The cosine theorem and its proof

Each of us spent many hours trying to solve one or another of the geometry problems. Of course, the question arises: why do you need to learn math at all? The question is especially relevant for geometry, whose knowledge, if useful, is very rare. But mathematics has an appointment 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 give students knowledge of the subject. Teachers set themselves the goal to teach children to think, reason, analyze and argue. This is what we find in geometry with its numerous axioms and theorems, corollaries and proofs.

The cosine theorem

Simultaneously with trigonometric functions and inequalities of algebra, we begin to study the angles, their significance, and their location. The cosine theorem is one of the first formulas that binds both sides of mathematical science in the student's understanding.

To find the side along the other two and the angle between them, the cosine theorem is applied. For a triangle with a right angle, the Pythagorean theorem also suits us, but if we talk about an arbitrary figure, then it can not be applied here.

The cosine theorem is as follows:

AC ² = AB ² + BC ² - 2 * AB * BC * 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 by the cosine of the angle they formed.

If you look more closely, this formula resembles the theorem of Pythagoras. Indeed, if we take the angle between the legs equal to 90, the value of its cosine will be 0. As a result, only the sum of the squares of the sides will remain, which is reflected by the Pythagorean theorem.

The cosine theorem: Proof.

From this expression we derive the formula AC ² and get:

AC ² = BC ² + AB ² - 2 * AB * BC * cos

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 determine the necessary dimensions of the building and the number of materials that will be required for its erection. Of course, most of the processes that previously required direct human participation and knowledge are automated to date. There are a huge number of programs that allow you to simulate similar projects on your computer. Their programming is also carried out taking into account all mathematical laws, properties and formulas.

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