I have this statement:
In the year $2750$ b.C, a Persian mathematician offers you 20,000 precious stones, if you discover the ratio between the leg of the right triangle and its hypotenuse.
He offers you some clues, and tells you that: The angle opposite the cathetus measures 48º, and the hypotenuse measures 21 meters.
In these times the idea of trigonometry did not exist yet.
It is trivial to perform an exercise like this with a calculator and the use of trigonometry.
My teacher said that the idea of doing this is to think about how the angles relate to the sides, to deduce how one comes to the idea that there can be a ratio between the hypotenuse, the leg and an angle. In order for ourselves to deduce that there is a reason between these elements and better understand trigonometry.
For this I have been investigated the origins of trigonometry, but I have only found history and the current equations (which I know).
So, how could the ratio between the hypotenuse and the leg, given the angle and the hypotenuse, be reached without using trigonometry? I have read that the ancient Egyptians fabricated trigonometric tables, measuring the position of certain stars, but I believe that this is not a quick path for me. Thanks in advance.
PD: mathematically I mean to find: $\frac{Oppositecathetus}{ hypotenuse} = k$, find $k$
Interesting that they are using meters in 2750 BC.
ABC is our right triangle AC = 21 meters
AFD is a 36 - 72 - 72 isosceles triangle
DGF is similar to AFD
AGD is a 36 - 36 - 108 isosceles triangle
DF = DG = AG
AED is a 30-60-90 right triangle.
CED is similar to ABC
That should be enough information to find CD + DB