# How to get the idea behind the relationship between angles and sides? [closed]

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 $k$

## closed as unclear what you're asking by tomasz, Jyrki Lahtonen, Jose Arnaldo Bebita-Dris, Namaste, Adrian KeisterAug 15 '18 at 12:40

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• So essentially you are asking how trigonometry came about? I don't really understand the question. – tomasz Aug 14 '18 at 22:28
• Exactly that is the focus of the question – Eduardo S. Aug 14 '18 at 22:44
• So you are NOT asking how to solve the Persian question without trigonometry? – Jens Aug 14 '18 at 22:49
• Yes, that is the exercise. I just explained, that my teacher's focus when asking this question, is what tomasz said. – Eduardo S. Aug 14 '18 at 23:18
• How could the ratio be found without trigonometry? By making an accurate drawing! Though this would only give you an approximate answer. I wonder whether the Persian mathematician was looking for what we would now call a rational number, i.e. a ratio of two integers? I know that Ancient Greek mathematicians were very keen on rational numbers... Perhaps an answer that was NOT an exact ratio of integers would not really be considered an answer at all... I imagine the Persian mathematicians could easily have made an accurate drawing but that would perhaps miss the point? – alcana Aug 14 '18 at 23:35

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