Each of us spent many hours on the solution of a geometry problem. Of course, the question arises, why do you need to learn mathematics at all? The question is especially relevant for geometry, knowledge of which, if useful, is very rare. But mathematics has a purpose for those who are not going to become workers in the exact sciences. It makes a person work and develop.
The original purpose of mathematics was not to give students knowledge about the subject. Teachers set themselves the goal of teaching children to think, reason, analyze and argue. This is exactly what we find in geometry with its many axioms and theorems, corollaries and proofs.
Cosine theorem
Simultaneously with trigonometric functions and inequalities, algebra begins to study angles, their meaning and finding. The cosine theorem is one of the first formulas that connects both sides of mathematical science in the understanding of the student.
To find a side by two others and the angle between them, the cosine theorem is used. For a triangle with a right angle, the Pythagorean theorem is also suitable for us, but if we talk about an arbitrary figure,then it cannot be applied here.
The cosine theorem looks like this:
AC 2=AB 2+ BC 2- 2 AB BC cos<ABS
The square of one side is equal to the sum of the other two sides squared, minus their product times two and the cosine of the angle they form.
If you look more closely, this formula resembles the Pythagorean theorem. Indeed, if we take the angle between the legs equal to 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 deduce the formula AC 2 and get:
AC 2 =SU 2 + AB 2 - 2ABBCcos <ABC
Thus, we see that the expression corresponds to the above formula, which indicates its truth. We can say that the cosine theorem has been proven. It is used for all kinds of triangles.
Use
In addition to the lessons in mathematics and physics, this theorem is widely used in architecture and construction, to calculate the required sides and angles. With its help, determine the required dimensions of the building and the amount of materials that will be required for its construction. Of course, most of the processes that previously required direct human participation and knowledge,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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