Triangular problems: how to find the hypotenuse knowing the angle and leg

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Triangular problems: how to find the hypotenuse knowing the angle and leg
Triangular problems: how to find the hypotenuse knowing the angle and leg
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Greeks started everything. Not current, but those that lived before. There were no calculators yet, and the need for calculations was already present. And almost every calculation ended up with right triangles. They gave a solution to many problems, one of which sounded like this: "How to find the hypotenuse, knowing the angle and leg?".

Right angle triangles

Despite the simplicity of definition, this figure on the plane can ask a lot of riddles. Many have experienced this for themselves, at least in the school curriculum. It's good that he himself gives answers to all questions.

But isn't it possible to further simplify this simple combination of sides and corners? It turned out it was possible. It is enough to make one angle right, i.e. equal to 90 °.

What are the angles

It would seem, what's the difference? Huge. If it is almost impossible to understand the whole variety of angles, then, having fixed one of them, it is easy to come to amazing conclusions. Which is what Pythagoras did.

Did he come up with the words "leg" and "hypotenuse" or is itsomeone else did it, it doesn't matter. The main thing is that they got their names for a reason, but thanks to their relationship with the right angle. Two sides were adjacent to it. These were the skates. The third was opposite, it became the hypotenuse.

So what?

At least that there was an opportunity to answer the question of how to find the hypotenuse by the leg and angle. Thanks to the concepts introduced by the ancient Greek, the logical construction of the relationship of sides and angles became possible.

Triangles themselves, including rectangular ones, were used during the construction of the pyramids. The famous Egyptian triangle with sides 3, 4 and 5 may have prompted Pythagoras to formulate the famous theorem. She, in turn, became the solution to the problem of how to find the hypotenuse, knowing the angle and leg

The squares of the sides turned out to be interconnected with each other. The merit of the ancient Greek is not that he noticed this, but that he was able to prove his theorem for all other triangles, not just the Egyptian one.

egyptian triangle

Now it's easy to calculate the length of one side, knowing the other two. But in life, for the most part, problems of a different kind arise when it is necessary to find out the hypotenuse, knowing the leg and angle. How to determine the width of a river without getting your feet wet? Easily. We build a triangle, one leg of which is the width of the river, the other is known to us from the construction. To know the opposite side… The followers of Pythagoras have already found the solution.

So, the task is: how to find the hypotenuse, knowing the angle and leg

In addition to the ratio of the squares of the sides, they discovered many morecurious relationship. New definitions were introduced to describe them: sine, cosine, tangent, cotangent and other trigonometry. The designations for the formulas were: Sin, Cos, Tg, Ctg. What it is is shown in the picture.

Relationships in a triangle

The values ​​of the functions, if the angle is known, have long been calculated and tabulated by the famous Russian scientist Bradis. For example, Sin30°=0.5. And so for each angle. Let us now return to the river, on one side of which we drew the SA line. We know its length: 30 meters. They did it themselves. On the opposite side there is a tree at point B. It will not be difficult to measure angle A, let it be 60 °.

In the table of sines we find the value for the angle 60° - this is 0.866. So, CA\AB=0.866. Therefore, AB is defined as CA:0.866=34.64. Now that 2 sides are known a right-angled triangle, it will not be difficult to calculate the third. Pythagoras did everything for us, you just need to substitute the numbers:

BC=√AB2 - AC2=√1199, 93 - 900=√299, 93=17, 32 meters.

That's how we killed two birds with one stone: figured out how to find the hypotenuse, knowing the angle and leg, and calculated the width of the river.

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