Positive and negative angles in trigonometry. Basic properties of trigonometric functions: even, odd, periodicity. Signs of the values ​​of trigonometric functions by quarters

Lesson type: systematization of knowledge and intermediate control.

Equipment: trigonometric circle, tests, task cards.

Lesson objectives: systematize the studied theoretical material according to the definitions of sine, cosine, tangent of an angle; check the degree of knowledge acquisition on this topic and application in practice.

Tasks:

• Generalize and consolidate the concepts of sine, cosine and tangent of an angle.
• Form a comprehensive understanding of trigonometric functions.
• To promote students’ desire and need to study trigonometric material; cultivate a culture of communication, the ability to work in groups and the need for self-education.

“Whoever does and thinks for himself from a young age,
Then it becomes more reliable, stronger, smarter.

(V. Shukshin)

DURING THE CLASSES

I. Organizational moment

The class is represented by three groups. Each group has a consultant.
The teacher announces the topic, goals and objectives of the lesson.

II. Updating knowledge (frontal work with the class)

1) Work in groups on tasks:

1. Formulate the definition of sin angle.

– What signs does sin α have in each coordinate quadrant?
– At what values ​​does the expression sin α make sense, and what values ​​can it take?

2. The second group is the same questions for cos α.

3. The third group prepares answers to the same questions tg α and ctg α.

At this time, three students work independently at the board using cards (representatives of different groups).

Card No. 1.

Practical work.
Using the unit circle, calculate the values ​​of sin α, cos α and tan α for angles of 50, 210 and – 210.

Card No. 2.

Determine the sign of the expression: tg 275; cos 370; sin 790; tg 4.1 and sin 2.

Card number 3.

1) Calculate:
2) Compare: cos 60 and cos 2 30 – sin 2 30

2) Orally:

a) A series of numbers is proposed: 1; 1.2; 3; , 0, , – 1. Among them there are redundant ones. What property of sin α or cos α can these numbers express (Can sin α or cos α take these values).
b) Does the expression make sense: cos (–); sin 2; tg 3: ctg (– 5); ; ctg0;
cotg(–π). Why?
c) Is there a smallest and highest value sin or cos, tg, ctg.
d) Is it true?
1) α = 1000 is the angle of the second quarter;
2) α = – 330 is the angle of the IV quarter.
e) The numbers correspond to the same point on the unit circle.

3) Work at the board

No. 567 (2; 4) – Find the value of the expression
No. 583 (1-3) Determine the sign of the expression

Homework: table in notebook. No. 567(1, 3) No. 578

III. Acquiring additional knowledge. Trigonometry in the palm of your hand

Teacher: It turns out that the values ​​of the sines and cosines of angles are “located” in the palm of your hand. Reach out your hand (either hand) and spread your fingers as far apart as possible (as in the poster). One student is invited. We measure the angles between our fingers.
Take a triangle where there is an angle of 30, 45 and 60 90 and apply the vertex of the angle to the hillock of the Moon in the palm of your hand. The Mount of the Moon is located at the intersection of the extensions of the little finger and thumb. We combine one side with the little finger, and the other side with one of the other fingers.
It turns out that there is an angle of 90 between the little finger and the thumb, 30 between the little and ring fingers, 45 between the little and middle fingers, and 60 between the little and index fingers. And this is true for all people without exception.

little finger No. 0 – corresponds to 0,
unnamed No. 1 – corresponds to 30,
average No. 2 – corresponds to 45,
index number 3 – corresponds to 60,
large No. 4 – corresponds to 90.

Thus, we have 4 fingers on our hand and remember the formula:

This is just a mnemonic rule. In general, the value of sin α or cos α must be known by heart, but sometimes this rule will help in difficult times.
Come up with a rule for cos (angles do not change, but are counted from the thumb). A physical pause associated with the signs sin α or cos α.

IV. Checking your knowledge of knowledge and skills

Independent work with feedback

Each student receives a test (4 options) and the answer sheet is the same for everyone.

Test

Option 1

1) At what angle of rotation will the radius take the same position as when turning through an angle of 50?
2) Find the value of the expression: 4cos 60 – 3sin 90.
3) Which number less than zero: sin 140, cos 140, sin 50, tg 50.

Option 2

1) At what angle of rotation will the radius take the same position as when turning by an angle of 10.
2) Find the value of the expression: 4cos 90 – 6sin 30.
3) Which number is greater than zero: sin 340, cos 340, sin 240, tg (– 240).

Option 3

1) Find the value of the expression: 2ctg 45 – 3cos 90.
2) Which number is less than zero: sin 40, cos (– 10), tan 210, sin 140.
3) Which quarter angle is angle α, if sin α > 0, cos α< 0.

Option 4

1) Find the value of the expression: tg 60 – 6ctg 90.
2) Which number is less than zero: sin(– 10), cos 140, tg 250, cos 250.
3) Which quarter angle is angle α, if ctg α< 0, cos α> 0.

(the key word is trigonometry)

V. Information from the history of trigonometry

Teacher: Trigonometry is a fairly important branch of mathematics for human life. Modern look trigonometry was introduced by the greatest mathematician of the 18th century, Leonhard Euler - Swiss by birth long years worked in Russia and was a member of the St. Petersburg Academy of Sciences. He introduced well-known definitions of trigonometric functions, formulated and proved well-known formulas, we will learn them later. Euler’s life is very interesting and I advise you to get acquainted with it through Yakovlev’s book “Leonard Euler”.

(Message from the guys on this topic)

VI. Summing up the lesson

Game "Tic Tac Toe"

The two most active students are participating. They are supported by groups. The solutions to the tasks are written down in a notebook.

Tasks

1) Find the error

a) sin 225 = – 1.1 c) sin 115< О
b) cos 1000 = 2 d) cos (– 115) > 0

2) Express the angle in degrees
3) Express the angle 300 in radians
4) What is the largest and smallest value the expression can have: 1+ sin α;
5) Determine the sign of the expression: sin 260, cos 300.
6) In which quarter of the number circle is the point located?
7) Determine the signs of the expression: cos 0.3π, sin 195, ctg 1, tg 390
8) Calculate:
9) Compare: sin 2 and sin 350

VII. Lesson reflection

Teacher: Where can we meet trigonometry?
In what lessons in 9th grade, and even now, do you use the concepts of sin α, cos α; tg α; ctg α and for what purpose?

Diverse. Some of them are about in which quarters the cosine is positive and negative, in which quarters the sine is positive and negative. Everything turns out to be simple if you know how to calculate the value of these functions in different angles and are familiar with the principle of plotting functions on a graph.

What are the cosine values?

If we consider it, we have the following aspect ratio, which determines it: the cosine of the angle A is the ratio of the adjacent leg BC to the hypotenuse AB (Fig. 1): cos a= BC/AB.

Using the same triangle you can find the sine of an angle, tangent and cotangent. The sine will be the ratio of the opposite side of the angle AC to the hypotenuse AB. The tangent of an angle is found if the sine of the desired angle is divided by the cosine of the same angle; Substituting the corresponding formulas for finding sine and cosine, we obtain that tg a= AC/BC. Cotangent, as a function inverse to tangent, will be found like this: ctg a= BC/AC.

That is, with the same angle values, it was discovered that in a right triangle the aspect ratio is always the same. It would seem that it has become clear where these values ​​come from, but why do we get negative numbers?

To do this, you need to consider the triangle in the Cartesian coordinate system, where there are both positive and negative values.

Clearly about the quarters, where is which

What are Cartesian coordinates? If we talk about two-dimensional space, we have two directed lines that intersect at point O - these are the abscissa axis (Ox) and the ordinate axis (Oy). From point O in the direction of the straight line there are positive numbers, and in the opposite direction - negative numbers. Ultimately, this directly determines in which quarters the cosine is positive and in which, accordingly, negative.

First quarter

If you place right triangle in the first quarter (from 0 o to 90 o), where the x and y axis have positive values(segments AO and BO lie on the axes where the values ​​have a “+” sign), then both sine and cosine will also have positive values, and they are assigned a value with a “plus” sign. But what happens if you move the triangle to the second quarter (from 90 o to 180 o)?

Second quarter

We see that along the y-axis the legs AO received a negative value. Cosine of angle a now has this side in relation to a minus, and therefore its final value becomes negative. It turns out that in which quarter the cosine is positive depends on the placement of the triangle in the Cartesian coordinate system. And in this case, the cosine of the angle receives a negative value. But for the sine nothing has changed, because to determine its sign you need the OB side, which in this case remained with the plus sign. Let's summarize the first two quarters.

To find out in which quarters the cosine is positive and in which it is negative (as well as sine and other trigonometric functions), you need to look at what sign is assigned to which side. For cosine of angle a The side AO is important, for the sine - OB.

The first quarter has so far become the only one that answers the question: “In which quarters are sine and cosine positive at the same time?” Let's see further whether there will be further coincidences in the sign of these two functions.

In the second quarter, the side AO began to have a negative value, which means the cosine also became negative. The sine is kept positive.

Third quarter

Now both sides AO and OB have become negative. Let us recall the relations for cosine and sine:

Cos a = AO/AB;

Sin a = VO/AV.

AB always has a positive sign in a given coordinate system, since it is not directed in either of the two directions defined by the axes. But the legs have become negative, which means the result for both functions is also negative, because if you perform multiplication or division operations with numbers, among which one and only one has a minus sign, then the result will also be with this sign.

The result at this stage:

1) In which quarter is the cosine positive? In the first of three.

2) In which quarter is the sine positive? In the first and second of three.

Fourth quarter (from 270 o to 360 o)

Here the side AO again acquires a plus sign, and therefore the cosine too.

For sine, things are still “negative”, because leg OB remains below the starting point O.

conclusions

In order to understand in which quarters the cosine is positive, negative, etc., you need to remember the relationship for calculating the cosine: the leg adjacent to the angle divided by the hypotenuse. Some teachers suggest remembering this: k(osine) = (k) angle. If you remember this “cheat”, then you automatically understand that sine is the ratio of the opposite leg of the angle to the hypotenuse.

It is quite difficult to remember in which quarters the cosine is positive and in which it is negative. There are many trigonometric functions, and they all have their own meanings. But still, as a result: positive values ​​for the sine are 1.2 quarters (from 0 o to 180 o); for cosine 1.4 quarters (from 0 o to 90 o and from 270 o to 360 o). In the remaining quarters the functions have minus values.

Perhaps it will be easier for someone to remember which sign is which by depicting the function.

For the sine it is clear that from zero to 180 o the ridge is above the line of sin(x) values, which means the function here is positive. For cosine it’s the same: in which quarter the cosine is positive (photo 7), and in which it is negative, you can see by moving the line above and below the cos(x) axis. As a result, we can remember two ways to determine the sign of the sine and cosine functions:

1. Based on an imaginary circle with a radius equal to one (although, in fact, it does not matter what the radius of the circle is, this is the example most often given in textbooks; this makes it easier to understand, but at the same time, unless it is stipulated that this It doesn’t matter, children can get confused).

2. By depicting the dependence of the function along (x) on the argument x itself, as in the last figure.

Using the first method, you can UNDERSTAND what exactly the sign depends on, and we explained this in detail above. Figure 7, constructed from these data, visualizes the resulting function and its sign in the best possible way.

If you are already familiar with trigonometric circle , and you just want to refresh your memory of certain elements, or you are completely impatient, then here it is:

Here we will analyze everything in detail step by step.

The trigonometric circle is not a luxury, but a necessity

Trigonometry Many people associate it with an impenetrable thicket. Suddenly, so many values ​​of trigonometric functions, so many formulas pile up... But it’s like, it didn’t work out at the beginning, and... off we go... complete misunderstanding...

It is very important not to give up values ​​of trigonometric functions, - they say, you can always look at the spur with a table of values.

If you are constantly looking at a table with the values ​​of trigonometric formulas, let's get rid of this habit!

He will help us out! You will work with it several times, and then it will pop up in your head. How is it better than a table? Yes, in the table you will find a limited number of values, but on the circle - EVERYTHING!

For example, say while looking at standard table of values ​​of trigonometric formulas , what is the sine equal to, say, 300 degrees, or -45.

No way?.. you can, of course, connect reduction formulas... And looking at the trigonometric circle, you can easily answer such questions. And you will soon know how!

And when deciding trigonometric equations and inequalities without trigonometric circle- nowhere at all.

Introduction to the trigonometric circle

Let's go in order.

First, let's write out this series of numbers:

And now this:

And finally this one:

Of course, it is clear that, in fact, in first place is , in second place is , and in last place is . That is, we will be more interested in the chain.

But how beautiful it turned out! If something happens, we will restore this “miracle ladder.”

And why do we need it?

This chain is the main values ​​of sine and cosine in the first quarter.

Let us draw a circle of unit radius in a rectangular coordinate system (that is, we take any radius in length, and declare its length to be unit).

From the “0-Start” beam we lay the corners in the direction of the arrow (see figure).

We get the corresponding points on the circle. So, if we project the points onto each of the axes, then we will get exactly the values ​​​​from the above chain.

Why is this, you ask?

Let's not analyze everything. Let's consider principle, which will allow you to cope with other, similar situations.

Triangle AOB is rectangular and contains . And we know that opposite the angle b lies a leg half the size of the hypotenuse (we have the hypotenuse = the radius of the circle, that is, 1).

This means AB= (and therefore OM=). And according to the Pythagorean theorem

I hope something is already becoming clear?

So point B will correspond to the value, and point M will correspond to the value

Same with the other values ​​of the first quarter.

As you understand, the familiar axis (ox) will be cosine axis, and the axis (oy) – axis of sines . Later.

To the left of zero along the cosine axis (below zero along the sine axis) there will, of course, be negative values.

So, here it is, the ALMIGHTY, without whom there is nowhere in trigonometry.

But we’ll talk about how to use the trigonometric circle in.

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

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

The cosine of angle α is the abscissa (x coordinate) of a point on a trigonometric circle, which occurs when the radius is rotated by angle α.

The tangent of the angle α is the ratio of sine to cosine. Or, which is the same thing, the ratio of the y coordinate to the x coordinate.

Notation: sin α = y ; cos α = x ; tg α = y : x .

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

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

1. sin α > 0 if angle α lies in the I or II coordinate quadrant. This is because, by definition, 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 angle α lies in the 1st or 4th coordinate quadrant. Because only there the x coordinate (aka abscissa) will be greater than zero;
3. tan α > 0 if angle α lies in the I or III coordinate quadrant. This follows from the definition: after all, tan α = y : x, therefore it is positive only where the signs of x and y coincide. This happens in the first coordinate quarter (here x > 0, y > 0) and the third coordinate quarter (x< 0, y < 0).

For clarity, let us note the signs of each trigonometric function - sine, cosine and tangent - on separate “radars”. We get the following picture:

Please note: in my discussions I never spoke about the fourth trigonometric function - cotangent. The fact is that the cotangent signs coincide with the tangent signs - there are no special rules there.

Now I propose to consider examples similar to problems B11 from trial Unified State Exam in mathematics, which took place on September 27, 2011. After all, The best way understanding theory is practice. It is advisable to have 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 calculated):

1. sin(3π/4);
2. cos(7π/6);
3. tg(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 this: first we convert all angles from radian measures to degrees (π → 180°), and then look at which coordinate quarter the resulting number lies in. 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 the angle from the third 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 the IV quarter, where the tangent takes negative values. Therefore tan (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 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 the factors have different signs. Since “minus by plus gives 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 sine: since 150° ∈ , we're talking about 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 have obtained 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. tg(3π/4)< 0. Аналогично, угол 300° ∈ — это IV четверть, т.е. cos (5π/3) >0. Since “minus by 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. the simplest angle. Therefore tan (π/6) > 0. Again we have two positive expressions - their product will also be positive. Therefore cot (4π/3) tg (π/6) > 0.

In conclusion, let's look at some more complex tasks. In addition to figuring out the sign of the trigonometric function, you will have to do a little math here - exactly as it is done in real problems B11. In principle, these are almost real problems that actually appear in the Unified State Examination 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. All that remains is to decide: plus or minus? By condition, 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. extract Square root: cos 2 α = 0.04 ⇒ cos α = ±0.2. By condition, angle α ∈ [π; 3π/2], i.e. We are talking about the third coordinate quarter. All cosines there 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. We look at the angle again: α ∈ is the IV coordinate quarter, in which, as we know, the sine will be negative. Thus, we conclude: sin α = −0.5.

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

Everything is the same, only for the tangent. Extract the square root: tan 2 α = 9 ⇒ tan α = ±3. But according to the condition, the angle α ∈ is the I coordinate quarter. All trigonometric functions, incl. tangent, there are positive, so tan α = 3. That's it!

This article will look at three basic properties of trigonometric functions: sine, cosine, tangent and cotangent.

The first property is the sign of the function depending on which quarter of the unit circle the angle α belongs to. The second property is periodicity. According to this property, the tigonometric function does not change its value when the angle changes by an integer number of revolutions. The third property determines how the values ​​of the functions sin, cos, tg, ctg change at opposite angles α and - α.

Yandex.RTB R-A-339285-1

Often in a mathematical text or in the context of a problem you can find the phrase: “the angle of the first, second, third or fourth coordinate quarter.” What it is?

Let's turn to the unit circle. It is divided into four quarters. Let's mark the starting point A 0 (1, 0) on the circle and, rotating it around the point O by an angle α, we will get to the point A 1 (x, y). Depending on which quarter the point A 1 (x, y) lies in, the angle α will be called the angle of the first, second, third and fourth quarter, respectively.

For clarity, here is an illustration.

The angle α = 30° lies in the first quarter. Angle - 210° is the second quarter angle. The 585° angle is the third quarter angle. The angle - 45° is the fourth quarter angle.

In this case, the angles ± 90 °, ± 180 °, ± 270 °, ± 360 ° do not belong to any quarter, since they lie on the coordinate axes.

Now consider the signs that sine, cosine, tangent and cotangent take, depending on which quadrant the angle lies in.

To determine the signs of the sine by quarters, recall the definition. Sine is the ordinate of point A 1 (x, y). The figure shows that in the first and second quarters it is positive, and in the third and quadruple it is negative.

Cosine is the abscissa of point A 1 (x, y). In accordance with this, we determine the signs of the cosine on the circle. The cosine is positive in the first and fourth quarters, and negative in the second and third quarters.

To determine the signs of tangent and cotangent by quarters, we also recall the definitions of these trigonometric functions. Tangent is the ratio of the ordinate of a point to the abscissa. This means, according to the rule for dividing numbers with different signs, when the ordinate and abscissa have the same signs, the tangent sign on the circle will be positive, and when the ordinate and abscissa have different signs- negative. The cotangent signs for quarters are determined in a similar way.

Important to remember!

1. The sine of angle α has a plus sign in the 1st and 2nd quarters, a minus sign in the 3rd and 4th quarters.
2. The cosine of angle α has a plus sign in the 1st and 4th quarters, a minus sign in the 2nd and 3rd quarters.
3. The tangent of the angle α has a plus sign in the 1st and 3rd quarters, a minus sign in the 2nd and 4th quarters.
4. The cotangent of angle α has a plus sign in the 1st and 3rd quarters, a minus sign in the 2nd and 4th quarters.

Periodicity property

The property of periodicity is one of the most obvious properties of trigonometric functions.

Periodicity property

When the angle changes by an integer number of full revolutions, the values ​​of the sine, cosine, tangent and cotangent of the given angle remain unchanged.

Indeed, when the angle changes by an integer number of revolutions, we will always get from the initial point A on the unit circle to point A 1 with the same coordinates. Accordingly, the values ​​of sine, cosine, tangent and cotangent will not change.

Mathematically, this property is written as follows:

sin α + 2 π z = sin α cos α + 2 π z = cos α t g α + 2 π z = t g α c t g α + 2 π z = c t g α

How is this property used in practice? The periodicity property, like reduction formulas, is often used to calculate the values ​​of sines, cosines, tangents and cotangents of large angles.

Let's give examples.

sin 13 π 5 = sin 3 π 5 + 2 π = sin 3 π 5

t g (- 689 °) = t g (31 ° + 360 ° (- 2)) = t g 31 ° t g (- 689 °) = t g (- 329 ° + 360 ° (- 1)) = t g (- 329 °)

Let's look again at the unit circle.

Point A 1 (x, y) is the result of rotating the initial point A 0 (1, 0) around the center of the circle by angle α. Point A 2 (x, - y) is the result of rotating the starting point by an angle - α.

Points A 1 and A 2 are symmetrical about the abscissa axis. In the case where α = 0 °, ± 180 °, ± 360 ° points A 1 and A 2 coincide. Let one point have coordinates (x, y) and the second - (x, - y). Let us recall the definitions of sine, cosine, tangent, cotangent and write:

sin α = y , cos α = x , t g α = y x , c t g α = x y sin - α = - y , cos - α = x , t g - α = - y x , c t g - α = x - y

This implies the property of sines, cosines, tangents and cotangents of opposite angles.

Property of sines, cosines, tangents and cotangents of opposite angles

sin - α = - sin α cos - α = cos α t g - α = - t g α c t g - α = - c t g α

According to this property, the equalities are true

sin - 48 ° = - sin 48 ° , c t g π 9 = - c t g - π 9 , cos 18 ° = cos - 18 °

The considered property is often used when solving practical problems in cases where you need to get rid of negative angle signs in the arguments of trigonometric functions.

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