Cosine in the second quarter. Signs of trigonometric functions

This article will cover three main properties 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.

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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 is it?

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 °

This property is often used in solving practical problems in cases where it is necessary to get rid of negative angle signs in the arguments of trigonometric functions.

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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, 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.

Finally, let's look at some more complex problems. 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!

Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were derived by astronomers to create an accurate calendar and orientation by the stars. These calculations related to spherical trigonometry, while in the school course they study the ratio of sides and angles of a plane triangle.

Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationships between the sides and angles of triangles.

During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazwi introduced functions such as tangent and cotangent, and compiled the first tables of values ​​for sines, tangents and cotangents. The concepts of sine and cosine were introduced by Indian scientists. Trigonometry received a lot of attention in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

The basic trigonometric functions of a numeric argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.

The formulas for calculating the values ​​of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants are equal in all directions,” since the proof is given using the example of an isosceles right triangle.

Sine, cosine and other dependencies establish the relationship between the acute angles and sides of any right triangle. Let us present formulas for calculating these quantities for angle A and trace the relationships between trigonometric functions:

As you can see, tg and ctg are inverse functions. If we imagine leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, we obtain the following formulas for tangent and cotangent:

Trigonometric circle

Graphically, the relationship between the mentioned quantities can be represented as follows:

The circle, in this case, represents everything possible values angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will have a “+” sign if α belongs to the 1st and 2nd quarters of the circle, that is, it is in the range from 0° to 180°. For 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 the quantities.

Values ​​of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values ​​of trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen at random. The designation π in the tables is for radians. Rad is the angle at which the length of a circle's arc corresponds to its radius. This value was introduced in order to establish a universal dependence; when calculating in radians, the actual length of the radius in cm does not matter.

Angles 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: sine and cosine

In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, 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 comparison table properties for sine and cosine:

Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with the signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs match, the function is even; otherwise, it is odd.

The introduction of radians and the listing of the basic properties of sine and cosine waves allow us to present the following pattern:

It is very easy to verify that the formula is correct. For example, for x = π/2, the sine is 1, as is the cosine of x = 0. The check can be done by consulting tables or by tracing function curves for given values.

Properties of tangentsoids and cotangentsoids

The graphs of the tangent and cotangent functions differ significantly from the sine and cosine functions. The values ​​tg and ctg are reciprocals of each other.

1. Y = tan x.
2. The tangent tends to the values ​​of y at x = π/2 + πk, but never reaches them.
3. The smallest positive period of the tangentoid is π.
4. Tg (- x) = - tg x, i.e. the function is odd.
5. Tg x = 0, for x = πk.
6. The function is increasing.
7. Tg x › 0, for x ϵ (πk, π/2 + πk).
8. Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
9. Derivative (tg x)’ = 1/cos 2 ⁡x.

Consider the graphic image of the cotangentoid below in the text.

Main properties of cotangentoids:

1. Y = cot x.
2. Unlike the sine and cosine functions, in the tangentoid Y can take on the values ​​of the set of all real numbers.
3. The cotangentoid tends to the values ​​of y at x = πk, but never reaches them.
4. The smallest positive period of a cotangentoid is π.
5. Ctg (- x) = - ctg x, i.e. the function is odd.
6. Ctg x = 0, for x = π/2 + πk.
7. The function is decreasing.
8. Ctg x › 0, for x ϵ (πk, π/2 + πk).
9. Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
10. Derivative (ctg x)’ = - 1/sin 2 ⁡x Correct

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 a 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 reverse side- negative. Ultimately, this directly determines in which quarters the cosine is positive and in which, accordingly, negative.

First quarter

If you place a 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 the 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.