In this article we will show how to give definitions of sine, cosine, tangent and cotangent of an angle and number in trigonometry. Here we will talk about notations, give examples of entries, and give graphic illustrations. In conclusion, let us draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.
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Definition of sine, cosine, tangent and cotangent
Let's see how the idea of sine, cosine, tangent and cotangent is formed in a school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in a right triangle is given. And later trigonometry is studied, which talks about sine, cosine, tangent and cotangent of the angle of rotation and number. Let us present all these definitions, give examples and give the necessary comments.
Acute angle in a right triangle
From the geometry course we know the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle. They are given as the ratio of the sides of a right triangle. Let us give their formulations.
Definition.
Sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse.
Definition.
Cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse.
Definition.
Tangent of an acute angle in a right triangle– this is the ratio of the opposite side to the adjacent side.
Definition.
Cotangent of an acute angle in a right triangle- this is the ratio of the adjacent side to the opposite side.
The designations for sine, cosine, tangent and cotangent are also introduced there - sin, cos, tg and ctg, respectively.
For example, if ABC is a right triangle with right angle C, then the sine of the acute angle A is equal to the ratio of the opposite side BC to the hypotenuse AB, that is, sin∠A=BC/AB.
These definitions allow you to calculate the values of sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from the known values of sine, cosine, tangent, cotangent and the length of one of the sides to find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is equal to 3 and the hypotenuse AB is equal to 7, then we could calculate the value of the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7.
Rotation angle
In trigonometry, they begin to look at the angle more broadly - they introduce the concept of angle of rotation. The magnitude of the rotation angle, unlike an acute angle, is not limited to 0 to 90 degrees; the rotation angle in degrees (and in radians) can be expressed by any real number from −∞ to +∞.
In this light, the definitions of sine, cosine, tangent and cotangent are given not of an acute angle, but of an angle of arbitrary size - the angle of rotation. They are given through the x and y coordinates of the point A 1, to which the so-called starting point A(1, 0) goes after its rotation by an angle α around the point O - the beginning of the rectangular Cartesian coordinate system and the center of the unit circle.
Definition.
Sine of rotation angleα is the ordinate of point A 1, that is, sinα=y.
Definition.
Cosine of the rotation angleα is called the abscissa of point A 1, that is, cosα=x.
Definition.
Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tanα=y/x.
Definition.
Cotangent of the rotation angleα is the ratio of the abscissa of point A 1 to its ordinate, that is, ctgα=x/y.
Sine and cosine are defined for any angle α, since we can always determine the abscissa and ordinate of the point, which is obtained by rotating the starting point by angle α. But tangent and cotangent are not defined for any angle. The tangent is not defined for angles α at which the starting point goes to a point with zero abscissa (0, 1) or (0, −1), and this occurs at angles 90°+180° k, k∈Z (π /2+π·k rad). Indeed, at such angles of rotation, the expression tgα=y/x does not make sense, since it contains division by zero. As for the cotangent, it is not defined for angles α at which the starting point goes to the point with the zero ordinate (1, 0) or (−1, 0), and this occurs for angles 180° k, k ∈Z (π·k rad).
So, sine and cosine are defined for any rotation angles, tangent is defined for all angles except 90°+180°k, k∈Z (π/2+πk rad), and cotangent is defined for all angles except 180° ·k , k∈Z (π·k rad).
The definitions include the designations already known to us sin, cos, tg and ctg, they are also used to designate sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the designations tan and cotcorresponding to tangent and cotangent). So the sine of a rotation angle of 30 degrees can be written as sin30°, the entries tg(−24°17′) and ctgα correspond to the tangent of the rotation angle −24 degrees 17 minutes and the cotangent of the rotation angle α. Recall that when writing the radian measure of an angle, the designation “rad” is often omitted. For example, the cosine of a rotation angle of three pi rad is usually denoted cos3·π.
In conclusion of this point, it is worth noting that when talking about sine, cosine, tangent and cotangent of the angle of rotation, the phrase “angle of rotation” or the word “rotation” is often omitted. That is, instead of the phrase “sine of the rotation angle alpha,” the phrase “sine of the alpha angle” or even shorter, “sine alpha,” is usually used. The same applies to cosine, tangent, and cotangent.
We will also say that the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle are consistent with the definitions just given for sine, cosine, tangent and cotangent of an angle of rotation ranging from 0 to 90 degrees. We will justify this.
Numbers
Definition.
Sine, cosine, tangent and cotangent of a number t is a number equal to the sine, cosine, tangent and cotangent of the rotation angle in t radians, respectively.
For example, the cosine of the number 8·π by definition is a number equal to the cosine of the angle of 8·π rad. And the cosine of an angle of 8·π rad is equal to one, therefore, the cosine of the number 8·π is equal to 1.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is associated with a point on the unit circle with the center at the origin of the rectangular coordinate system, and sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's look at this in more detail.
Let us show how a correspondence is established between real numbers and points on a circle:
- the number 0 is assigned the starting point A(1, 0);
- the positive number t is associated with a point on the unit circle, which we will get to if we move along the circle from the starting point in a counterclockwise direction and walk a path of length t;
- the negative number t is associated with a point on the unit circle, which we will get to if we move along the circle from the starting point in a clockwise direction and walk a path of length |t| .
Now we move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point on the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1) ).
Definition.
Sine of the number t is the ordinate of the point on the unit circle corresponding to the number t, that is, sint=y.
Definition.
Cosine of the number t is called the abscissa of the point of the unit circle corresponding to the number t, that is, cost=x.
Definition.
Tangent of the number t is the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of a number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost.
Definition.
Cotangent of the number t is the ratio of the abscissa to the ordinate of a point on the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is this: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t: ctgt=cost/sint.
Here we note that the definitions just given are consistent with the definition given at the beginning of this paragraph. Indeed, the point on the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point by an angle of t radians.
It is still worth clarifying this point. Let's say we have the entry sin3. How can we understand whether we are talking about the sine of the number 3 or the sine of the rotation angle of 3 radians? This is usually clear from the context, otherwise it is likely not of fundamental importance.
Trigonometric functions of angular and numeric argument
According to the definitions given in the previous paragraph, each angle of rotation α corresponds to a very specific value sinα, as well as the value cosα. In addition, all rotation angles other than 90°+180°k, k∈Z (π/2+πk rad) correspond to tgα values, and values other than 180°k, k∈Z (πk rad ) – values of ctgα . Therefore sinα, cosα, tanα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.
We can speak similarly about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a very specific value sint, as well as cost. In addition, all numbers other than π/2+π·k, k∈Z correspond to values tgt, and numbers π·k, k∈Z - values ctgt.
The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.
It is usually clear from the context whether we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can think of the independent variable as both a measure of the angle (angular argument) and a numeric argument.
However, at school we mainly study numerical functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking specifically about functions, then it is advisable to consider trigonometric functions as functions of numerical arguments.
Relationship between definitions from geometry and trigonometry
If we consider the rotation angle α ranging from 0 to 90 degrees, then the definitions of sine, cosine, tangent and cotangent of the rotation angle in the context of trigonometry are fully consistent with the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's justify this.
Let us depict the unit circle in the rectangular Cartesian coordinate system Oxy. Let's mark the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get point A 1 (x, y). Let us drop the perpendicular A 1 H from point A 1 to the Ox axis.
It is easy to see that in a right triangle, the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of point A 1, that is, |OH|=x, the length of the leg A 1 H opposite to the angle is equal to the ordinate of point A 1, that is, |A 1 H|=y, and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of an acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y. And by definition from trigonometry, the sine of the rotation angle α is equal to the ordinate of point A 1, that is, sinα=y. This shows that determining the sine of an acute angle in a right triangle is equivalent to determining the sine of the rotation angle α when α is from 0 to 90 degrees.
Similarly, it can be shown that the definitions of cosine, tangent and cotangent of an acute angle α are consistent with the definitions of cosine, tangent and cotangent of the rotation angle α.
Bibliography.
- Geometry. 7-9 grades: textbook for general education institutions / [L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev, etc.]. - 20th ed. M.: Education, 2010. - 384 p.: ill. - ISBN 978-5-09-023915-8.
- Pogorelov A.V. Geometry: Textbook. for 7-9 grades. general education institutions / A. V. Pogorelov. - 2nd ed. - M.: Education, 2001. - 224 p.: ill. - ISBN 5-09-010803-X.
- Algebra and elementary functions: Textbook for students of 9th grade of secondary school / E. S. Kochetkov, E. S. Kochetkova; Edited by Doctor of Physical and Mathematical Sciences O. N. Golovin. - 4th ed. M.: Education, 1969.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Mordkovich A. G. Algebra and the beginnings of analysis. Grade 10. In 2 parts. Part 1: textbook for general education institutions (profile level) / A. G. Mordkovich, P. V. Semenov. - 4th ed., add. - M.: Mnemosyne, 2007. - 424 p.: ill. ISBN 978-5-346-00792-0.
- Algebra and the beginning of mathematical analysis. 10th grade: textbook. for general education institutions: basic and profile. levels /[Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; edited by A. B. Zhizhchenko. - 3rd ed. - I.: Education, 2010.- 368 p.: ill.- ISBN 978-5-09-022771-1.
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Instructions
Video on the topic
note
When calculating the sides of a right triangle, knowledge of its characteristics can play a role:
1) If the leg of a right angle lies opposite an angle of 30 degrees, then it is equal to half the hypotenuse;
2) The hypotenuse is always longer than any of the legs;
3) If a circle is circumscribed around a right triangle, then its center must lie in the middle of the hypotenuse.
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 size of one of the acute angles of the triangle.
Instructions
Let us know one of the legs and the angle adjacent to it. To be specific, let these be the side |AB| and angle α. Then we can use the formula for the trigonometric cosine - cosine ratio of the adjacent leg to. Those. in our notation cos α = |AB| / |AC|. From this we obtain the length of the hypotenuse |AC| = |AB| / cos α.
If we know the side |BC| and angle α, then we will use the formula to calculate 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 find that the length of the hypotenuse is |AC| = |BC| / cos α.
For clarity, let's look at an example. Let the length of the leg |AB| be given. = 15. And angle α = 60°. We get |AC| = 15 / cos 60° = 15 / 0.5 = 30.
Let's look at 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 tan α = |BC| / |AC|, we get |BC| = |AB| * tan α = 15 * tan 60° = 15 * √3. Next, we apply the Pythagorean theorem, we get 15^2 + (15 * √3)^2 = 30^2 => 225 + 675 = 900. Check completed.
Helpful advice
After calculating the hypotenuse, check whether the resulting value satisfies the Pythagorean theorem.
Sources:
- Table of prime numbers from 1 to 10000
Legs are the two short sides of a right triangle that make up the vertex whose size 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 make it possible to calculate the length of the leg if several other parameters are known.
Instructions
Use the Pythagorean theorem for leg (A) if you know the length of the other two sides (B and C) of the right triangle. This theorem states that the sum of the squared lengths of the legs is equal to the square of the hypotenuse. It follows from this that the length of each leg is equal to the square root of the lengths of the hypotenuse and the second leg: A=√(C²-B²).
Use the definition of the direct trigonometric function “sine” for an acute angle if you know the magnitude of the angle (α) lying opposite the leg being calculated and the length of the hypotenuse (C). This states that the sine of this known 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 quantities, you can also use the cosecant and calculate the required 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 cosine function if, in addition to the length of the hypotenuse (C), the magnitude of the acute angle (β) adjacent to the desired one is also known. The cosine of this angle is 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 to 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 if we use 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).
Video on the topic
The word “kathet” came into Russian from Greek. In exact translation, it means a plumb line, that is, perpendicular to the surface of the earth. In mathematics, legs are the sides that form a right angle of a right triangle. The side opposite this angle is called the hypotenuse. The term “cathet” is also used in architecture and welding technology.
The secant of this angle is obtained by dividing the hypotenuse by the adjacent leg, that is, secCAB = c/b. The result is the reciprocal of the cosine, that is, it can be expressed using the formula secCAB=1/cosSAB.
The cosecant is equal to the quotient of the hypotenuse divided by the opposite side and is the reciprocal of the sine. It can be calculated using the formula cosecCAB=1/sinCAB
Both legs are connected to each other and by a cotangent. In this case, the tangent will be the ratio of side a to side b, that is, the opposite side to the adjacent side. This relationship can be expressed by the formula tgCAB=a/b. Accordingly, the inverse ratio will be the cotangent: ctgCAB=b/a.
The relationship between the sizes of the hypotenuse and both legs was determined by the ancient Greek Pythagoras. People still use the theorem and his name. It says that the square of the hypotenuse is equal to the sum of the squares of the legs, that is, c2 = a2 + b2. Accordingly, each leg will be equal to the square root of the difference between the squares of the hypotenuse and the other leg. This formula can be written as b=√(c2-a2).
The length of the leg can also be expressed through the relationships known to you. According to the theorems of sines and cosines, a leg is equal to the product of the hypotenuse and one of these functions. It can be expressed as and or cotangent. Leg a can be found, for example, using the formula a = b*tan CAB. In exactly the same way, depending on the given tangent or , the second leg is determined.
The term "cathet" is also used in architecture. It is applied to the Ionic capital and plumb through the middle of its back. That is, in this case, this term is perpendicular to a given line.
In welding technology there is a “fillet weld leg”. As in other cases, this is the shortest distance. Here we are talking about the gap between one of the parts being welded to the border of the seam located on the surface of the other part.
Video on the topic
Sources:
- what are leg and hypotenuse in 2019
Sinus acute angle α of a right triangle is the ratio opposite leg to hypotenuse.
It is denoted as follows: sin α.
Cosine The acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is designated as follows: cos α.
Tangent acute angle α is the ratio of the opposite side to the adjacent side.
It is designated as follows: tg α.
Cotangent acute angle α is the ratio of the adjacent side to the opposite side.
It is designated as follows: ctg α.
The sine, cosine, tangent and cotangent of an angle depend only on the size of the angle.
Rules:
Basic trigonometric identities in a right triangle:
(α – acute angle opposite to the leg b and adjacent to the leg a . Side With – hypotenuse. β – second acute angle).
b | sin 2 α + cos 2 α = 1 | |
a | 1 | |
b | 1 | |
a | 1 1 | |
sin α |
As the acute angle increasessin α andtan α increase, andcos α decreases.
For any acute angle α:
sin (90° – α) = cos α
cos (90° – α) = sin α
Example-explanation:
Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.
Let's find out 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 the acute angles is 90º, then angle B = 60º:
B = 90º – 30º = 60º.
2) Let's calculate sin A. We know that the sine is equal to the ratio of the opposite side to the hypotenuse. For angle A, the opposite side is side BC. So:
BC 3 1
sin A = -- = - = -
AB 6 2
3) Now let's 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 by 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.
It follows from this 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 make sure of this 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 information about trigonometry, see the Algebra section)
Average level
Right triangle. The Complete Illustrated Guide (2019)
RIGHT TRIANGLE. FIRST LEVEL.
In problems, the right angle is not at all necessary - the lower left, so you need to learn to recognize a right triangle in this form,
and in this
and in this 
What's good about a right triangle? Well..., firstly, there are special beautiful names for its sides.
Attention to the drawing!
Remember and don't confuse: there are two legs, and there is only one hypotenuse(one and only, unique and longest)!
Well, we’ve 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 a lot of benefit to those who know it. And the best thing about it is that it is simple.
So, Pythagorean theorem:
Do you remember the joke: “Pythagorean pants are equal on all sides!”?
Let's draw these same Pythagorean pants and look at them. 
Doesn't it look like some kind of 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, or more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:
"Sum areas of squares, built on the legs, is equal to square area, built on the hypotenuse."
Does it really sound a little different? And so, when Pythagoras drew the statement of his theorem, this is exactly the picture that came 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 children can better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty came up with 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 remember everything in words??! And we can rejoice that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to remember it better:
It should be easy now:
| The square of the hypotenuse is equal to the sum of the squares of the legs. |
Well, the most important theorem about right triangles has been discussed. If you are interested in how it is proven, read the following levels of theory, and now let's go further... 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 I really don’t want to, do I? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:
Why is everything just 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.
Actually it sounds like this:
What about the angle? Is there a leg that is opposite the corner, that is, an opposite (for an angle) leg? Of course have! This is a leg!
What about the angle? Look carefully. Which leg is adjacent to the corner? Of course, the leg. This means that for the angle the leg is adjacent, and
Now, pay attention! Look what we got:
See how cool it is:
Now let's move on to tangent and cotangent.
How can I write this down in words now? What is the leg in relation to the angle? Opposite, of course - it “lies” opposite the corner. What about the leg? Adjacent to the corner. So what have we got?
See how the numerator and denominator have swapped places?
And now the corners again and made an exchange:
Summary
Let's briefly write down everything we've learned.
 |
Pythagorean theorem: |
The main theorem about right triangles is the Pythagorean theorem.
Pythagorean theorem
By the way, do you remember well what legs and hypotenuse are? If not very good, 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 can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.
See how cleverly we divided its sides into lengths and!
Now let's connect the marked dots
Here we, however, noted something else, but you yourself look at the drawing and think why this is so.
What is the area of the larger square? Right, . What about a smaller area? Certainly, . The total area of the four corners remains. Imagine that we took them two at a time and leaned them against each other with their hypotenuses. What happened? Two rectangles. This means that the area of the “cuts” 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 side 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 side to the adjacent side.
The cotangent of an acute angle is equal to the ratio of the adjacent side to the opposite side.
And once again all this in the form of a tablet:
It is very comfortable!
Signs of equality of right triangles
I. On two sides
II. By leg and hypotenuse
III. By hypotenuse and acute angle
IV. Along the leg and acute angle
a) 
b) 
Attention! It is very important here that the legs are “appropriate”. For example, if it goes like this:
THEN 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 it was opposite.
Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Take a look at the topic “and pay attention to the fact that for equality of “ordinary” triangles, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides. But for the equality of right triangles, only two corresponding elements are enough. Great, right?
The situation is approximately the same with the signs of similarity of right triangles.
Signs of similarity of right triangles
I. Along an acute angle
II. On two sides
III. By leg and hypotenuse
Median in a right triangle
Why is this so?
Instead of a right triangle, consider a whole rectangle.
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 turned out that
- - median:
Remember this fact! Helps a lot!
What’s even more surprising is that the opposite is also true.
What good can be obtained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture
Look carefully. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But there is only one point in the triangle, the distances from which from all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCLE. So what happened?
So let's start with this “besides...”.
Let's look at and.
But similar triangles have all equal angles!
The same can be said about and
Now let's draw it together:
What benefit can be derived from this “triple” similarity?
Well, for example - two formulas for the height of a right triangle.
Let us write down the relations of the corresponding parties:
To find the height, we solve the proportion and get the 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:
You need to remember both of these formulas very well and use the one that is more convenient. 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 sides:
- by leg and hypotenuse: or
- along the leg and 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 acute corner: or
- from the proportionality of 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 side 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 side to the adjacent side:
- The cotangent of an acute angle of a right triangle is the ratio of the adjacent side to the opposite side: .
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:
- via legs: