the study of triangles unwittingly raises the question of calculating the relationship between their sides and angles.In geometry theorem of sines and cosines gives the most complete answer to this problem.The abundance of various mathematical expressions and formulas, laws, theories and regulations are such that different extraordinary harmony, brevity and simplicity of filing a prisoner in them.Sines is a prime example of such a mathematical formulation.If the verbal interpretation and there is still a certain obstacle in the understanding of mathematical rules, when looking at a mathematical formula all at once falls into place.
first information about this theorem were found in the form of a proof of it in the framework of mathematical work, Nasir al-Din al-Tusi, dating back to the thirteenth century.
Approaching closer to the relationship between the sides and angles of any triangle, it is worth noting that the sine theorem allows us to solve a lot of mathematical problems, and the geometry of the law finds application in a variety of practical human activity.
itself sine theorem states that for any triangle characteristic proportional to the sine of the opposite sides of the corners.There is also a second part of this theorem, according to which the ratio of either side of the triangle to the sine of the opposite corner is the diameter of the circle described about the triangle under consideration.
as the formula is an expression looks like
a / sinA = b / sinB = c / sinC = 2R
has sine theorem proof, which in various versions of textbooks available in a rich variety of versions.
For example, consider one of the proofs, giving an explanation of the first part of the theorem.To do this, we will ask to prove faithful expression a sinC = c sinA.
In an arbitrary triangle ABC, construct the height BH.In one embodiment, the construct H will lie on the segment AC, and the other outside it, depending on the magnitude of the angles at the vertices of the triangles.In the first case, the height can be expressed through the corners and sides of the triangle as a sinC = BH and BH sinA = c, which is the required evidence.
Where the H-point is outside the segment AC, can get the following solutions:
HV = a sinC and HV = c sin (180-A) = c sinA;
or HV = a sin (180-C) = a sinC and HV = c sinA.
As you can see, regardless of design options, we arrive at the desired result.
proof of the second part of the theorem will require us to describe a circle around the triangle.Through one of the heights of the triangle, for example B, construct a circle diameter.The resulting point on the circle D is connected to one of the height of the triangle, let it be a point A of a triangle.
If we consider the resulting triangle ABD and ABC, we can see the equality of angles C and D (they are based on one arc).And considering that the angle A is equal to ninety degrees to the sin D = c / 2R, or sin C = c / 2R, as required.
sines is the starting point for a wide range of different tasks.A special attraction is the practical application of it, as a consequence of the theorem we are able to relate the values of the sides of the triangle, opposite angles and the radius (diameter) of a circle circumscribed around the triangle.The simplicity and accessibility of a formula that describes this mathematical expression, makes extensive use of this theorem to solve problems using a variety of mechanical devices countable (slide rules, tables, and so forth.), But even the arrival of a person in the service of powerful computing devices did not reduce the relevance of the theorem.
This theorem is not only part of the required course of high school geometry, but later used in some industries practice.
