When studying triangles, the question involuntarily arises of calculating the relationship between their sides and angles. In geometry, the cosine and sine theorem gives the most complete answer to solve this problem. In abundance of various mathematical expressions and formulas, laws, theorems and rules, there are those that are distinguished by extraordinary harmony, conciseness and ease of presentation of the meaning contained in them. The sine theorem is a prime example of such a mathematical formulation. If in the verbal interpretation a certain obstacle also arises in understanding this mathematical rule, then when you look at the mathematical formula everything immediately falls into place.
The first information about this theorem was discovered in the form of proof of it in the framework of the mathematical work of Nasir ad-Din At-Tusi, dated to the thirteenth century.
Approaching closer to the consideration of the aspect ratio and angles in any triangle, it is worth noting that the sine theorem allows you to solve a lot of mathematical problems, while this law of geometry finds application in various types of practical human activity.
The sine theorem itself states that for any triangle the proportionality of the sides to the sines of opposite angles is characteristic. There is also the second part of this theorem, according to which the ratio of either side of the triangle to the sine of the opposite angle is equal to the diameter of the circle described around the triangle in question.
In the form of a formula, this expression looks like
a / sinA = b / sinB = c / sinC = 2R
It has a sine theorem proof, which is proposed in various versions of textbooks in a rich variety of versions.
As an example, we consider one of the proofs that provide an explanation of the first part of the theorem. To this end, we set ourselves the goal of proving the validity of the expression a sinC = c sinA.
In an arbitrary triangle ABC, we construct the height BH. In one construction variant, H will lie on the segment AC, and in 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 in terms of the angles and sides of the triangle, as BH = a sinC and BH = c sinA, which is the required proof.
In the case when the point H is outside the segment AC, we can obtain the following solutions:
BH = a sinC and BH = c sin (180-A) = c sinA;
or BH = a sin (180-C) = a sinC and BH = c sinA.
As you can see, regardless of the construction options, we come to the desired result.
The 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, we construct the diameter of the circle. We connect the resulting point on the circle D with one of the height of the triangle, let it be point A of the triangle.
If we consider the resulting triangles ABD and ABC, then we can notice the equality of the angles C and D (they rely on the same arc). And given that the angle A is ninety degrees, then sin D = c / 2R, or sin C = c / 2R, as required.
The sine theorem is the starting point for solving a wide range of different problems. Of particular attractiveness is its practical application, as a consequence of the theorem we get the opportunity to relate to each other the values of the sides of the triangle, opposite angles and radius (diameter) of the circle circumscribed around the triangle. The simplicity and accessibility of the formula describing this mathematical expression made it possible to widely use this theorem for solving problems using various mechanical calculating devices (slide rules, tables, etc.), but even the arrival of powerful computing devices in the human service did not reduce the relevance of this theorem.
This theorem is not only included in the required course of secondary school geometry, but is also subsequently applied in some branches of practical activity.