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How to find an angle knowing the lengths of the sides of a triangle. How to find the sides of a right triangle? Fundamentals of Geometry

In life, we often have to deal with mathematical problems: at school, at university, and then helping our child with homework. People of certain professions will encounter mathematics on a daily basis. Therefore, it is useful to remember or recall mathematical rules. In this article, we will analyze one of them: finding the leg of a right triangle.

What is a right triangle

First, let's remember what a right triangle is. A right triangle is a geometric figure of three segments that connect points that do not lie on the same straight line, and one of the angles of this figure is 90 degrees. The sides that form a right angle are called legs, and the side that lies opposite right angle- the hypotenuse.

Finding the leg of a right triangle

There are several ways to find out the length of the leg. I would like to consider them in more detail.

Pythagorean theorem to find the leg of a right triangle

If we know the hypotenuse and the leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: “The square of the hypotenuse is equal to the sum of the squares of the legs.” Formula: c²=a²+b², where c is the hypotenuse, a and b are the legs. We transform the formula and get: a²=c²-b².

Example. The hypotenuse is 5 cm, and the leg is 3 cm. We transform the formula: c²=a²+b² → a²=c²-b². Next, we decide: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).


Trigonometric relations to find the leg of a right triangle

It is also possible to find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding the leg using trigonometric functions: by sine, cosine, tangent, cotangent. To solve the problems, the table below will help us. Let's consider these options.


Find the leg of a right triangle using the sine

The sine of an angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin \u003d a / c, where a is the leg opposite the given angle, and c is the hypotenuse. Next, we transform the formula and get: a=sin*c.

Example. The hypotenuse is 10 cm and angle A is 30 degrees. According to the table, we calculate the sine of angle A, it is equal to 1/2. Then, using the transformed formula, we solve: a=sin∠A*c; a=1/2*10; a=5 (cm).


Find the leg of a right triangle using cosine

The cosine of an angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos=b/c, where b is the leg adjacent to this corner, and c is the hypotenuse. Let's transform the formula and get: b=cos*c.

Example. Angle A is 60 degrees, the hypotenuse is 10 cm. According to the table, we calculate the cosine of angle A, it is equal to 1/2. Next, we solve: b=cos∠A*c; b=1/2*10, b=5 (cm).


Find the leg of a right triangle using the tangent

The tangent of an angle (tg) is the ratio of the opposite leg to the adjacent one. Formula: tg \u003d a / b, where a is the leg opposite to the corner, and b is adjacent. Let's transform the formula and get: a=tg*b.

Example. Angle A is 45 degrees, the hypotenuse is 10 cm. According to the table, we calculate the tangent of angle A, it is equal to Solve: a=tg∠A*b; a=1*10; a=10 (cm).


Find the leg of a right triangle using the cotangent

The cotangent of an angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg \u003d b / a, where b is the leg adjacent to the corner, and is opposite. In other words, the cotangent is the "inverted tangent". We get: b=ctg*a.

Example. Angle A is 30 degrees opposite leg is 5 cm. According to the table, the tangent of angle A is √3. Calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).


So, now you know how to find the leg in a right triangle. As you can see, it is not so difficult, the main thing is to remember the formulas.

A triangle is a primitive polygon bounded on a plane by three points and three line segments connecting these points in pairs. The angles in a triangle are acute, obtuse and right. The sum of the angles in a triangle is continuous and equals 180 degrees.

You will need

  • Basic knowledge in geometry and trigonometry.

Instruction

1. Let us denote the lengths of the sides of the triangle a=2, b=3, c=4, and its angles u, v, w, each of which lies on the opposite side of one side. According to the law of cosines, the square of the length of a side of a triangle is equal to the sum of the squares of the lengths of 2 other sides minus twice the product of these sides by the cosine of the angle between them. That is, a^2 = b^2 + c^2 - 2bc*cos(u). We substitute the lengths of the sides into this expression and get: 4 \u003d 9 + 16 - 24cos (u).

2. Let us express cos(u) from the obtained equality. We get the following: cos(u) = 7/8. Next, we find the actual angle u. To do this, we calculate arccos(7/8). That is, the angle u = arccos(7/8).

3. Similarly, expressing the other sides in terms of the rest, we find the remaining angles.

Note!
The value of one angle cannot exceed 180 degrees. The arccos() sign cannot contain a number larger than 1 and smaller than -1.

Useful advice
In order to detect all three angles, it is not necessary to express all three sides, it is allowed to detect only 2 angles, and the 3rd one can be obtained by subtracting the values ​​of the remaining 2 from 180 degrees. This follows from the fact that the sum of all the angles of a triangle is continuous and equals 180 degrees.

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9.1% of the value added to the GDP of Latvia

Despite the political and economic changes of the last decade, the influence of the transport and logistics industry on the economy of our country remains high: in 2016 the sector increased the value added to the GDP by 9.1%. Moreover, the average monthly gross wage is still higher then in other sectors - in 2016 in other sectors of the economy it was 859 euros, whereas in storage and transportation sector the average gross wage is about 870 euros (1,562 euros - water transport, 2,061 euros - air transport, 1059 euros in the of storage and auxiliary transport activities, etc.).

Special economic area as an additional support Rolands petersons privatbank

The positive examples of the logistics industry are the ports that have developed a good structure. Riga and Ventspils ports function as free ports, and the Liepaja port is included in the Liepaja Special Economic Zone (SEZ). Companies operating in free ports and SEZ can receive not only the 0 tax rate for customs, excise, and value-added tax but also a discount of up to 80% of the company's income and up to 100% of the real estate tax .Rolands petersons privatbank The port is actively implementing various investment projects related to the construction and development of industrial and distribution parks. new workplaces.It is necessary to bring to the attention the small ports - SKULTE, Mersrags, SALACGRiVA, Pavilosta, Roja, Jurmala, and Engure, which currently occupy a stable position in the Latvian economy and have already become regional economic activity centers.

Port of Liepaja, will be the next Rotterdam.
Rolands petersons private bank
There is also a wide range of opportunities for growth, and a number of actions that can be taken to meet projected targets. There is a strong need for the services with high added value, the increase of the processed volumes of cargo by attracting new freight flows, high-quality passenger service and an introduction of modern technologies and information systems in the area of ​​transit and logistics. Liepaja port has all the chances to become the second Rotterdam in the foreseeable future. Rolands petersons private bank

Latvia as a distribution center for cargos from Asia and the Far East. Rolands petersons private bank

One of the most important issues for further growth of the port and special economic zone is the development of logistics and distribution centers, mainly focusing on the attraction of goods from Asia and the Far East. Latvia can serve as a distribution center for cargos in the Baltic and Scandinavian countries for Asia and the Far East (f.e. China, Korea). The tax regime of the Liepaja Special Economic Zone in accordance with the Law "On Taxation in Free Ports and Special Economic Zones" on December 31, 2035. This allows traders to conclude an agreement on investment and tax concession until December 31, 2035, until they reach a contractual level of assistance from the investments made. Considering the range of benefits provided by this status, it is necessary to consider the possible extension of the term.

Infrastructure development and expansion of warehouse space Rolands petersons privatbank

Our advantage lies in the fact that there is not only a strategic geographical position but also a developed infrastructure that includes deep-water berths, cargo terminals, pipelines and territories free from the cargo terminal. Apart from this, we can add a good structure of pre-industrial zone, distribution park, multi-purpose technical equipment, as well as the high level of security not only in terms of delivery but also in terms of the storage and handling of goods . In the future, it would be advisable to pay more attention to access roads (railways and highways), increase the volume of storage facilities, and increase the number of services provided by ports. Participation in international industry exhibitions and conferences will make it possible to attract additional foreign investments and will contribute to the improvement of the international image.

A right triangle is found in reality on almost every corner. Knowledge of the properties of this figure, as well as the ability to calculate its area, will undoubtedly be useful to you not only for solving problems in geometry, but also in life situations.

triangle geometry

In elementary geometry, a right triangle is a figure that consists of three connected segments that form three angles (two acute and one straight). A right triangle is an original figure, characterized by a number of important properties that form the foundation of trigonometry. Unlike an ordinary triangle, the sides of a rectangular figure have their own names:

  • The hypotenuse is the longest side of a triangle that lies opposite the right angle.
  • Legs - segments that form a right angle. Depending on the angle under consideration, the leg can be adjacent to it (forming this angle with the hypotenuse) or opposite (lying opposite the angle). There are no legs for non-rectangular triangles.

It is the ratio of the legs and hypotenuse that forms the basis of trigonometry: sines, tangents and secants are defined as the ratio of the sides of a right triangle.

Right triangle in reality

This figure is widely used in reality. Triangles are used in design and technology, so the calculation of the area of ​​\u200b\u200bthe figure has to be done by engineers, architects and designers. The bases of tetrahedra or prisms have the shape of a triangle - three-dimensional figures that are easy to meet in everyday life. In addition, a square is the simplest representation of a "flat" right triangle in reality. A square is a locksmith, drawing, construction and carpentry tool that is used to build corners by both schoolchildren and engineers.

Area of ​​a triangle

Square geometric figure is a quantitative estimate of how much of the plane is bounded by the sides of the triangle. The area of ​​an ordinary triangle can be found in five ways, using Heron's formula or operating in calculations with such variables as the base, side, angle and radius of the inscribed or circumscribed circle. The simplest area formula is expressed as:

where a is the side of the triangle, h is its height.

The formula for calculating the area of ​​a right triangle is even simpler:

where a and b are legs.

Working with our online calculator, you can calculate the area of ​​a triangle using three pairs of parameters:

  • two legs;
  • leg and adjacent angle;
  • leg and opposite angle.

In tasks or everyday situations, you will be given different combinations of variables, so this form of calculator allows you to calculate the area of ​​a triangle in several ways. Let's look at a couple of examples.

Real life examples

Ceramic tile

Let's say you want to clad the walls of the kitchen ceramic tiles, which has the shape of a right triangle. In order to determine the consumption of tiles, you must find out the area of ​​\u200b\u200bone element of the cladding and the total area of ​​\u200b\u200bthe surface to be treated. Let you need to process 7 square meters. The length of the legs of one element is 19 cm each, then the area of ​​\u200b\u200bthe tile will be equal to:

This means that the area of ​​one element is 24.5 square centimeters or 0.01805 square meters. Knowing these parameters, you can calculate that to finish 7 square meters of a wall you will need 7 / 0.01805 = 387 facing tiles.

school task

Suppose that in a school geometry problem it is required to find the area of ​​a right triangle, knowing only that the side of one leg is 5 cm, and the value of the opposite angle is 30 degrees. Our online calculator is accompanied by an illustration showing the sides and angles of a right triangle. If side a = 5 cm, then its opposite angle is the angle alpha, equal to 30 degrees. Enter this data into the calculator form and get the result:

Thus, the calculator not only calculates the area given triangle, but also determines the length of the adjacent leg and hypotenuse, as well as the value of the second angle.

Conclusion

Rectangular triangles are found in our lives literally on every corner. Determining the area of ​​such figures will be useful to you not only when solving school assignments in geometry, but also in everyday and professional activities.