Sine, cosine, tangent, cotangent of an acute angle. Trigonometric functions. Rules for finding trigonometric functions: sine, cosine, tangent and cotangent

We will begin our study of trigonometry with the right triangle. Let's define what sine and cosine are, as well as tangent and cotangent of an acute angle. This is the basics of trigonometry.

Let us remind you that right angle is an angle equal to 90 degrees. In other words, half a turned angle.

Sharp corner- less than 90 degrees.

Obtuse angle- greater than 90 degrees. In relation to such an angle, “obtuse” is not an insult, but a mathematical term :-)

Let's draw a right triangle. A right angle is usually denoted by . Please note that the side opposite the corner is indicated by the same letter, only small. Thus, the side opposite angle A is designated .

The angle is denoted by the corresponding Greek letter.

Hypotenuse of a right triangle is the side opposite the right angle.

Legs- sides lying opposite acute angles.

The leg lying opposite the angle is called opposite(relative to angle). The other leg, which lies on one of the sides of the angle, is called adjacent.

Sinus The acute angle in a right triangle is the ratio of the opposite side to the hypotenuse:

Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:

Tangent acute angle in a right triangle - the ratio of the opposite side to the adjacent:

Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of the angle to its cosine:

Cotangent acute angle in a right triangle - the ratio of the adjacent side to the opposite (or, which is the same, the ratio of cosine to sine):

Note the basic relationships for sine, cosine, tangent, and cotangent below. They will be useful to us when solving problems.

Let's prove some of them.

Okay, we have given definitions and written down formulas. But why do we still need sine, cosine, tangent and cotangent?

We know that the sum of the angles of any triangle is equal to.

We know the relationship between parties right triangle. This is the Pythagorean theorem: .

It turns out that knowing two angles in a triangle, you can find the third. Knowing the two sides of a right triangle, you can find the third. This means that the angles have their own ratio, and the sides have their own. But what should you do if in a right triangle you know one angle (except the right angle) and one side, but you need to find the other sides?

This is what people in the past encountered when making maps of the area and the starry sky. After all, it is not always possible to directly measure all sides of a triangle.

Sine, cosine and tangent - they are also called trigonometric angle functions- give relationships between parties And corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.

We will also draw a table of the values ​​of sine, cosine, tangent and cotangent for “good” angles from to.

Please note the two red dashes in the table. At appropriate angle values, tangent and cotangent do not exist.

Let's look at several trigonometry problems from the FIPI Task Bank.

1. In a triangle, the angle is , . Find .

The problem is solved in four seconds.

Because the , .

2. In a triangle, the angle is , , . Find .

Let's find it using the Pythagorean theorem.

The problem is solved.

Often in problems there are triangles with angles and or with angles and. Remember the basic ratios for them by heart!

For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.

A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.

We looked at problems solving right triangles - that is, finding unknown sides or angles. But that's not all! There are many problems in the Unified State Examination in mathematics that involve sine, cosine, tangent or cotangent of an external angle of a triangle. More on this in the next article.

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 relationships 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 all possible values ​​of the 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 meaning 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 the comparative table of 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 coincide, 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

I think you deserve more than this. Here is my key to trigonometry:

• Draw the dome, wall and ceiling
• Trigonometric functions are nothing but percentages of these three forms.

Metaphor for sine and cosine: dome

Instead of just looking at the triangles themselves, imagine them in action by finding a specific real-life example.

Imagine you are in the middle of a dome and want to hang a movie projector screen. You point your finger at the dome at a certain angle “x”, and the screen should be suspended from this point.

The angle you point to determines:

• sine(x) = sin(x) = screen height (from floor to dome mounting point)
• cosine(x) = cos(x) = distance from you to the screen (by floor)
• hypotenuse, the distance from you to the top of the screen, always the same, equal to the radius of the dome

Do you want the screen to be as large as possible? Hang it directly above you.

Do you want the screen to hang as far away from you as possible? Hang it straight perpendicular. The screen will have zero height in this position and will hang furthest away, as you asked.

Height and distance from the screen are inversely proportional: the closer the screen hangs, the greater its height.

Sine and cosine are percentages

No one during my years of study, alas, explained to me that the trigonometric functions sine and cosine are nothing more than percentages. Their values ​​range from +100% to 0 to -100%, or from a positive maximum to zero to a negative maximum.

Let's say I paid a tax of 14 rubles. You don't know how much it is. But if you say that I paid 95% in tax, you will understand that I was simply fleeced.

Absolute height doesn't mean anything. But if the sine value is 0.95, then I understand that the TV is hanging almost on the top of your dome. Very soon it will reach its maximum height in the center of the dome, and then begin to decline again.

How can we calculate this percentage? It's very simple: divide the current screen height by the maximum possible (the radius of the dome, also called the hypotenuse).

That's why we are told that “cosine = opposite side / hypotenuse.” It's all about getting interest! It is best to define sine as “the percentage of the current height from the maximum possible.” (The sine becomes negative if your angle points “underground.” The cosine becomes negative if the angle points toward the dome point behind you.)

Let's simplify the calculations by assuming we are at the center of the unit circle (radius = 1). We can skip the division and just take the sine equal to the height.

Each circle is essentially a single circle, scaled up or down to the desired size. So determine the unit circle connections and apply the results to your specific circle size.

Experiment: take any corner and see what percentage of height to width it displays:

The graph of the growth of the sine value is not just a straight line. The first 45 degrees cover 70% of the height, but the last 10 degrees (from 80° to 90°) cover only 2%.

This will make it clearer to you: if you walk in a circle, at 0° you rise almost vertically, but as you approach the top of the dome, the height changes less and less.

Tangent and secant. Wall

One day a neighbor built a wall right next to each other to your dome. Cried your view from the window and a good price for resale!

But is it possible to somehow win in this situation?

Of course yes. What if we hung a movie screen right on our neighbor's wall? You target the angle (x) and get:

• tan(x) = tan(x) = screen height on the wall
• distance from you to the wall: 1 (this is the radius of your dome, the wall is not moving anywhere from you, right?)
• secant(x) = sec(x) = “length of the ladder” from you standing in the center of the dome to the top of the suspended screen

Let's clarify a couple of points regarding the tangent, or screen height.

• it starts at 0, and can go infinitely high. You can stretch the screen higher and higher on the wall to create an endless canvas for watching your favorite movie! (For such a huge one, of course, you will have to spend a lot of money).
• tangent is just a larger version of sine! And while the increase in sine slows down as you move towards the top of the dome, the tangent continues to grow!

Sekansu also has something to brag about:

• The secant starts at 1 (the ladder is on the floor, from you to the wall) and starts to rise from there
• The secant is always longer than the tangent. The slanted ladder you use to hang your screen should be longer than the screen itself, right? (With unrealistic sizes, when the screen is sooooo long and the ladder needs to be placed almost vertically, their sizes are almost the same. But even then the secant will be a little longer).

Remember, the values ​​are percent. If you decide to hang the screen at an angle of 50 degrees, tan(50)=1.19. Your screen is 19% larger than the distance to the wall (dome radius).

(Enter x=0 and check your intuition - tan(0) = 0 and sec(0) = 1.)

Cotangent and cosecant. Ceiling

Incredibly, your neighbor has now decided to build a roof over your dome. (What's wrong with him? Apparently he doesn't want you to spy on him while he's walking around the yard naked...)

Well, it's time to build an exit to the roof and talk to your neighbor. You choose the angle of inclination and begin construction:

• the vertical distance between the roof outlet and the floor is always 1 (the radius of the dome)
• cotangent(x) = cot(x) = distance between the top of the dome and the exit point
• cosecant(x) = csc(x) = length of your path to the roof

Tangent and secant describe the wall, and COtangent and COsecant describe the ceiling.

Our intuitive conclusions this time are similar to the previous ones:

• If you take the angle equal to 0°, your exit to the roof will last forever, since it will never reach the ceiling. Problem.
• The shortest “ladder” to the roof will be obtained if you build it at an angle of 90 degrees to the floor. The cotangent will be equal to 0 (we do not move along the roof at all, we exit strictly perpendicularly), and the cosecant will be equal to 1 (“the length of the ladder” will be minimal).

Visualize connections

If all three cases are drawn in a dome-wall-ceiling combination, the result will be the following:

Well, it’s still the same triangle, increased in size to reach the wall and the ceiling. We have vertical sides (sine, tangent), horizontal sides (cosine, cotangent) and “hypotenuses” (secant, cosecant). (By the arrows you can see where each element reaches. The cosecant is the total distance from you to the roof).

A little magic. All triangles share the same equalities:

From the Pythagorean theorem (a 2 + b 2 = c 2) we see how the sides of each triangle are connected. In addition, the “height to width” ratios should also be the same for all triangles. (Simply move from the largest triangle to the smaller one. Yes, the size has changed, but the proportions of the sides will remain the same).

Knowing which side in each triangle is equal to 1 (the radius of the dome), we can easily calculate that “sin/cos = tan/1”.

I have always tried to remember these facts through simple visualization. In the picture you clearly see these dependencies and understand where they come from. This technique is much better than memorizing dry formulas.

Don't forget about other angles

Psst... Don't get stuck on one graph, thinking that the tangent is always less than 1. If you increase the angle, you can reach the ceiling without reaching the wall:

Pythagorean connections always work, but the relative sizes may vary.

(You may have noticed that the sine and cosine ratios are always the smallest because they are contained within the dome).

To summarize: what do we need to remember?

For most of us, I'd say this will be enough:

• trigonometry explains the anatomy of mathematical objects such as circles and repeating intervals
• The dome/wall/roof analogy shows the relationship between different trigonometric functions
• The trigonometric functions result in percentages, which we apply to our script.

You don't need to memorize formulas like 1 2 + cot 2 = csc 2 . They are only suitable for stupid tests in which knowledge of a fact is passed off as understanding it. Take a minute to draw a semicircle in the form of a dome, a wall and a roof, label the elements, and all the formulas will come to you on paper.

Application: Inverse Functions

Any trigonometric function takes an angle as an input parameter and returns the result as a percentage. sin(30) = 0.5. This means that an angle of 30 degrees takes up 50% of the maximum height.

The inverse trigonometric function is written as sin -1 or arcsin. Asin is also often written in various programming languages.

If our height is 25% of the dome's height, what is our angle?

In our table of proportions you can find a ratio where the secant is divided by 1. For example, the secant by 1 (hypotenuse to the horizontal) will be equal to 1 divided by the cosine:

Let's say our secant is 3.5, i.e. 350% of the radius of a unit circle. What angle of inclination to the wall does this value correspond to?

Appendix: Some examples

A boring task. Let's complicate the banal “find the sine” to “What is the height as a percentage of the maximum (hypotenuse)?”

First, notice that the triangle is rotated. There's nothing wrong with that. The triangle also has a height, it is indicated in green in the figure.

What is the hypotenuse equal to? According to the Pythagorean theorem, we know that:

3 2 + 4 2 = hypotenuse 2 25 = hypotenuse 2 5 = hypotenuse

Fine! Sine is the percentage of the height of the triangle's longest side, or hypotenuse. In our example, the sine is 3/5 or 0.60.

Of course, we can go several ways. Now we know that the sine is 0.60, we can simply find the arcsine:

Asin(0.6)=36.9

Here's another approach. Note that the triangle is “facing the wall,” so we can use the tangent instead of the sine. The height is 3, the distance to the wall is 4, so the tangent is ¾ or 75%. We can use the arctangent to go from a percentage value back to an angle:

Tan = 3/4 = 0.75 atan(0.75) = 36.9 Example: Will you swim to the shore?

You are in a boat and you have enough fuel to travel 2 km. You are now 0.25 km from the coast. At what maximum angle to the shore can you swim to it so that you have enough fuel? Addition to the problem statement: we only have a table of arc cosine values.

What we have? The coastline can be represented as a “wall” in our famous triangle, and the “length of the ladder” attached to the wall is the maximum possible distance to be covered by boat to the shore (2 km). A secant appears.

First, you need to go to percentages. We have 2 / 0.25 = 8, that is, we can swim a distance that is 8 times the straight distance to the shore (or to the wall).

The question arises: “What is the secant of 8?” But we cannot answer it, since we only have arc cosines.

We use our previously derived dependencies to relate the secant to the cosine: “sec/1 = 1/cos”

The secant of 8 is equal to the cosine of ⅛. An angle whose cosine is ⅛ is equal to acos(1/8) = 82.8. And this is the largest angle we can afford on a boat with the specified amount of fuel.

Not bad, right? Without the dome-wall-ceiling analogy, I would have gotten lost in a bunch of formulas and calculations. Visualizing the problem greatly simplifies the search for a solution, and it is also interesting to see which trigonometric function will ultimately help.

For each problem, think like this: Am I interested in the dome (sin/cos), wall (tan/sec), or ceiling (cot/csc)?

And trigonometry will become much more enjoyable. Easy calculations for you!

What is sine, cosine, tangent, cotangent of an angle will help you understand a right triangle.

What are the sides of a right triangle called? That's right, hypotenuse and legs: the hypotenuse is the side that lies opposite the right angle (in our example this is the side \(AC\)); legs are the two remaining sides \(AB\) and \(BC\) (those adjacent to the right angle), and if we consider the legs relative to the angle \(BC\), then leg \(AB\) is the adjacent leg, and leg \(BC\) is opposite. So, now let’s answer the question: what are sine, cosine, tangent and cotangent of an angle?

Sine of angle– this is the ratio of the opposite (distant) leg to the hypotenuse.

In our triangle:

\[ \sin \beta =\dfrac(BC)(AC) \]

Cosine of angle– this is the ratio of the adjacent (close) leg to the hypotenuse.

In our triangle:

\[ \cos \beta =\dfrac(AB)(AC) \]

Tangent of the angle– this is the ratio of the opposite (distant) side to the adjacent (close).

In our triangle:

\[ tg\beta =\dfrac(BC)(AB) \]

Cotangent of angle– this is the ratio of the adjacent (close) leg to the opposite (far).

In our triangle:

\[ ctg\beta =\dfrac(AB)(BC) \]

These definitions are necessary remember! To make it easier to remember which leg to divide into what, you need to clearly understand that in tangent And cotangent only the legs sit, and the hypotenuse appears only in sinus And cosine. And then you can come up with a chain of associations. For example, this one:

Cosine→touch→touch→adjacent;

Cotangent→touch→touch→adjacent.

First of all, you need to remember that sine, cosine, tangent and cotangent as the ratios of the sides of a triangle do not depend on the lengths of these sides (at the same angle). Do not believe? Then make sure by looking at the picture:

Consider, for example, the cosine of the angle \(\beta \) . By definition, from a triangle \(ABC\) : \(\cos \beta =\dfrac(AB)(AC)=\dfrac(4)(6)=\dfrac(2)(3) \), but we can calculate the cosine of the angle \(\beta \) from the triangle \(AHI \) : \(\cos \beta =\dfrac(AH)(AI)=\dfrac(6)(9)=\dfrac(2)(3) \). You see, the lengths of the sides are different, but the value of the cosine of one angle is the same. Thus, the values ​​of sine, cosine, tangent and cotangent depend solely on the magnitude of the angle.

If you understand the definitions, then go ahead and consolidate them!

For the triangle \(ABC \) shown in the figure below, we find \(\sin \ \alpha ,\ \cos \ \alpha ,\ tg\ \alpha ,\ ctg\ \alpha \).

\(\begin(array)(l)\sin \ \alpha =\dfrac(4)(5)=0.8\\\cos \ \alpha =\dfrac(3)(5)=0.6\\ tg\ \alpha =\dfrac(4)(3)\\ctg\ \alpha =\dfrac(3)(4)=0.75\end(array) \)

Well, did you get it? Then try it yourself: calculate the same for the angle \(\beta \) .

Answers: \(\sin \ \beta =0.6;\ \cos \ \beta =0.8;\ tg\ \beta =0.75;\ ctg\ \beta =\dfrac(4)(3) \).

Unit (trigonometric) circle

Understanding the concepts of degrees and radians, we considered a circle with a radius equal to \(1\) . Such a circle is called single. It will be very useful when studying trigonometry. Therefore, let's look at it in a little more detail.

As you can see, this circle is constructed in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin of coordinates, the initial position of the radius vector is fixed along the positive direction of the \(x\) axis (in our example, this is the radius \(AB\)).

Each point on the circle corresponds to two numbers: the coordinate along the \(x\) axis and the coordinate along the \(y\) axis. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, we need to remember about the considered right triangle. In the figure above, you can see two whole right triangles. Consider the triangle \(ACG\) . It is rectangular because \(CG\) is perpendicular to the \(x\) axis.

What is \(\cos \ \alpha \) from the triangle \(ACG \)? That's right \(\cos \ \alpha =\dfrac(AG)(AC) \). In addition, we know that \(AC\) is the radius of the unit circle, which means \(AC=1\) . Let's substitute this value into our formula for cosine. Here's what happens:

\(\cos \ \alpha =\dfrac(AG)(AC)=\dfrac(AG)(1)=AG \).

What is \(\sin \ \alpha \) from the triangle \(ACG \) equal to? Well, of course, \(\sin \alpha =\dfrac(CG)(AC)\)! Substitute the value of the radius \(AC\) into this formula and get:

\(\sin \alpha =\dfrac(CG)(AC)=\dfrac(CG)(1)=CG \)

So, can you tell what coordinates the point \(C\) belonging to the circle has? Well, no way? What if you realize that \(\cos \ \alpha \) and \(\sin \alpha \) are just numbers? What coordinate does \(\cos \alpha \) correspond to? Well, of course, the coordinate \(x\)! And what coordinate does \(\sin \alpha \) correspond to? That's right, coordinate \(y\)! So the point \(C(x;y)=C(\cos \alpha ;\sin \alpha) \).

What then are \(tg \alpha \) and \(ctg \alpha \) equal to? That’s right, let’s use the corresponding definitions of tangent and cotangent and get that \(tg \alpha =\dfrac(\sin \alpha )(\cos \alpha )=\dfrac(y)(x) \), A \(ctg \alpha =\dfrac(\cos \alpha )(\sin \alpha )=\dfrac(x)(y) \).

What if the angle is larger? For example, like in this picture:

What has changed in this example? Let's figure it out. To do this, let's turn again to a right triangle. Consider a right triangle \(((A)_(1))((C)_(1))G \) : angle (as adjacent to angle \(\beta \) ). What is the value of sine, cosine, tangent and cotangent for an angle \(((C)_(1))((A)_(1))G=180()^\circ -\beta \ \)? That's right, we adhere to the corresponding definitions of trigonometric functions:

\(\begin(array)(l)\sin \angle ((C)_(1))((A)_(1))G=\dfrac(((C)_(1))G)(( (A)_(1))((C)_(1)))=\dfrac(((C)_(1))G)(1)=((C)_(1))G=y; \\\cos \angle ((C)_(1))((A)_(1))G=\dfrac(((A)_(1))G)(((A)_(1)) ((C)_(1)))=\dfrac(((A)_(1))G)(1)=((A)_(1))G=x;\\tg\angle ((C )_(1))((A)_(1))G=\dfrac(((C)_(1))G)(((A)_(1))G)=\dfrac(y)( x);\\ctg\angle ((C)_(1))((A)_(1))G=\dfrac(((A)_(1))G)(((C)_(1 ))G)=\dfrac(x)(y)\end(array) \)

Well, as you can see, the value of the sine of the angle still corresponds to the coordinate \(y\) ; the value of the cosine of the angle - coordinate \(x\) ; and the values ​​of tangent and cotangent to the corresponding ratios. Thus, these relations apply to any rotation of the radius vector.

It has already been mentioned that the initial position of the radius vector is along the positive direction of the \(x\) axis. So far we have rotated this vector counterclockwise, but what happens if we rotate it clockwise? Nothing extraordinary, you will also get an angle of a certain value, but only it will be negative. Thus, when rotating the radius vector counterclockwise, we get positive angles, and when rotating clockwise – negative.

So, we know that the whole revolution of the radius vector around the circle is \(360()^\circ \) or \(2\pi \) . Is it possible to rotate the radius vector by \(390()^\circ \) or by \(-1140()^\circ \)? Well, of course you can! In the first case, \(390()^\circ =360()^\circ +30()^\circ \), thus, the radius vector will make one full revolution and stop at the position \(30()^\circ \) or \(\dfrac(\pi )(6) \) .

In the second case, \(-1140()^\circ =-360()^\circ \cdot 3-60()^\circ \), that is, the radius vector will make three full revolutions and stop at the position \(-60()^\circ \) or \(-\dfrac(\pi )(3) \) .

Thus, from the above examples we can conclude that angles that differ by \(360()^\circ \cdot m \) or \(2\pi \cdot m \) (where \(m \) is any integer ), correspond to the same position of the radius vector.

The figure below shows the angle \(\beta =-60()^\circ \) . The same image corresponds to the corner \(-420()^\circ ,-780()^\circ ,\ 300()^\circ ,660()^\circ \) etc. This list can be continued indefinitely. All these angles can be written by the general formula \(\beta +360()^\circ \cdot m\) or \(\beta +2\pi \cdot m \) (where \(m \) is any integer)

\(\begin(array)(l)-420()^\circ =-60+360\cdot (-1);\\-780()^\circ =-60+360\cdot (-2); \\300()^\circ =-60+360\cdot 1;\\660()^\circ =-60+360\cdot 2.\end(array) \)

Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values ​​are:

\(\begin(array)(l)\sin \ 90()^\circ =?\\\cos \ 90()^\circ =?\\\text(tg)\ 90()^\circ =? \\\text(ctg)\ 90()^\circ =?\\\sin \ 180()^\circ =\sin \ \pi =?\\\cos \ 180()^\circ =\cos \ \pi =?\\\text(tg)\ 180()^\circ =\text(tg)\ \pi =?\\\text(ctg)\ 180()^\circ =\text(ctg)\ \pi =?\\\sin \ 270()^\circ =?\\\cos \ 270()^\circ =?\\\text(tg)\ 270()^\circ =?\\\text (ctg)\ 270()^\circ =?\\\sin \ 360()^\circ =?\\\cos \ 360()^\circ =?\\\text(tg)\ 360()^ \circ =?\\\text(ctg)\ 360()^\circ =?\\\sin \ 450()^\circ =?\\\cos \ 450()^\circ =?\\\text (tg)\ 450()^\circ =?\\\text(ctg)\ 450()^\circ =?\end(array) \)

Here's a unit circle to help you:

Having difficulties? Then let's figure it out. So we know that:

\(\begin(array)(l)\sin \alpha =y;\\cos\alpha =x;\\tg\alpha =\dfrac(y)(x);\\ctg\alpha =\dfrac(x )(y).\end(array)\)

From here, we determine the coordinates of the points corresponding to certain angle measures. Well, let's start in order: the corner in \(90()^\circ =\dfrac(\pi )(2) \) corresponds to a point with coordinates \(\left(0;1 \right) \) , therefore:

\(\sin 90()^\circ =y=1 \) ;

\(\cos 90()^\circ =x=0 \) ;

\(\text(tg)\ 90()^\circ =\dfrac(y)(x)=\dfrac(1)(0)\Rightarrow \text(tg)\ 90()^\circ \)- does not exist;

\(\text(ctg)\ 90()^\circ =\dfrac(x)(y)=\dfrac(0)(1)=0 \).

Further, adhering to the same logic, we find out that the corners in \(180()^\circ ,\ 270()^\circ ,\ 360()^\circ ,\ 450()^\circ (=360()^\circ +90()^\circ)\ \ ) correspond to points with coordinates \(\left(-1;0 \right),\text( )\left(0;-1 \right),\text( )\left(1;0 \right),\text( )\left(0 ;1 \right) \), respectively. Knowing this, it is easy to determine the values ​​of trigonometric functions at the corresponding points. Try it yourself first, and then check the answers.

Answers:

\(\displaystyle \sin \180()^\circ =\sin \ \pi =0 \)

\(\displaystyle \cos \180()^\circ =\cos \ \pi =-1\)

\(\text(tg)\ 180()^\circ =\text(tg)\ \pi =\dfrac(0)(-1)=0 \)

\(\text(ctg)\ 180()^\circ =\text(ctg)\ \pi =\dfrac(-1)(0)\Rightarrow \text(ctg)\ \pi \)- does not exist

\(\sin \270()^\circ =-1\)

\(\cos \ 270()^\circ =0 \)

\(\text(tg)\ 270()^\circ =\dfrac(-1)(0)\Rightarrow \text(tg)\ 270()^\circ \)- does not exist

\(\text(ctg)\ 270()^\circ =\dfrac(0)(-1)=0 \)

\(\sin \360()^\circ =0\)

\(\cos \360()^\circ =1\)

\(\text(tg)\ 360()^\circ =\dfrac(0)(1)=0 \)

\(\text(ctg)\ 360()^\circ =\dfrac(1)(0)\Rightarrow \text(ctg)\ 2\pi \)- does not exist

\(\sin \ 450()^\circ =\sin \ \left(360()^\circ +90()^\circ \right)=\sin \ 90()^\circ =1 \)

\(\cos \ 450()^\circ =\cos \ \left(360()^\circ +90()^\circ \right)=\cos \ 90()^\circ =0 \)

\(\text(tg)\ 450()^\circ =\text(tg)\ \left(360()^\circ +90()^\circ \right)=\text(tg)\ 90() ^\circ =\dfrac(1)(0)\Rightarrow \text(tg)\ 450()^\circ \)- does not exist

\(\text(ctg)\ 450()^\circ =\text(ctg)\left(360()^\circ +90()^\circ \right)=\text(ctg)\ 90()^ \circ =\dfrac(0)(1)=0 \).

Thus, we can make the following table:

There is no need to remember all these values. It is enough to remember the correspondence between the coordinates of points on the unit circle and the values ​​of trigonometric functions:

\(\left. \begin(array)(l)\sin \alpha =y;\\cos \alpha =x;\\tg \alpha =\dfrac(y)(x);\\ctg \alpha =\ dfrac(x)(y).\end(array) \right\)\ \text(You must remember or be able to display it!! \) !}

But the values ​​of the trigonometric functions of angles in and \(30()^\circ =\dfrac(\pi )(6),\ 45()^\circ =\dfrac(\pi )(4)\) given in the table below, you must remember:

Don’t be scared, now we’ll show you one example of a fairly simple memorization of the corresponding values:

To use this method, it is vital to remember the sine values ​​for all three measures of angle ( \(30()^\circ =\dfrac(\pi )(6),\ 45()^\circ =\dfrac(\pi )(4),\ 60()^\circ =\dfrac(\pi )(3)\)), as well as the value of the tangent of the angle in \(30()^\circ \) . Knowing these \(4\) values, it is quite simple to restore the entire table - the cosine values ​​are transferred in accordance with the arrows, that is:

\(\begin(array)(l)\sin 30()^\circ =\cos \ 60()^\circ =\dfrac(1)(2)\ \ \\\sin 45()^\circ = \cos \ 45()^\circ =\dfrac(\sqrt(2))(2)\\\sin 60()^\circ =\cos \ 30()^\circ =\dfrac(\sqrt(3 ))(2)\ \end(array) \)

\(\text(tg)\ 30()^\circ \ =\dfrac(1)(\sqrt(3)) \), knowing this, you can restore the values ​​for \(\text(tg)\ 45()^\circ , \text(tg)\ 60()^\circ \). The numerator "\(1 \)" will correspond to \(\text(tg)\ 45()^\circ \ \) and the denominator "\(\sqrt(\text(3)) \)" will correspond to \(\text (tg)\ 60()^\circ \ \) . Cotangent values ​​are transferred in accordance with the arrows indicated in the figure. If you understand this and remember the diagram with the arrows, then it will be enough to remember only \(4\) values ​​from the table.

Coordinates of a point on a circle

Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation? Well, of course you can! Let's derive a general formula for finding the coordinates of a point. For example, here is a circle in front of us:

We are given that point \(K(((x)_(0));((y)_(0)))=K(3;2) \)- center of the circle. The radius of the circle is \(1.5\) . It is necessary to find the coordinates of the point \(P\) obtained by rotating the point \(O\) by \(\delta \) degrees.

As can be seen from the figure, the coordinate \(x\) of the point \(P\) corresponds to the length of the segment \(TP=UQ=UK+KQ\) . The length of the segment \(UK\) corresponds to the coordinate \(x\) of the center of the circle, that is, it is equal to \(3\) . The length of the segment \(KQ\) can be expressed using the definition of cosine:

\(\cos \ \delta =\dfrac(KQ)(KP)=\dfrac(KQ)(r)\Rightarrow KQ=r\cdot \cos \ \delta \).

Then we have that for the point \(P\) the coordinate \(x=((x)_(0))+r\cdot \cos \ \delta =3+1.5\cdot \cos \ \delta \).

Using the same logic, we find the value of the y coordinate for the point \(P\) . Thus,

\(y=((y)_(0))+r\cdot \sin \ \delta =2+1.5\cdot \sin \delta \).

So, in general, the coordinates of points are determined by the formulas:

\(\begin(array)(l)x=((x)_(0))+r\cdot \cos \ \delta \\y=((y)_(0))+r\cdot \sin \ \delta \end(array) \), Where

\(((x)_(0)),((y)_(0)) \) - coordinates of the center of the circle,

\(r\) - radius of the circle,

\(\delta \) - rotation angle of the vector radius.

As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are equal to zero and the radius is equal to one:

\(\begin(array)(l)x=((x)_(0))+r\cdot \cos \ \delta =0+1\cdot \cos \ \delta =\cos \ \delta \\y =((y)_(0))+r\cdot \sin \ \delta =0+1\cdot \sin \ \delta =\sin \ \delta \end(array) \)

Trigonometry is a branch of mathematical science that studies trigonometric functions and their use in geometry. The development of trigonometry began in ancient Greece. During the Middle Ages, scientists from the Middle East and India made important contributions to the development of this science.

This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the basic trigonometric functions: sine, cosine, tangent and cotangent. Their meaning is explained and illustrated in the context of geometry.

Yandex.RTB R-A-339285-1

Initially, the definitions of trigonometric functions whose argument is an angle were expressed in terms of the ratio of the sides of a right triangle.

Definitions of trigonometric functions

The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.

Cosine of the angle (cos α) - the ratio of the adjacent leg to the hypotenuse.

Angle tangent (t g α) - the ratio of the opposite side to the adjacent side.

Angle cotangent (c t g α) - the ratio of the adjacent side to the opposite side.

These definitions are given for the acute angle of a right triangle!

Let's give an illustration.

In triangle ABC with right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.

The definitions of sine, cosine, tangent and cotangent allow you to calculate the values ​​of these functions from the known lengths of the sides of the triangle.

Important to remember!

The range of values ​​of sine and cosine is from -1 to 1. In other words, sine and cosine take values ​​from -1 to 1. The range of values ​​of tangent and cotangent is the entire number line, that is, these functions can take on any values.

The definitions given above apply to acute angles. In trigonometry, the concept of a rotation angle is introduced, the value of which, unlike an acute angle, is not limited to 0 to 90 degrees. The rotation angle in degrees or radians is expressed by any real number from - ∞ to + ∞.

In this context, we can define sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Let us imagine a unit circle with its center at the origin of the Cartesian coordinate system.

The initial point A with coordinates (1, 0) rotates around the center of the unit circle through a certain angle α and goes to point A 1. The definition is given in terms of the coordinates of point A 1 (x, y).

Sine (sin) of the rotation angle

The sine of the rotation angle α is the ordinate of point A 1 (x, y). sin α = y

Cosine (cos) of the rotation angle

The cosine of the rotation angle α is the abscissa of point A 1 (x, y). cos α = x

Tangent (tg) of the rotation angle

The tangent of the angle of rotation α is the ratio of the ordinate of point A 1 (x, y) to its abscissa. t g α = y x

Cotangent (ctg) of the rotation angle

The cotangent of the rotation angle α is the ratio of the abscissa of point A 1 (x, y) to its ordinate. c t g α = x y

Sine and cosine are defined for any rotation angle. This is logical, because the abscissa and ordinate of a point after rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is undefined when a point after rotation goes to a point with zero abscissa (0, 1) and (0, - 1). In such cases, the expression for tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with cotangent. The difference is that the cotangent is not defined in cases where the ordinate of a point goes to zero.

Important to remember!

Sine and cosine are defined for any angles α.

Tangent is defined for all angles except α = 90° + 180° k, k ∈ Z (α = π 2 + π k, k ∈ Z)

Cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)

When solving practical examples, do not say “sine of the angle of rotation α”. The words “angle of rotation” are simply omitted, implying that it is already clear from the context what is being discussed.

Numbers

What about the definition of sine, cosine, tangent and cotangent of a number, and not the angle of rotation?

Sine, cosine, tangent, cotangent of a number

Sine, cosine, tangent and cotangent of a number t is a number that is respectively equal to sine, cosine, tangent and cotangent in t radian.

For example, the sine of the number 10 π is equal to the sine of the rotation angle of 10 π rad.

There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's take a closer look at it.

Any real number t a point on the unit circle is associated with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are determined through the coordinates of this point.

The starting point on the circle is point A with coordinates (1, 0).

Positive number t

Negative number t corresponds to the point to which the starting point will go if it moves around the circle counterclockwise and passes the path t.

Now that the connection between a number and a point on a circle has been established, we move on to the definition of sine, cosine, tangent and cotangent.

Sine (sin) of t

Sine of a number t- ordinate of a point on the unit circle corresponding to the number t. sin t = y

Cosine (cos) of t

Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x

Tangent (tg) of t

Tangent of a number t- the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t. t g t = y x = sin t cos t

The latest definitions are in accordance with and do not contradict the definition given at the beginning of this paragraph. Point on the circle corresponding to the number t, coincides with the point to which the starting point goes after turning by an angle t radian.

Trigonometric functions of angular and numeric argument

Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) correspond to a certain tangent value. Cotangent, as stated above, is defined for all α except α = 180° k, k ∈ Z (α = π k, k ∈ Z).

We can say that sin α, cos α, t g α, c t g α are functions of the angle alpha, or functions of the angular argument.

Similarly, we can talk about sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a certain value of the sine or cosine of a number t. All numbers other than π 2 + π · k, k ∈ Z, correspond to a tangent value. Cotangent, similarly, is defined for all numbers except π · k, k ∈ Z.

Basic functions of trigonometry

Sine, cosine, tangent and cotangent are the basic trigonometric functions.

It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.

Let's return to the definitions given at the very beginning and the alpha angle, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent, and cotangent are entirely consistent with the geometric definitions given by the aspect ratios of a right triangle. Let's show it.

Let's take a unit circle with a center in a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right triangle, the angle A 1 O H is equal to the angle of rotation α, the length of the leg O H is equal to the abscissa of the point A 1 (x, y). The length of the leg opposite the angle is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.

In accordance with the definition from geometry, the sine of angle α is equal to the ratio of the opposite side to the hypotenuse.

sin α = A 1 H O A 1 = y 1 = y

This means that determining the sine of an acute angle in a right triangle through the aspect ratio is equivalent to determining the sine of the rotation angle α, with alpha lying in the range from 0 to 90 degrees.

Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.

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