Right triangle. Complete illustrated guide (2019)

Where the tasks for solving a right-angled triangle were considered, I promised to present a technique for memorizing the definitions of sine and cosine. Using it, you will always quickly remember which leg belongs to the hypotenuse (adjacent or opposite). I decided not to put it off indefinitely, the necessary material is below, please read it 😉

The fact is that I have repeatedly observed how students in grades 10-11 have difficulty remembering these definitions. They remember very well that the leg refers to the hypotenuse, but which one- forget and confused. The price of a mistake, as you know in the exam, is a lost score.

The information that I will present directly to mathematics has nothing to do. It is associated with figurative thinking, and with the methods of verbal-logical connection. That's right, I myself, once and for all remembereddefinition data. If you still forget them, then with the help of the presented techniques it is always easy to remember.

Let me remind you the definitions of sine and cosine in a right triangle:

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

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

So, what associations does the word cosine evoke in you?

Probably everyone has their ownRemember the link:

Thus, you will immediately have an expression in your memory -

«… ratio of ADJACENT leg to hypotenuse».

The problem with the definition of cosine is solved.

If you need to remember the definition of the sine in a right triangle, then remembering the definition of the cosine, you can easily establish that the sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse. After all, there are only two legs, if the adjacent leg is “occupied” by the cosine, then only the opposite side remains for the sine.

What about tangent and cotangent? Same confusion. Students know that this is the ratio of legs, but the problem is to remember which one refers to which - either opposite to adjacent, or vice versa.

Definitions:

Tangent an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one:

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

How to remember? There are two ways. One also uses a verbal-logical connection, the other - a mathematical one.

MATHEMATICAL METHOD

There is such a definition - the tangent of an acute angle is the ratio of the sine of an angle to its cosine:

* Remembering the formula, you can always determine that the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one.

Likewise.The cotangent of an acute angle is the ratio of the cosine of an angle to its sine:

So! Remembering these formulas, you can always determine that:

- the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent

- the cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite one.

VERBAL-LOGICAL METHOD

About tangent. Remember the link:

That is, if you need to remember the definition of the tangent, using this logical connection, you can easily remember what it is

"... the ratio of the opposite leg to the adjacent"

If it comes to cotangent, then remembering the definition of tangent, you can easily voice the definition of cotangent -

"... the ratio of the adjacent leg to the opposite"

There is an interesting technique for memorizing tangent and cotangent on the site " Mathematical tandem " , look.

METHOD UNIVERSAL

You can just grind.But as practice shows, thanks to verbal-logical connections, a person remembers information for a long time, and not only mathematical.

I hope the material was useful to you.

Sincerely, Alexander Krutitskikh

P.S: I would be grateful if you tell about the site in social networks.

Instruction

Method 1. Using the Pythagorean theorem. The theorem says: the square of the hypotenuse is equal to the sum squares of legs. It follows that any of the sides of a right-angled triangle can be calculated knowing its other two sides (Fig. 2)

Method 2. It follows from the fact that the median drawn from to the hypotenuse forms 3 similar triangles among themselves (Fig. 3). In this figure, triangles ABC, BCD and ACD are similar.

Example 6: Using unit circles to find coordinates

First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the reference angle, and give them signs corresponding to the y- and x-values of the quadrant. Next, we will find the cosine and sine of the given angle.

Sieve angle, angle triangle and cube root

Polygons that can be built with a compass and straightedge include.

Note: the sieve angle cannot be plotted with a compass and straightedge. Multiplying the side length of a cube by the cube root of 2 gives the side length of a cube with double the volume. With the help of the innovative theory of the French mathematician Évariste Galois, it can be shown that for all three classical problems, construction with a circle and a ruler is impossible.

The hypotenuse is the side in a right triangle that is opposite the 90 degree angle. In order to calculate its length, it is enough to know the length of one of the legs and the value of one of the acute angles of the triangle.

Keep in mind: the three-component angle and cube root construction are not possible with a compass and straightedge.

On the other hand, the solution of the equation of the third degree according to the Cardano formula can be represented by dividing the angle and the cube root. In the future, we build some angle with a circle and a ruler. However, after the triangle of this angle and the determination of the cube root, the completion of the construction of the sieve square can be done with the help of a compass and straightedge.

Construction of a lattice deck according to this calculation

The algebraic formulation of the construction problem leads to an equation whose structural analysis will provide additional information about the construction of the ternary structure. The one-to-one ratio of an angle to its cosine is used here: if the magnitude of the angle is known, the length of the cosine of the angle can be uniquely constructed on the unit circle and vice versa.

Instruction

With a known leg and an acute angle of a right triangle, then the size of the hypotenuse can be equal to the ratio of the leg to the cosine / sine of this angle, if this angle is opposite / adjacent to it:

h = C1(or C2)/sinα;

h = С1(or С2)/cosα.

Example: Given a right triangle ABC with hypotenuse AB and right angle C. Let angle B be 60 degrees and angle A 30 degrees. The length of leg BC is 8 cm. Find the length of hypotenuse AB. To do this, you can use any of the methods suggested above:

This one-to-one task allows you to go from the definition of the angle to the definition of the cosine of the angle. In the following, 3 φ denotes the angle to be divided. Thus, φ is the angle, the value of which must be determined for given 3 φ. Starting with compounds known from trigonometry.

Follows at a given angle 3 φ. An algebraic consideration of the solvability of a three-dimensional equation leads directly to the question of the possibility of constructing solutions and, consequently, to the question of the possibility or impossibility of a constructive triple angle of a given angle.

AB=BC/cos60=8 cm.

AB = BC/sin30 = 8 cm.

The hypotenuse is the side of a right triangle that is opposite the right angle. It is the longest side of a right triangle. You can calculate it using the Pythagorean theorem or using the formulas of trigonometric functions.

The value of the exit angle has a great influence on the possibility of linking the third angle, since this, as an absolute term, decisively determines the type of solutions in the three-dimensional equation. If a triangulation equation has at least one real solution that can be obtained by rational operations or a square root pattern for a given initial angle, that solution is constructive.

Breidenbach formulated as a criterion that the three-second angle can only be interpreted in a rational solution of a three-part equation. If such a solution is not available, the problem of three-part construction is irreconcilable with the compass and ruler. Cluster analysis is a general technique for assembling small groups from a large data set. Similar to discriminant analysis, cluster analysis is also used to classify observations in groups. On the other hand, discriminatory analysis requires knowledge of the group memberships in the cases used to derive the classification rule.

Instruction

The legs are called the sides of a right triangle adjacent to a right angle. In the figure, the legs are designated as AB and BC. Let the lengths of both legs be given. Let's denote them as |AB| and |BC|. In order to find the length of the hypotenuse |AC|, we use the Pythagorean theorem. According to this theorem, the sum of the squares of the legs is equal to the square of the hypotenuse, i.e. in the notation of our drawing |AB|^2 + |BC|^2 = |AC|^2. From the formula we get that the length of the hypotenuse AC is found as |AC| = √(|AB|^2 + |BC|^2) .

Cluster analysis is a more primitive method because it makes no assumptions about the number of groups or group membership. Classification Cluster analysis provides a way to discover potential relationships and create a systematic structure across a large number of variables and observations. Hierarchical cluster analysis is the main statistical method for finding relatively homogeneous clusters of cases based on measured characteristics. It starts with each case as a separate cluster.

The clusters are then merged sequentially, the number of clusters decreasing with each step until only one cluster remains. The clustering method uses differences between objects to form clusters. Hierarchical cluster analysis is best for small samples.

Consider an example. Let the lengths of legs |AB| = 13, |BC| = 21. By the Pythagorean theorem, we get that |AC|^2 = 13^2 + 21^2 = 169 + 441 = 610. from number 610: |AC| = √610. Using the table of squares of integers, we find out that the number 610 is not a perfect square of any integer. In order to get the final value of the length of the hypotenuse, let's try to take out a full square from under the sign of the root. To do this, we decompose the number 610 into factors. 610 \u003d 2 * 5 * 61. According to the table of prime numbers, we see that 61 is a prime number. Therefore, further reduction of the number √610 is impossible. We get the final answer |AC| = √610.
If the square of the hypotenuse were, for example, 675, then √675 = √(3 * 25 * 9) = 5 * 3 * √3 = 15 * √3. If such a cast is possible, perform a reverse check - square the result and compare with the original value.

Hierarchical cluster analysis is just one way to observe the formation of homogeneous variable groups. There is no specific way to set the number of clusters for your analysis. You may need to look at the dendrogram as well as the characteristics of the clusters and then adjust the number in steps to get a good cluster solution.

When variables are measured on different scales, you have three ways to standardize the variables. As a result, all variables with approximately equal proportions contribute to the distance measurement, even if you may lose information about the variance of the variables.

Let us know one of the legs and the angle adjacent to it. For definiteness, let it be the leg |AB| and angle α. Then we can use the formula for the trigonometric function cosine - the cosine of the angle is equal to the ratio of the adjacent leg to the hypotenuse. Those. in our notation cos α = |AB| / |AC|. From here we get the length of the hypotenuse |AC| = |AB| / cosα.
If we know the leg |BC| and angle α, then we use the formula for calculating the sine of the angle - the sine of the angle is equal to the ratio of the opposite leg to the hypotenuse: sin α = |BC| / |AC|. We get that the length of the hypotenuse is found as |AC| = |BC| / cosα.

Euclidean distance: Euclidean distance is the most common measurement method. Squared Euclidean Distance: Squared Euclidean distance focuses attention on objects that are farther apart. City Block Distance: Both city blocks and Euclidean distance are special cases of the Minkowski metric. While Euclidean distance corresponds to the length of the shortest path between two points, city block distance is the sum of the distances along each dimension. Pearson's correlation distance The difference between 1 and the cosine coefficient of two observations The cosine coefficient is the cosine of the angle between the two vectors. Jaccard distance The difference between 1 and the Jacquard coefficient for two observations For binary data, the Jaccard coefficient is equal to the ratio of the amount of overlap and the sum of the two observations. Nearest Neighbor This method assumes that the distance between two clusters corresponds to the distance between features in their nearest neighborhood. Best Neighbor In this method, the distance between two clusters corresponds to the maximum distance between two objects in different clusters. Group Average: With this method, the distance between two clusters corresponds to the average distance between all pairs of objects in different clusters. This method is generally recommended as it contains a higher amount of information. Median This method is identical to the centroid method, except that it is unweighted. Then, for each case, the quadratic Euclidean distance to the cluster means is calculated. The cluster to be merged is the one that increases the sum at least. That is, this method minimizes the increase in the total sum of squared distances within clusters. This method tends to create smaller clusters.

This is a geometric distance in multidimensional space.
It is only suitable for continuous variables.
Cosine Distance The cosine of the angle between two value vectors.
This method is recommended when drawing drawn clusters.
If the drawn clusters form unique "clumps", the method is suitable.
A cluster centroid is a midpoint in a multidimensional space.
It should not be used if the cluster sizes are very different.
Ward Mean values for all variables are computed for each cluster.
These distances are summed for all cases.

The idea is to minimize the distance between the data and the corresponding cluster of clusters.

For clarity, consider an example. Let the length of the leg |AB| = 15. And the angle α = 60°. We get |AC| = 15 / cos 60° = 15 / 0.5 = 30.
Consider how you can check your result using the Pythagorean theorem. To do this, we need to calculate the length of the second leg |BC|. Using the formula for the tangent of the angle tg α = |BC| / |AC|, we obtain |BC| = |AB| * tg α = 15 * tg 60° = 15 * √3. Next, we apply the Pythagorean theorem, we get 15^2 + (15 * √3)^2 = 30^2 => 225 + 675 = 900. The verification is done.

The sine function is defined from the concept of the sine, given that the angle must always be expressed in radians. We can observe several characteristics of the sinusoidal function.

Your domain contains all real.
In this case, the function is said to be periodic, with period 2π.

The cosine function is defined from the concept of cosine, given that the angle must always be expressed in radians.

We can observe several characteristics of the cosine function. Thus, this is a periodic period of 2π. . The restriction does not remove the generality of the formula, because we can always reduce the angles of the second, third, and fourth quadrants to the first. An exercise. - Calculate the sine of 15º without using a calculator.

After calculating the hypotenuse, check whether the resulting value satisfies the Pythagorean theorem.

Sources:

Table of prime numbers from 1 to 10000

Legs name the two short sides of a right triangle that make up its vertex, the value of which is 90 °. The third side in such a triangle is called the hypotenuse. All these sides and angles of the triangle are interconnected by certain relationships that allow you to calculate the length of the leg if several other parameters are known.

Cosine of the sum of two angles

Cosine of the difference of two angles

To get the formula, we can proceed in the same way as in the previous section, but we will see another very simple demonstration based on the Pythagorean theorem. Simplifying and changing the sign, we have Tangent sum and difference of two angles.

An exercise. In today's article, we'll look at a very specific subset: trigonometric functions. To enjoy everything that math has to offer, we must import it. We'll see other import styles in the next article, each with its own advantages and disadvantages. But with this simple instruction, you already have access to the entire math module namespace filled with dozens of functions, including the ones we'll be dealing with today.

Instruction

Use the Pythagorean theorem to calculate the length of the leg (A) if you know the length of the other two sides (B and C) of a right triangle. This theorem states that the sum of the lengths of the legs squared is equal to the square of the hypotenuse. It follows from this that the length of each of the legs is equal to the square root of the difference between the squares of the lengths of the hypotenuse and the second leg: A=√(C²-B²).

Basically, we will need to calculate the sine, cosine and tangent of the angle, as well as its inverse functions. Additionally, we would like to be able to work in both radians and degrees so that we can also use the appropriate conversion functions.

You should keep in mind that these functions expect the argument to be provided in radians, not degrees. To this end, you will be interested to know that you have the following constant. So we can use this expression instead of a numeric value.

There is no direct function for the cosecant, secant and cotangent as this is not necessary as they are simply the inverse of the sine, cosine and tangent respectively. As before, the returned angle is also in radians. Another useful function of mathematics allows us to know the value of the hypotenuse of a right triangle given its legs, which allows us to calculate the square root of the sum of the squares of them.

Use the definition of the direct trigonometric function "sine" for an acute angle, if you know the value of the angle (α) opposite the calculated leg, and the length of the hypotenuse (C). This definition states that the sine of this known angle is equal to the ratio of the length of the desired leg to the length of the hypotenuse. This means that the length of the desired leg is equal to the product of the length of the hypotenuse and the sine of the known angle: A=C∗sin(α). For the same known values, you can use the definition of the cosecant function and calculate the desired length by dividing the length of the hypotenuse by the cosecant of the known angle A=C/cosec(α).

Use the definition of the direct trigonometric function cosine if, in addition to the length of the hypotenuse (C), the value of the acute angle (β) adjacent to the desired leg is also known. The cosine of this angle is defined as the ratio of the lengths of the desired leg and the hypotenuse, and from this we can conclude that the length of the leg is equal to the product of the length of the hypotenuse and the cosine of the known angle: A=C∗cos(β). You can use the definition of the secant function and calculate the desired value by dividing the length of the hypotenuse by the secant of the known angle A=C/sec(β).

Derive the required formula from a similar definition for the derivative of the trigonometric function tangent, if, in addition to the value of the acute angle (α) lying opposite the desired leg (A), the length of the second leg (B) is known. The tangent of the angle opposite the desired leg is the ratio of the length of this leg to the length of the second leg. This means that the desired value will be equal to the product of the length of the known leg and the tangent of the known angle: A=B∗tg(α). From these same known quantities, another formula can be derived using the definition of the cotangent function. In this case, to calculate the length of the leg, it will be necessary to find the ratio of the length of the known leg to the cotangent of the known angle: A=B/ctg(α).

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 parties.

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:

The square of the hypotenuse is equal to the sum of the squares of the legs.

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 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

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

- 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:

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.

An angle is denoted by the corresponding Greek letter.

Hypotenuse A right triangle is the side opposite the 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.

Sinus acute angle α of a right triangle is the ratio opposite catheter to the hypotenuse.
It is denoted as follows: sin α.

Cosine acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is denoted as follows: cos α.

Tangent acute angle α is the ratio of the opposite leg to the adjacent leg.
It is denoted as follows: tg α.

Cotangent acute angle α is the ratio of the adjacent leg to the opposite one.
It is designated as follows: ctg α.

The sine, cosine, tangent and cotangent of an angle depend only on the magnitude of the angle.

Rules:

Basic trigonometric identities in a right triangle:

(α - acute angle opposite the leg b and adjacent to the leg a . Side with - hypotenuse. β - the second acute angle).

b sinα = - c	sin 2 α + cos 2 α = 1
a cosα = - c	1 1 + tg 2 α = -- cos 2 α
b tgα = - a	1 1 + ctg 2 α = -- sin2α
a ctgα = - b	1 1 1 + -- = -- tg 2 α sin 2 α
sinα tgα = -- cosα

As the acute angle increasessinα andtg α increase, andcos α decreases.

For any acute angle α:

sin (90° - α) = cos α

cos (90° - α) = sin α

Explanatory example:

Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.

Find the sine of angle A and the cosine of angle B.

Decision .

1) First, we find the value of angle B. Everything is simple here: since in a right triangle the sum of acute angles is 90º, then angle B \u003d 60º:

B \u003d 90º - 30º \u003d 60º.

2) Calculate sin A. We know that the sine is equal to the ratio of the opposite leg to the hypotenuse. For angle A, the opposite leg is side BC. So:

BC 3 1
sin A = -- = - = -
AB 6 2

3) Now we calculate cos B. We know that the cosine is equal to the ratio of the adjacent leg to the hypotenuse. For angle B, the adjacent leg is the same side BC. This means that we again need to divide BC into AB - that is, perform the same actions as when calculating the sine of angle A:

BC 3 1
cos B = -- = - = -
AB 6 2

The result is:
sin A = cos B = 1/2.

sin 30º = cos 60º = 1/2.

From this it follows that in a right triangle the sine of one acute angle is equal to the cosine of another acute angle - and vice versa. This is exactly what our two formulas mean:
sin (90° - α) = cos α
cos (90° - α) = sin α

Let's check it out again:

1) Let α = 60º. Substituting the value of α into the sine formula, we get:
sin (90º - 60º) = cos 60º.
sin 30º = cos 60º.

2) Let α = 30º. Substituting the value of α into the cosine formula, we get:
cos (90° - 30º) = sin 30º.
cos 60° = sin 30º.

(For more on trigonometry, see the Algebra section)

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