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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 right triangle. Pythagoras proved it in perfect time immemorial, and since then she has brought many benefits to those who know her. 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 squares of 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 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? Of course, . 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:

Sinus acute angle 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 peaks triangles are 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 right angle, is equal to half of the hypotenuse: .

Area of ​​a right triangle:

• through the catheters:

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:

Main 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.

Solution .

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)

Lecture: Sine, cosine, tangent, cotangent of an arbitrary angle

Sine, cosine of an arbitrary angle

To understand what is trigonometric functions, we turn to a circle with a unit radius. Given circle is centered at the origin in the coordinate plane. To determine the given functions, we will use the radius vector OR, which starts at the center of the circle, and the point R is a point on the circle. This radius vector forms an angle alpha with the axis OH. Since the circle has a radius equal to one, then OR = R = 1.

If from the point R drop a perpendicular on the axis OH, then we get a right triangle with hypotenuse equal to one.

If the radius vector moves clockwise, then this direction is called negative, but if it moves counter-clockwise - positive.

The sine of an angle OR, is the ordinate of the point R vectors on a circle.

That is, to obtain the value of the sine of a given angle alpha, it is necessary to determine the coordinate At on surface.

How given value was received? Since we know that the sine of an arbitrary angle in a right triangle is the ratio of the opposite leg to the hypotenuse, we get that

And since R=1, then sin(α) = y 0 .

In the unit circle, the ordinate value cannot be less than -1 and greater than 1, which means that

Sinus accepts positive value in the first and second quarters of the unit circle, and negative in the third and fourth.

Cosine of an angle given circle formed by the radius vector OR, is the abscissa of the point R vectors on a circle.

That is, to obtain the value of the cosine of a given angle alpha, it is necessary to determine the coordinate X on surface.

The cosine of an arbitrary angle in a right triangle is the ratio of the adjacent leg to the hypotenuse, we get that

And since R=1, then cos(α) = x 0 .

In the unit circle, the value of the abscissa cannot be less than -1 and greater than 1, which means that

The cosine is positive in the first and fourth quadrants of the unit circle, and negative in the second and third.

tangentarbitrary angle the ratio of sine to cosine is calculated.

If we consider a right triangle, then this is the ratio of the opposite leg to the adjacent one. If we are talking about the unit circle, then this is the ratio of the ordinate to the abscissa.

Judging by these relationships, it can be understood that the tangent cannot exist if the value of the abscissa is zero, that is, at an angle of 90 degrees. The tangent can take all other values.

The tangent is positive in the first and third quarters of the unit circle, and negative in the second and fourth.

Reference data for tangent (tg x) and cotangent (ctg x). Geometric definition, properties, graphs, formulas. Table of tangents and cotangents, derivatives, integrals, series expansions. Expressions through complex variables. Connection with hyperbolic functions.

Geometric definition

|BD| - the length of the arc of a circle centered at point A.
α is the angle expressed in radians.

Tangent ( tgα) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the adjacent leg |AB| .

Cotangent ( ctgα) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| .

Tangent

Where n- whole.

AT Western literature tangent is defined as follows:
.
;
;
.

Graph of the tangent function, y = tg x

Cotangent

Where n- whole.

In Western literature, the cotangent is denoted as follows:
.
The following notation has also been adopted:
;
;
.

Graph of the cotangent function, y = ctg x

Properties of tangent and cotangent

Periodicity

Functions y= tg x and y= ctg x are periodic with period π.

Parity

The functions tangent and cotangent are odd.

Domains of definition and values, ascending, descending

The functions tangent and cotangent are continuous on their domain of definition (see the proof of continuity). The main properties of the tangent and cotangent are presented in the table ( n- integer).

	y= tg x	y= ctg x
Scope and continuity
Range of values	-∞ < y < +∞	-∞ < y < +∞
Ascending		-
Descending	-
Extremes	-	-
Zeros, y= 0
Points of intersection with the y-axis, x = 0	y= 0	-

Formulas

Expressions in terms of sine and cosine

; ;
; ;
;

Formulas for tangent and cotangent of sum and difference

The rest of the formulas are easy to obtain, for example

Product of tangents

The formula for the sum and difference of tangents

This table shows the values ​​of tangents and cotangents for some values ​​of the argument.

Expressions in terms of complex numbers

Expressions in terms of hyperbolic functions

;
;

Derivatives

; .

.
Derivative of the nth order with respect to the variable x of the function :
.
Derivation of formulas for tangent > > > ; for cotangent > > >

Integrals

Expansions into series

To get the expansion of the tangent in powers of x, you need to take several terms of the expansion in power series for functions sin x and cos x and divide these polynomials into each other , . This results in the following formulas.

At .

at .
where B n- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
where .
Or according to the Laplace formula:

Inverse functions

The inverse functions to tangent and cotangent are arctangent and arccotangent, respectively.

Arctangent, arctg

, where n- whole.

Arc tangent, arcctg

, where n- whole.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.
G. Korn, Handbook of Mathematics for Researchers and Engineers, 2012.

In this article, we will take a comprehensive look at . Basic trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, and allow you to find any of these trigonometric functions through a known other.

We immediately list the main trigonometric identities, which we will analyze in this article. We write them down in a table, and below we give the derivation of these formulas and give the necessary explanations.

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Relationship between sine and cosine of one angle

Sometimes they talk not about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind . The explanation for this fact is quite simple: the equalities are obtained from the basic trigonometric identity after dividing both of its parts by and respectively, and the equalities and follow from the definitions of sine, cosine, tangent, and cotangent. We will discuss this in more detail in the following paragraphs.

That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.

Before proving the basic trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.

The basic trigonometric identity is very often used in transformation of trigonometric expressions. It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often, the basic trigonometric identity is used in reverse order: the unit is replaced by the sum of the squares of the sine and cosine of any angle.

Tangent and cotangent through sine and cosine

Identities connecting the tangent and cotangent with the sine and cosine of one angle of the form and immediately follow from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, the sine is the ordinate of y, the cosine is the abscissa of x, the tangent is the ratio of the ordinate to the abscissa, that is, , and the cotangent is the ratio of the abscissa to the ordinate, that is, .

Due to this obviousness of the identities and often the definitions of tangent and cotangent are given not through the ratio of the abscissa and the ordinate, but through the ratio of the sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.

To conclude this section, it should be noted that the identities and hold for all such angles for which the trigonometric functions in them make sense. So the formula is valid for any other than (otherwise the denominator will be zero, and we did not define division by zero), and the formula - for all , different from , where z is any .

Relationship between tangent and cotangent

An even more obvious trigonometric identity than the previous two is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it takes place for any angles other than , otherwise either the tangent or the cotangent is not defined.

Proof of the formula very simple. By definition and from where . The proof could have been carried out in a slightly different way. Since and , then .

So, the tangent and cotangent of one angle, at which they make sense, is.