range of cosine- segment from -1 before 1 , since any number of this segment on the x-axis is a projection of some point on the circle, but no point outside this segment is a projection of any of these points.

cosine period is equal to . After all, every time the position of the point representing the number is exactly repeated.

Cosine sign:

1. cosine is zero at , where n- any integer;

2. cosine is positive at , where n- any integer;

3. cosine is negative at , where n- any integer.

Cosine- function even. First, the domain of this function is the set of all numbers, and therefore is symmetrical with respect to the origin. And secondly, if we postpone two opposite numbers from the beginning: x and -x, then their abscissas - cosines - will be equal. I.e

for anyone x.

1. Cosine increases on segments , where n- any integer.

2. Cosine decreases on segments , where n- any integer.

at ;

at .

Tangent

tangent number is the ratio of the sine of this number to the cosine of this number:.

tangent angle in a radian is called the tangent of a number a.

Tangent is a number function. Her domain- the set of all numbers whose cosine is not equal to zero, since there are no other restrictions on the definition of the tangent. And since the cosine is zero at , then , where .

Tangent Range

Tangent period x(not equal), differing from each other by , and draw a straight line through them, then this straight line will pass through the origin and intersect the line of tangents at some point t. So it turns out that, that is, the number is the period of the tangent.

Tangent sign: tangent is the ratio of sine to cosine. So he

1. is zero when the sine is zero, that is, when , where n- any integer.

2. is positive when the sine and cosine have the same sign. This happens only in the first and third quarters, that is, when , where a- any integer.

3. is negative when the sine and cosine have different signs. This happens only in the second and fourth quarters, that is, when , where a- any integer.

Tangent- function odd. First, the domain of definition of this function is symmetrical with respect to the origin. And secondly, . Due to the oddness of the sine and the evenness of the cosine, the numerator of the resulting fraction is equal to, and its denominator is equal to, which means that this fraction itself is equal to.

So it turned out that .

Means, the tangent increases in each section of its domain of definition, that is, on all intervals of the form , where a- any integer.

Cotangent

Cotangent number is the ratio of the cosine of this number to the sine of this number: . Cotangent angle in a radian is called the cotangent of a number a. Cotangent is a number function. Her domain- the set of all numbers whose sine is not equal to zero, since there are no other restrictions on the definition of the cotangent. And since the sine is zero at , then , where

Cotangent Range is the set of all real numbers.

Cotangent period is equal to . After all, if we take any two possible values x(not equal ), differing from each other by , and draw a straight line through them, then this straight line will pass through the origin and intersect the line of cotangents at some point t. So it turns out that, that is, that the number is the period of the cotangent.

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Counting angles on a trigonometric circle.

Attention!
There are additional
material in Special Section 555.
For those who strongly "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 chin-china. Added numbers of quarters (in the corners of a large square) - from the first to the fourth. And then suddenly who does not 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 frills.

And added a green arrow. With a plus. What does she mean? Let me remind you that the fixed side of the corner always nailed to the positive axis OH. So, if we twist the moving side of the corner plus arrow, i.e. in ascending quarter numbers, the angle will be considered positive. For example, the picture shows a positive angle of +60°.

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

Here, usually, the first misunderstandings begin. How so!? And if the negative angle on the circle coincides with the positive!? And in general, it turns out that the same position of the movable side (or a point on the numerical 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 the negative angle is -250°.

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

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

And now a question for you. How do I know that the position of the 110° angle is the same as the position of the -250° angle?
I will hint that this is due to the full turnover. In 360°... Not clear? Then we draw a circle. We draw on paper. Marking the corner about 110°. And believe how much remains until a full turn. Just 250° remains...

Got it? And now - attention! If the angles 110° and -250° occupy the circle same position, then what? Yes, the fact that 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 it is necessary to simplify expressions, and as a basis for the subsequent development of reduction formulas and other intricacies of trigonometry.

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

I think you understand what negative angles are. It's quite simple. Counter-clockwise is a positive count. Along the way, it's negative. Consider angle positive or negative depends on us. From our desire. Well, and more from the task, of course ... I hope you understand how to move in trigonometric functions from negative to positive angles and vice versa. Draw a circle, an approximate angle, and see how much is missing before a full turn, i.e. up to 360°.

Angles greater than 360°.

Let's deal with angles that are greater than 360 °. And such things happen? There are, of course. How to draw them on a circle? Not a problem! Suppose we need to understand in which quarter an angle of 1000 ° will fall? Easily! We make one full turn counterclockwise (the angle was given to us positive!). Rewind 360°. Well, let's move on! Another turn - it has already turned out 720 °. How much is left? 280°. It is not enough for a full turn ... But the angle is more than 270 ° - and this is the border between the third and fourth quarter. So our angle of 1000° falls into the fourth quarter. Everything.

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 turns, are, strictly speaking, various 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 necessary? Why do we need to translate angles from one to another? Yes, all for the same.) In order to simplify expressions. Simplification of expressions, in fact, is the main task of school mathematics. Well, along the way, the head is training.)

Well, shall we practice?)

We answer questions. Simple at first.

1. In which quarter does the angle -325° fall?

2. In which quarter does the angle 3000° fall?

3. In which quarter does the angle -3000° fall?

There is a problem? Or insecurity? We go to Section 555, Practical work with a trigonometric circle. There, in the first lesson of this very "Practical work ..." everything is detailed ... In such questions of uncertainty shouldn't!

4. What is the sign of sin555°?

5. What is the sign of tg555°?

Determined? Fine! Doubt? It is necessary to Section 555 ... By the way, there you will learn how to draw tangent and cotangent on a trigonometric circle. A very useful thing.

And now the smarter questions.

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

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

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

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

What, questions 6-9 puzzled? Get used to it, there are not such formulations on the exam ... So be it, I will translate it. Only for you!

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

I will give the answers in order (in violation of our rules). But what to do, there are only two signs, and only four quarters ... You will not scatter in options.

6. sin57°.

7.cos(-57°).

8.cos57°.

9.-sin(-57°)

I suppose that the answers to questions 6-9 confused some people. Especially -sin(-57°), right?) Indeed, in the elementary rules for counting angles there is room for errors ... That is why I had to make a lesson: "How to determine the signs of functions and give angles on a trigonometric circle?" In Section 555. There tasks 4 - 9 are sorted out. 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". Learn how to easily and correctly convert degrees to radians and vice versa. And we will be surprised to find that this elementary information on the site enough already to solve some non-standard trigonometry puzzles!

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

The sign of the trigonometric function depends solely on the coordinate quarter in which the numeric argument is located. Last time we learned how to translate arguments from a radian measure into a degree measure (see the lesson “ Radian and degree measure of an angle”), and then determine this same coordinate quarter. Now let's deal, in fact, with the definition of the sign of the sine, cosine and tangent.

The sine of angle α is the ordinate (coordinate y) of a point on a trigonometric circle, which occurs when the radius is rotated through angle α.

The cosine of the angle α is the abscissa (x coordinate) of a point on a trigonometric circle that occurs when the radius rotates through the angle α.

The tangent of the angle α is the ratio of the sine to the cosine. Or, equivalently, the ratio of the y-coordinate to the x-coordinate.

Notation: sin α = y ; cosα = x; tgα = y : x .

All these definitions are familiar to you from the high school algebra course. However, we are not interested in the definitions themselves, but in the consequences that arise on the trigonometric circle. Take a look:

Blue indicates the positive direction of the OY axis (y-axis), red indicates the positive direction of the OX axis (abscissa). On this "radar" the signs of trigonometric functions become obvious. In particular:

1. sin α > 0 if the angle α lies in the I or II coordinate quarter. This is because, by definition, a sine is an ordinate (y coordinate). And the y coordinate will be positive precisely in the I and II coordinate quarters;
2. cos α > 0 if the angle α lies in the I or IV coordinate quarter. Because only there the x coordinate (it is also the abscissa) will be greater than zero;
3. tg α > 0 if the angle α lies in the I or III coordinate quadrant. This follows from the definition: after all, tg α = y : x , so it is positive only where the signs of x and y coincide. This happens in the 1st coordinate quarter (here x > 0, y > 0) and the 3rd coordinate quarter (x< 0, y < 0).

For clarity, we note the signs of each trigonometric function - sine, cosine and tangent - on separate "radar". We get the following picture:

Note: in my reasoning, I never spoke about the fourth trigonometric function - the cotangent. The fact is that the signs of the cotangent coincide with the signs of the tangent - there are no special rules there.

Now I propose to consider examples similar to tasks B11 from the trial exam in mathematics, which took place on September 27, 2011. After all, the best way to understand the theory is practice. Preferably a lot of practice. Of course, the conditions of the tasks were slightly changed.

Task. Determine the signs of trigonometric functions and expressions (the values ​​of the functions themselves do not need to be considered):

1. sin(3π/4);
2. cos(7π/6);
3. tan (5π/3);
4. sin(3π/4) cos(5π/6);
5. cos (2π/3) tg (π/4);
6. sin(5π/6) cos(7π/4);
7. tan (3π/4) cos (5π/3);
8. ctg (4π/3) tg (π/6).

The action plan is as follows: first, we convert all angles from radian measure to degree measure (π → 180°), and then look in which coordinate quarter the resulting number lies. Knowing the quarters, we can easily find the signs - according to the rules just described. We have:

1. sin (3π/4) = sin (3 180°/4) = sin 135°. Since 135° ∈ , this is an angle from the II coordinate quadrant. But the sine in the second quarter is positive, so sin (3π/4) > 0;
2. cos (7π/6) = cos (7 180°/6) = cos 210°. Because 210° ∈ , this is an angle from the III coordinate quadrant in which all cosines are negative. Therefore, cos (7π/6)< 0;
3. tg (5π/3) = tg (5 180°/3) = tg 300°. Since 300° ∈ , we are in quadrant IV, where the tangent takes negative values. Therefore tg (5π/3)< 0;
4. sin (3π/4) cos (5π/6) = sin (3 180°/4) cos (5 180°/6) = sin 135° cos 150°. Let's deal with the sine: because 135° ∈ , this is the second quarter, in which the sines are positive, i.e. sin (3π/4) > 0. Now we work with the cosine: 150° ∈ - again the second quarter, the cosines there are negative. Therefore cos (5π/6)< 0. Наконец, следуя правилу «плюс на минус дает знак минус», получаем: sin (3π/4) · cos (5π/6) < 0;
5. cos (2π/3) tg (π/4) = cos (2 180°/3) tg (180°/4) = cos 120° tg 45°. We look at the cosine: 120° ∈ is the II coordinate quarter, so cos (2π/3)< 0. Смотрим на тангенс: 45° ∈ — это I четверть (самый обычный угол в тригонометрии). Тангенс там положителен, поэтому tg (π/4) >0. Again we got a product in which factors of different signs. Since “a minus times a plus gives a minus”, we have: cos (2π/3) tg (π/4)< 0;
6. sin (5π/6) cos (7π/4) = sin (5 180°/6) cos (7 180°/4) = sin 150° cos 315°. We work with the sine: since 150° ∈ , we are talking about the II coordinate quarter, where the sines are positive. Therefore, sin (5π/6) > 0. Similarly, 315° ∈ is the IV coordinate quarter, the cosines there are positive. Therefore, cos (7π/4) > 0. We got the product of two positive numbers - such an expression is always positive. We conclude: sin (5π/6) cos (7π/4) > 0;
7. tg (3π/4) cos (5π/3) = tg (3 180°/4) cos (5 180°/3) = tg 135° cos 300°. But the angle 135° ∈ is the second quarter, i.e. tan (3π/4)< 0. Аналогично, угол 300° ∈ — это IV четверть, т.е. cos (5π/3) >0. Since “a minus plus gives a minus sign”, we have: tg (3π/4) cos (5π/3)< 0;
8. ctg (4π/3) tg (π/6) = ctg (4 180°/3) tg (180°/6) = ctg 240° tg 30°. We look at the cotangent argument: 240° ∈ is the III coordinate quarter, therefore ctg (4π/3) > 0. Similarly, for the tangent we have: 30° ∈ is the I coordinate quarter, i.e. easiest corner. Therefore, tg (π/6) > 0. Again, we got two positive expressions - their product will also be positive. Therefore ctg (4π/3) tg (π/6) > 0.

Finally, let's look at a few more complex problems. In addition to finding out the sign of the trigonometric function, here you have to do a little calculation - just like it is done in real problems B11. In principle, these are almost real tasks that are really found in the exam in mathematics.

Task. Find sin α if sin 2 α = 0.64 and α ∈ [π/2; π].

Since sin 2 α = 0.64, we have: sin α = ±0.8. It remains to decide: plus or minus? By assumption, the angle α ∈ [π/2; π] is the II coordinate quarter, where all sines are positive. Therefore, sin α = 0.8 - the uncertainty with signs is eliminated.

Task. Find cos α if cos 2 α = 0.04 and α ∈ [π; 3π/2].

We act similarly, i.e. we take the square root: cos 2 α = 0.04 ⇒ cos α = ±0.2. By assumption, the angle α ∈ [π; 3π/2], i.e. we are talking about the III coordinate quarter. There, all cosines are negative, so cos α = −0.2.

Task. Find sin α if sin 2 α = 0.25 and α ∈ .

We have: sin 2 α = 0.25 ⇒ sin α = ±0.5. Again we look at the angle: α ∈ is the IV coordinate quarter, in which, as you know, the sine will be negative. Thus, we conclude: sin α = −0.5.

Task. Find tg α if tg 2 α = 9 and α ∈ .

Everything is the same, only for the tangent. We take the square root: tg 2 α = 9 ⇒ tg α = ±3. But by the condition, the angle α ∈ is the I coordinate quadrant. All trigonometric functions, incl. tangent, there are positive, so tg α = 3. That's it!