Let's recall the school mathematics course and talk about what a tangent is and how to find the tangent of an angle. First, let's define what is called a tangent. AT right triangle tangent acute angle is the ratio of the opposite leg to the adjacent one. The adjacent leg is the one that participates in the formation of the angle, the opposite one is the one that is located opposite the angle.

Also, the tangent of an acute angle is the ratio of the sine of this angle to its cosine. For understanding, we recall what is the sine and cosine of an angle. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse, the cosine is the ratio of the adjacent leg to the hypotenuse.

There is also a cotangent, it is the opposite of the tangent. The cotangent is the ratio of the adjacent leg to the opposite leg and, accordingly, the ratio of the cosine of an angle to its sine.

Sine, cosine, tangent and cotangent are trigonometric functions of an angle, they show the relationship between the angles and sides of a triangle, help to calculate the sides of a triangle.

Calculate the tangent of an acute angle

How to find the tangent in a triangle? In order not to waste time looking for the tangent, you can find special tables where the trigonometric functions of many angles are indicated. In school puzzles in geometry, certain angles are very common, and teachers are asked to remember the values ​​​​of their sines, cosines, tangents and cotangents. We offer you a small plate with the desired values ​​for these angles.

If the angle whose tangent needs to be found is not presented in this table, then you can use the two formulas that we presented above in verbal form.

The first way to calculate the tangent of an angle is to divide the length of the opposite leg by the length of the adjacent one. Let's say the opposite leg is 4, and the adjacent leg is 8. To find the tangent, you need 4:8. The tangent of the angle will be ½ or 0.5.

The second way to calculate the tangent is to divide the value of the sine of a given angle by the value of its cosine. For example, we are given an angle of 45 degrees. Its sin = square root of two divided by two; its cos is the same number. Now we divide the sine by the cosine and get the tangent equal to one.

It happens that you need to use this particular formula, but only one element is known - either sine or cosine. In this case, it will be useful to recall the formula

sin2 α + cos2 α = 1. This is the basic trigonometric identity. By expressing an unknown element in terms of a known one, one can find out its meaning. And knowing the sine and cosine, it is not difficult to find the tangent.

And if geometry is clearly not your calling, but to make homework still need, then you can use the online calculator for calculating the tangent of an angle.

We told you about simple examples how to find tangent. However, the conditions of the tasks are more difficult and it is not always possible to quickly find out all the necessary data. In this case, the Pythagorean theorem and various trigonometric functions will help you.

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

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

Yandex.RTB R-A-339285-1

Initially, the definitions of trigonometric functions, whose argument is an angle, were expressed through 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.

The cosine of the angle (cos α) is the ratio of the adjacent leg to the hypotenuse.

The tangent of the angle (t g α) is the ratio of the opposite leg to the adjacent one.

The cotangent of the angle (c t g α) is the ratio of the adjacent leg to the opposite one.

These definitions are given for an 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 make it possible to calculate the values ​​of these functions from the known lengths of the sides of a triangle.

Important to remember!

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

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

In this context, one can define the sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Imagine a unit circle centered at the origin of the Cartesian coordinate system.

The starting point A with coordinates (1 , 0) rotates around the center of the unit circle by some angle α and goes to point A 1 . The definition is given through the coordinates of the point A 1 (x, y).

Sine (sin) of the rotation angle

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

Cosine (cos) of the angle of rotation

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

Tangent (tg) of rotation angle

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

Cotangent (ctg) of rotation angle

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

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

Important to remember!

Sine and cosine are defined for any angles α.

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

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

When deciding practical examples don't say "sine of the angle of rotation α". The words "angle of rotation" are simply omitted, implying that from the context it is already clear what is at stake.

Numbers

What about the definition of the 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 a number is called, which is respectively equal to the sine, cosine, tangent and cotangent in t radian.

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

There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. Let's consider it in more detail.

Any real number t a point on the unit circle is put in correspondence with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are defined in terms of 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 move if it moves counterclockwise around the circle and passes the path t .

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

Sine (sin) of the number t

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

The latter definitions are consistent with and do not contradict the definition given at the beginning of this section. Point on a circle corresponding to a number t, coincides with the point to which the starting point passes after turning through the angle t radian.

Trigonometric functions of angular and numerical 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) corresponds to a certain value of the tangent. The cotangent, as mentioned above, is defined for all α, except for 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, one can speak of sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a specific value of the sine or cosine of a number t. All numbers other than π 2 + π · k , k ∈ Z, correspond to the value of the tangent. The cotangent is similarly 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 data at the very beginning of the definitions and the angle alpha, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent, and cotangent are in full agreement with the geometric definitions given by the ratios of the sides of a right triangle. Let's show it.

Take a unit circle centered on a rectangular Cartesian coordinate system. Let's turn starting point A (1, 0) at an angle of up to 90 degrees and draw from the resulting point A 1 (x, y) perpendicular to the x-axis. 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 corner 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 the angle α is equal to the ratio of the opposite leg to the hypotenuse.

sin α \u003d A 1 H O A 1 \u003d y 1 \u003d y

This means that the definition of the sine of an acute angle in a right triangle through the aspect ratio is equivalent to the definition of the sine of the angle of rotation 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.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

The ratio of the opposite leg to the hypotenuse is called sine of an acute angle right triangle.

\sin \alpha = \frac(a)(c)

Cosine of an acute angle of a right triangle

The ratio of the nearest leg to the hypotenuse is called cosine of an acute angle right triangle.

\cos \alpha = \frac(b)(c)

Tangent of an acute angle of a right triangle

The ratio of the opposite leg to the adjacent leg is called acute angle tangent right triangle.

tg \alpha = \frac(a)(b)

Cotangent of an acute angle of a right triangle

The ratio of the adjacent leg to the opposite leg is called cotangent of an acute angle right triangle.

ctg \alpha = \frac(b)(a)

Sine of an arbitrary angle

The ordinate of the point on the unit circle to which the angle \alpha corresponds is called sine of an arbitrary angle rotation \alpha .

\sin \alpha=y

Cosine of an arbitrary angle

The abscissa of a point on the unit circle to which the angle \alpha corresponds is called cosine of an arbitrary angle rotation \alpha .

\cos \alpha=x

Tangent of an arbitrary angle

The ratio of the sine of an arbitrary rotation angle \alpha to its cosine is called tangent of an arbitrary angle rotation \alpha .

tg \alpha = y_(A)

tg \alpha = \frac(\sin \alpha)(\cos \alpha)

Cotangent of an arbitrary angle

The ratio of the cosine of an arbitrary rotation angle \alpha to its sine is called cotangent of an arbitrary angle rotation \alpha .

ctg \alpha =x_(A)

ctg \alpha = \frac(\cos \alpha)(\sin \alpha)

An example of finding an arbitrary angle

If \alpha is some angle AOM , where M is a point on the unit circle, then

\sin \alpha=y_(M) , \cos \alpha=x_(M) , tg \alpha=\frac(y_(M))(x_(M)), ctg \alpha=\frac(x_(M))(y_(M)).

For example, if \angle AOM = -\frac(\pi)(4), then: the ordinate of the point M is -\frac(\sqrt(2))(2), the abscissa is \frac(\sqrt(2))(2) and that's why

\sin \left (-\frac(\pi)(4) \right)=-\frac(\sqrt(2))(2);

\cos \left (\frac(\pi)(4) \right)=\frac(\sqrt(2))(2);

tg;

ctg \left (-\frac(\pi)(4) \right)=-1.

Table of values ​​of sines of cosines of tangents of cotangents

The values ​​of the main frequently encountered angles are given in the table:

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

Recall that right angle is an angle equal to 90 degrees. In other words, half of the unfolded corner.

Sharp corner- less than 90 degrees.

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

Let's draw a right triangle. A right angle is usually denoted . Note that the side opposite the corner is denoted by the same letter, only small. So, the side lying opposite the angle A is denoted.

The angle is indicated by the corresponding Greek letter.

Hypotenuse right triangle is the side opposite right angle.

Legs- sides opposite sharp corners.

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

Sinus acute angle in a right triangle is the ratio of the opposite leg 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 leg to the adjacent:

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

Cotangent acute angle in a right triangle - the ratio of the adjacent leg to the opposite (or, equivalently, the ratio of cosine to sine):

Pay attention to the basic ratios for sine, cosine, tangent and cotangent, which are given below. They will be useful to us in solving problems.

Let's prove some of them.

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

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

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 one. Knowing two sides in a right triangle, you can find the third. So, for angles - their ratio, for sides - their own. But what to do if in a right triangle one angle (except for a right one) and one side are known, but you need to find other sides?

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

Sine, cosine and tangent - they are also called trigonometric functions of the angle- give the ratio 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 sine, cosine, tangent and cotangent values ​​for "good" angles from to.

Notice the two red dashes in the table. For the corresponding values ​​of the angles, the tangent and cotangent do not exist.

Let's analyze several problems in trigonometry from the Bank of FIPI tasks.

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

The problem is solved in four seconds.

Insofar as , .

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

Let's find by the Pythagorean theorem.

Problem solved.

Often in problems there are triangles with angles and or with angles and . Memorize 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 considered problems for solving right triangles - that is, for finding unknown sides or angles. But that's not all! In the variants of the exam in mathematics, there are many tasks where the sine, cosine, tangent or cotangent of the outer angle of the triangle appears. More on this in the next article.

Middle level

Right triangle. Complete illustrated guide (2019)

RIGHT TRIANGLE. FIRST LEVEL.

In problems, a right angle is not at all necessary - the lower left one, so you need to learn how to recognize a right triangle in this form,

and in such

and in such

What is good about a right triangle? Well... first of all, there are special beautiful names for his sides.

Attention to the drawing!

Remember and do not confuse: legs - two, and the hypotenuse - only one(the only, unique and longest)!

Well, we discussed the names, now the most important thing: the Pythagorean theorem.

Pythagorean theorem.

This theorem is the key to solving many problems involving a right triangle. It was proved by Pythagoras in completely immemorial times, and since then it has brought many benefits to those who know it. And the best thing about her is that she is simple.

So, Pythagorean theorem:

Do you remember the joke: “Pythagorean pants are equal on all sides!”?

Let's draw these very Pythagorean pants and look at them.

Does it really look like shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:

"Sum area of ​​squares, built on the legs, is equal to square area built on the hypotenuse.

Doesn't it sound a little different, doesn't it? And so, when Pythagoras drew the statement of his theorem, just such a picture turned out.

In this picture, the sum of the areas of the small squares is equal to the area of ​​the large square. And so that the children better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty invented this joke about Pythagorean pants.

Why are we now formulating the Pythagorean theorem

Did Pythagoras suffer and talk about squares?

You see, in ancient times there was no ... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to memorize everything with words??! And we can be glad that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to better remember:

Now it should be easy:

Well, the most important theorem about a right triangle was discussed. If you are interested in how it is proved, read the next levels of theory, and now let's move on ... into the dark forest ... of trigonometry! To the terrible words sine, cosine, tangent and cotangent.

Sine, cosine, tangent, cotangent in a right triangle.

In fact, everything is not so scary at all. Of course, the "real" definition of sine, cosine, tangent and cotangent should be looked at in the article. But you really don't want to, do you? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:

Why is it all about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!

1.
It actually sounds like this:

What about the angle? Is there a leg that is opposite the corner, that is, the opposite leg (for the corner)? Of course have! This is a cathet!

But what about the angle? Look closely. Which leg is adjacent to the corner? Of course, the cat. So, for the angle, the leg is adjacent, and

And now, attention! Look what we got:

See how great it is:

Now let's move on to tangent and cotangent.

How to put it into words now? What is the leg in relation to the corner? Opposite, of course - it "lies" opposite the corner. And the cathet? Adjacent to the corner. So what did we get?

See how the numerator and denominator are reversed?

And now again the corners and made the exchange:

Summary

Let's briefly write down what we have learned.

Pythagorean theorem:

The main right triangle theorem is the Pythagorean theorem.

Pythagorean theorem

By the way, do you remember well what the legs and hypotenuse are? If not, then look at the picture - refresh your knowledge

It is quite possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true. How would you prove it? Let's do like the ancient Greeks. Let's draw a square with a side.

You see how cunningly we divided its sides into segments of lengths and!

Now let's connect the marked points

Here we, however, noted something else, but you yourself look at the picture and think about why.

What is the area of ​​the larger square? Correctly, . What about the smaller area? Certainly, . The total area of ​​the four corners remains. Imagine that we took two of them and leaned against each other with hypotenuses. What happened? Two rectangles. So, the area of ​​"cuttings" is equal.

Let's put it all together now.

Let's transform:

So we visited Pythagoras - we proved his theorem in an ancient way.

Right triangle and trigonometry

For a right triangle, the following relations hold:

The sine of an acute angle is equal to the ratio of the opposite leg to the hypotenuse

The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.

The tangent of an acute angle is equal to the ratio of the opposite leg to the adjacent leg.

The cotangent of an acute angle is equal to the ratio of the adjacent leg to the opposite leg.

And once again, all this in the form of a plate:

It is very comfortable!

Signs of equality of right triangles

I. On two legs

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

Attention! Here it is very important that the legs are "corresponding". For example, if it goes like this:

THEN THE TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.

Need to in both triangles the leg was adjacent, or in both - opposite.

Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Look at the topic “and pay attention to the fact that for the equality of “ordinary” triangles, you need the equality of their three elements: two sides and an angle between them, two angles and a side between them, or three sides. But for the equality of right-angled triangles, only two corresponding elements are enough. It's great, right?

Approximately the same situation with signs of similarity of right triangles.

Signs of similarity of right triangles

I. Acute corner

II. On two legs

III. By leg and hypotenuse

Median in a right triangle

Why is it so?

Consider a whole rectangle instead of a right triangle.

Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?

And what follows from this?

So it happened that

1. - median:

Remember this fact! Helps a lot!

What is even more surprising is that the converse is also true.

What good can be gained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture

Look closely. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But in a triangle there is only one point, the distances from which about all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCUM DEscribed. So what happened?

So let's start with this "besides...".

Let's look at i.

But in similar triangles all angles are equal!

The same can be said about and

Now let's draw it together:

What use can be drawn from this "triple" similarity.

Well, for example - two formulas for the height of a right triangle.

We write the relations of the corresponding parties:

To find the height, we solve the proportion and get first formula "Height in a right triangle":

So, let's apply the similarity: .

What will happen now?

Again we solve the proportion and get the second formula:

Both of these formulas must be remembered very well and the one that is more convenient to apply. Let's write them down again.

Pythagorean theorem:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:.

Signs of equality of right triangles:

• on two legs:
• along the leg and hypotenuse: or
• along the leg and the adjacent acute angle: or
• along the leg and the opposite acute angle: or
• by hypotenuse and acute angle: or.

Signs of similarity of right triangles:

• one sharp corner: or
• from the proportionality of the two legs:
• from the proportionality of the leg and hypotenuse: or.

Sine, cosine, tangent, cotangent in a right triangle

• The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:
• The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
• The tangent of an acute angle of a right triangle is the ratio of the opposite leg to the adjacent one:
• The cotangent of an acute angle of a right triangle is the ratio of the adjacent leg to the opposite:.

Height of a right triangle: or.

In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .

Area of ​​a right triangle:

• through the catheters: