Each of us sat for many hours over the solution of a particular problem in geometry. Of course, the question arises, why do you need to learn math at all? The question is especially relevant for geometry, the knowledge of which, if useful, is very rare. But mathematics has a purpose for those who are not going to become an employee of the exact sciences. It makes a person work and develop.
The original purpose of mathematics was not to equip students with knowledge of the subject. Teachers set themselves the goal of teaching children to think, reason, analyze and reason. This is exactly what we find in geometry with its many axioms and theorems, corollaries and proofs.
Cosine theorem
Along with trigonometric functions and inequalities, algebras begin to study angles, their meaning and location. The cosine theorem is one of the first formulas that connects both sides of mathematical science in the understanding of the student.
To find the side in two others and the angle between them, the cosine theorem is used. For a triangle with a right angle, Pythagorean theorem is also suitable, but if we talk about an arbitrary figure, then it cannot be applied here.
The cosine theorem is as follows:
AC 2 = 2 + 2 - 2 * * * cos <
The square of one side is equal to the sum of the other two sides, taken in the square, minus their product, multiplied by two and the cosine of the angle formed by them.
If you look more closely, this formula resembles the Pythagorean theorem. Indeed, if we take the angle between the legs as 90, then the value of its cosine will be 0. As a result, only the sum of the squares of the sides will remain, which reflects the Pythagorean theorem.
Cosine Theorem: Proof
From this expression we derive the formula AC
2 and get:
AC 2 = Sun 2 + AB 2 - 2 * AB * BC * cos <ABC
Thus, we see that the expression corresponds to the above formula, which indicates its truth. We can say that the cosine theorem is proved. It is used for all kinds of triangles.
Using
In addition to lessons in mathematics and physics, this theorem is widely used in architecture and construction, to calculate the necessary sides and angles. With its help, the necessary dimensions of the building and the amount of materials that are required for its construction are determined. Of course, most of the processes that previously required direct human participation and knowledge are automated today. There are a huge number of programs that allow you to simulate such projects on a computer. Their programming is also carried out taking into account all mathematical laws, properties and formulas.
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