Calculate the angle given the length of the sides. How to find the sides of a right triangle? Fundamentals of Geometry
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.
whose side lengths (a, b, c) are known, use the cosine theorem. She states that the square of the length of any of the sides is equal to the sum of the squares of the lengths of the other two, from which the double product of the lengths of the same two sides and the cosine of the angle between them is subtracted. You can use this theorem to calculate the angle at any of the vertices, it is important to know only its location relative to the sides. For example, to find the angle α that lies between sides b and c, the theorem must be written as follows: a² = b² + c² - 2*b*c*cos(α).
Express the cosine of the desired angle from the formula: cos(α) = (b²+c²-a²)/(2*b*c). Apply the inverse cosine function to both parts of the equation - the arc cosine. It allows you to restore the value of the angle in degrees by the value of the cosine: arccos(cos(α)) = arccos((b²+c²-a²)/(2*b*c)). The left side can be simplified and the calculation of the angle between sides b and c will take on the final form: α = arccos((b²+c²-a²)/2*b*c).
When finding the magnitudes of acute angles in a right triangle, knowing the lengths of all sides is not necessary, two of them are enough. If these two sides are legs (a and b), divide the length of the one that lies opposite the desired angle (α) by the length of the other. So you get the value of the tangent of the desired angle tg (α) = a / b, and applying the inverse function - arc tangent to both parts of the equality - and simplifying, as in the previous step, the left side, derive the final formula: α = arctg (a / b ).
If the known sides are the leg (a) and the hypotenuse (c), to calculate the angle (β) formed by these sides, use the cosine function and its inverse - the arc cosine. The cosine is determined by the ratio of the leg length to the hypotenuse, and the final formula can be written as follows: β = arccos(a/c). To calculate the same initial acute angle (α) lying opposite the known leg, use the same ratio, replacing the arccosine with the arcsine: α = arcsin(a/c).
Sources:
- triangle formula with 2 sides
Tip 2: How to find the angles of a triangle by the lengths of its sides
There are several options for finding the values of all angles in a triangle, if the lengths of its three are known. parties. One way is to use two different area formulas triangle. To simplify the calculations, you can also apply the sine theorem and the theorem on the sum of angles triangle.
Instruction
Use, for example, two formulas for calculating the area triangle, one of which involves only three of his known parties s (Gerona), and in the other - two parties s and the sine of the angle between them. Using different pairs in the second formula parties, you can determine the magnitude of each of the angles triangle.
Solve the problem in general terms. Heron's formula determines the area triangle, as the square root of the product of the semiperimeter (half of all parties) on the difference between the semiperimeter and each of parties. If we replace the sum parties, then the formula can be written as follows: S=0.25∗√(a+b+c)∗(b+c-a)∗(a+c-b)∗(a+b-c).C another parties s area triangle can be expressed as half the product of its two parties by the sine of the angle between them. For example, for parties a and b with an angle γ between them, this formula can be written as follows: S=a∗b∗sin(γ). Replace the left side of the equation with Heron's formula: 0.25∗√(a+b+c)∗(b+c-a)∗(a+c-b)∗(a+b-c)=a∗b∗sin(γ). Derive from this equation the formula for
More precisely, from the very name of the “right-angled” triangle, it becomes clear that one angle in it is 90 degrees. The remaining angles can be found by recalling simple theorems and the properties of triangles.
You will need
- Table of sines and cosines, Bradis table
Instruction
1. Let's denote the angles of the triangle with the letters A, B and C, as shown in the figure. Angle BAC is equal to 90º, the other two angles are denoted by letters α and β. The legs of the triangle will be denoted by the letters a and b, and the hypotenuse by the letter c.
2. Then sinα = b/c, and cosα = a/c. Similarly for the second acute angle of the triangle: sinβ = a/c, and cosβ = b/c. Depending on which sides we know, we calculate the sines or cosines of the angles and we look at the Bradis table for the value of α and β.
3. Having found one of the corners, it is allowed to recall that the sum internal corners triangle is 180º. This means that the sum of α and β is equal to 180º - 90º = 90º. Then, having calculated the value for α from the tables, we can use the following formula to find β: β = 90º - α
4. If one of the sides of the triangle is unfamiliar, then we apply the Pythagorean theorem: a² + b² = c². We derive from it an expression for an unfamiliar side through the other two and substitute it into the formula for finding the sine or cosine of one of the angles.
Tip 2: How to find the hypotenuse in a right triangle
The hypotenuse is the side in a right triangle that lies opposite the right angle. The hypotenuse is the longest side in a right triangle. The remaining sides in a right triangle are called legs.
You will need
- Basic knowledge of geometry.
Instruction
1. The square of the length of the hypotenuse is equal to the sum of the squares of the legs. That is, in order to find the square of the length of the hypotenuse, you need to square the length of the legs and add.
2. The length of the hypotenuse is equal to the square root of the square of its length. In order to find its length, we extract the square root of a number equal to the sum of the squares of the legs. The resulting number will be the length of the hypotenuse.
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Note!
The length of the hypotenuse is correct, so when extracting the root, the radical expression must be larger than zero.
Useful advice
In an isosceles right triangle, the length of the hypotenuse can be calculated by multiplying the leg by the root of 2.
Tip 3: How to detect an acute angle in a right triangle
Straight carbonic the triangle is perhaps one of the most famous, from a historical point of view, geometric shapes. Pythagorean “trousers” can only compete with “Eureka!” Archimedes.
You will need
- - drawing of a triangle;
- - ruler;
- - protractor.
Instruction
1. As usual, the vertices of the corners of a triangle are indicated by capital Latin letters (A, B, C), and the opposite sides by small Latin letters (a, b, c) or by the names of the triangle vertices that form this side (AC, BC, AB).
2. The sum of the angles of a triangle is 180 degrees. in a rectangular triangle one angle (right) will invariably be 90 degrees, and the rest will be acute, i.e. less than 90 degrees all. In order to determine which angle in a rectangular triangle is straight, measure the sides of the triangle with the help of a ruler and determine the largest. It is called the hypotenuse (AB) and is located opposite the right angle (C). The remaining two sides form a right angle and are called legs (AC, BC).
3. Once you have determined which angle is acute, you can either measure the angle with a protractor, or calculate with the support of mathematical formulas.
4. In order to determine the value of the angle with the support of the protractor, align its top (denoted by the letter A) with a special mark on the ruler in the center of the protractor, the AC leg must coincide with its upper edge. Mark on the semicircular part of the protractor the point through which the hypotenuse AB passes. The value at this point corresponds to the angle value in degrees. If 2 values \u200b\u200bare indicated on the protractor, then for an acute angle it is necessary to choose a smaller one, for a blunt one - a large one.
6. Find the resulting value in the Bradis reference tables and determine which angle the resulting numerical value corresponds to. Our grandmothers used this method.
7. Nowadays, it’s enough to take a calculator with a function for calculating trigonometric formulas. Let's say the built-in Windows calculator. Launch the “Calculator” application, in the “View” menu item, select the “Engineering” item. Calculate the sine of the desired angle, say sin(A) = BC/AB = 2/4 = 0.5
8. Switch the calculator to the inverse function mode by clicking on the INV button on the calculator display, then click on the button for calculating the arcsine function (marked as sin to the minus one degree on the display). A further inscription will appear in the calculation window: asind (0.5) = 30. That is, the value of the desired angle is 30 degrees.
Tip 4: How to find the unknown side in a triangle
The method for calculating the unknown side of a triangle depends not only on the conditions of the assignment, but also on what it is done for. A similar task is faced not only by schoolchildren in geometry lessons, but also by engineers working in various industries, interior designers, cutters and representatives of many other professions. The accuracy of calculations for different purposes may be different, but their rule remains the same as in the school problem book.
You will need
- – a triangle with given parameters;
- - calculator;
- - pen;
- - pencil;
- - protractor;
- - paper;
- - a computer with AutoCAD software;
- - theorems of sines and cosines.
Instruction
1. Draw a triangle corresponding to the conditions of the task. A triangle can be built on three sides, two sides and an angle between them, or a side and two adjacent angles. The thesis of work in a notebook and on a computer in the AutoCAD program is identical in this regard. So in the task it is strictly necessary to indicate the dimensions of one or 2 sides and one or 2 corners.
2. When building on two sides and an angle, draw a segment on the sheet equal to the lead side. With the support of the protractor, set this corner aside and draw a second side, postponing the size given in the condition. If you are given one side and two corners adjacent to it, draw first side, then from the 2 ends of the resulting segment, set aside the corners and draw the other two sides. Label the triangle as ABC.
3. In the AutoCAD program, it is more comfortable for everyone to build an incorrect triangle with the help of the Segment tool. You will find it through the main tab, preferring the Drawing window. Set the coordinates of the side you know, after that - the final point of the second given segment.
4. Determine the type of triangle. If it is rectangular, then the unfamiliar side is calculated using the Pythagorean theorem. The hypotenuse is square root from the sum of the squares of the legs, that is, c=?a2+b2. Accordingly, each of their legs will be equal to the square root of the difference between the squares of the hypotenuse and the famous leg: a=?c2-b2.
5. To calculate the unknown side of a triangle given a side and two included angles, use the sine theorem. The a side is related to sin?, as the b side is to sin?. ? and? in this case, opposite angles. An angle that is not given by the conditions of the problem can be found by remembering that the sum of the interior angles of a triangle is 180°. Subtract from it the sum of the 2 angles you know. Discover unknown to you side b, solving the proportion by the usual method, that is, by multiplying the famous side and on sin? and dividing this product by sin?. You get the formula b=a*sin?/sin?.
6. If you are famous for the sides a and b and the angle? between them, use the law of cosines. The unfamiliar side c will be equal to the square root of the sum of the squares of the other 2 sides, minus twice the product of these same sides, multiplied by the cosine of the angle between them. That is c=?a2+b2-2ab*cos?.
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Tip 5: How to calculate the angle in a right triangle
Straight carbonic a triangle consists of two acute angles, the value of which depends on the lengths of the sides, as well as one angle of invariably constant value of 90 °. Calculate the size of an acute angle in degrees is allowed using trigonometric functions or theorems on the sum of angles at the vertices of a triangle in Euclidean space.
Instruction
1. Use trigonometric functions if only the dimensions of the sides of a triangle are given in the conditions of the problem. Let's say, according to the lengths of 2 legs (short sides adjacent to a right angle), it is possible to calculate any of the 2 acute angles. The tangent of that angle (?), the one adjacent to leg A, can be found by dividing the length of the opposite side (leg B) by the length of side A: tg (?) = B / A. And knowing the tangent, it is possible to calculate the corresponding angle value in degrees. For this, the arctangent function is prepared: ? = arctg(tg(?)) = arctg(B/A).
2. Using the same formula, it is possible to detect the value of another acute angle lying on the opposite leg A. Primitively change the designations of the sides. But it is also possible to do this vice versa, with the help of another pair of trigonometric functions - cotangent and arc cotangent. The cotangent of the angle b is determined by dividing the length of the adjacent leg A by the length of the opposite leg B: tg(?) = A/B. And the arc tangent will help to extract from the obtained value of the angle in degrees: ? = arcctg(ctg(?)) = arcctg(A/B).
3. If in the initial conditions the length of one of the legs (A) and the hypotenuse (C) is given, then to calculate the angles, use the functions that are inverse to sine and cosine - arcsine and arccosine. The sine of an acute angle? is equal to the ratio of the length of the leg B lying opposite it to the length of the hypotenuse C: sin (?) \u003d B / C. So, to calculate the value of this angle in degrees, use the following formula: = arcsin(V/C).
4. What is the value of the cosine of an angle? is determined by the ratio of the length of the leg A adjacent to this vertex of the triangle to the length of the hypotenuse C. This means that to calculate the angle in degrees, by analogy with the previous formula, you need to apply the following equation: = arccos(A/C).
5. The theorem on the sum of the angles of a triangle makes it inappropriate to use trigonometric functions if the value of one of the acute angles is given in the conditions of the problem. In this case, to calculate the unknown angle (?), easily subtract from 180° the values of 2 known angles - right (90°) and acute (?): = 180° – 90° – ? = 90° -?.
Note!
The height h divides the triangle ABC into two right triangles similar to it. Here the sign of similarity of triangles in three corners works.
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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.
Online calculator.
Solution of triangles.
The solution of a triangle is the finding of all its six elements (that is, three sides and three angles) by any three given elements that define the triangle.
This math program finds side \(c \), angles \(\alpha \) and \(\beta \) given user-specified sides \(a, b \) and the angle between them \(\gamma \)
The program not only gives the answer to the problem, but also displays the process of finding a solution.
This online calculator can be useful for high school students in preparing for tests and exams, when testing knowledge before the Unified State Examination, and for parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with a detailed solution.
In this way, you can conduct your own training and/or the training of your younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.
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Rules for entering numbers
Numbers can be set not only whole, but also fractional.
The integer and fractional parts in decimal fractions can be separated by either a dot or a comma.
For example, you can enter decimals so 2.5 or so 2.5
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A bit of theory.
Sine theorem
Theorem
The sides of a triangle are proportional to the sines of the opposite angles:
$$ \frac(a)(\sin A) = \frac(b)(\sin B) = \frac(c)(\sin C) $$
Cosine theorem
Theorem
Let in triangle ABC AB = c, BC = a, CA = b. Then
The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides times the cosine of the angle between them.
$$ a^2 = b^2+c^2-2ba \cos A $$
Solving Triangles
The solution of a triangle is the finding of all its six elements (i.e. three parties and three angles) according to some three given elements that define the triangle.
Consider three problems for solving a triangle. In this case, we will use the following notation for the sides of the triangle ABC: AB = c, BC = a, CA = b.
Solution of a triangle given two sides and an angle between them
Given: \(a, b, \angle C \). Find \(c, \angle A, \angle B \)
Solution
1. By the law of cosines we find \(c\):
$$ \cos A = \frac( b^2+c^2-a^2 )(2bc) $$
3. \(\angle B = 180^\circ -\angle A -\angle C \)
Solution of a triangle given a side and adjacent angles
Given: \(a, \angle B, \angle C \). Find \(\angle A, b, c \)
Solution
1. \(\angle A = 180^\circ -\angle B -\angle C \)
$$ b = a \frac(\sin B)(\sin A), \quad c = a \frac(\sin C)(\sin A) $$
Solving a Triangle with Three Sides
Given: \(a, b, c\). Find \(\angle A, \angle B, \angle C \)
Solution
1. According to the cosine theorem, we get:
$$ \cos A = \frac(b^2+c^2-a^2)(2bc) $$
2. Similarly, we find the angle B.
3. \(\angle C = 180^\circ -\angle A -\angle B \)
Solving a triangle given two sides and an angle opposite a known side
Given: \(a, b, \angle A \). Find \(c, \angle B, \angle C \)
Solution
1. By the sine theorem we find \(\sin B \) we get:
$$ \frac(a)(\sin A) = \frac(b)(\sin B) \Rightarrow \sin B = \frac(b)(a) \cdot \sin A $$
Let's introduce the notation: \(D = \frac(b)(a) \cdot \sin A \). Depending on the number D, the following cases are possible:
If D > 1, such a triangle does not exist, because \(\sin B \) cannot be greater than 1
If D = 1, there is a unique \(\angle B: \quad \sin B = 1 \Rightarrow \angle B = 90^\circ \)
If D If D 2. \(\angle C = 180^\circ -\angle A -\angle B \)
3. Using the sine theorem, we calculate the side c:
$$ c = a \frac(\sin C)(\sin A) $$
