The Greeks started everything. Not the current ones, but those that lived before. There were no calculators yet, and there was already a need for computing. And almost any calculation led ultimately to right triangles. They gave a solution to many problems, one of which was: "How to find the hypotenuse, knowing the angle and leg?".
Right angle triangles
Despite the simplicity of determination, this figure on the plane can ask many puzzles. Many have experienced this for themselves, at least in the school curriculum. It’s good that he himself answers all the questions.
But is it possible to simplify this simple combination of sides and angles? It turned out you can. It is enough to make one corner straight, i.e. equal to 90 °.
It would seem, what's the difference? Awesome. If it is almost impossible to figure out the whole variety of angles, then, having fixed one of them, it is easy to come to amazing conclusions. What did Pythagoras do.
Whether he coined the words “cathetus” and “hypotenuse” or whether someone else did it - it does not matter. The main thing is that they got names not just like that, but due to their relationship with a right angle. Two sides were adjacent to it. These were the legs. The third was opposite, she became a hypotenuse.
So what?
At least that there was an opportunity to answer the question of how to find the hypotenuse in terms of leg and angle. Thanks to the concepts introduced by the ancient Greek, logical constructions of the relations of sides and angles became possible.
Triangles themselves, including rectangular ones, were also used in the construction of the pyramids. The famous Egyptian triangle with sides 3, 4 and 5 may have pushed 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 mutually connected 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, and not just the Egyptian one.
Now it has become easy to calculate the length of one side, knowing the other two. But in life, for the most part, tasks of a different kind arise, when you need to know the hypotenuse, knowing the leg and angle. How to determine the width of the river without soaking your feet? Easy. We are building a triangle, one leg of which is the width of the river, the other is known to us by construction. To know the opposite side ... The solution was already found by the followers of Pythagoras.
So, the task: how to find the hypotenuse, knowing the angle and leg
In addition to the relationship of the squares of the sides, they found many other interesting relationships. To describe them, we introduced new definitions: sine, cosine, tangent, cotangent, and other trigonometry. The designations for the formulas were as follows: Sin, Cos, Tg, Ctg. What it is is shown in the figure.
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 every corner. Let us now return to the river, on one bank of which we drew the SA line. Its length is known to us: 30 meters. They did it themselves. On the opposite side is a tree at point B. Measure the angle A will not be difficult, let it be 60 °.
In the table of sines we find the value for the angle of 60 ° - this is 0.866. Therefore, CA \ AB = 0.866. Therefore, AB is defined as CA: 0.866 = 34.64. Now, when 2 sides of a right triangle are known, it will not be difficult to calculate the third. Pythagoras did everything for us, we just need to substitute the numbers:
BC = √ AB 2 - AC 2 = √ 1199.93 - 900 = √299.93 = 17.32 meters.
So we killed two birds with one stone: we figured out how to find the hypotenuse, knowing the angle and leg, and calculated the width of the river.