ie right angle triangles with whole number sides. The most famous of these is 3, 4, 5,
pictured below. It is unique, and no other triangle with these proportions can be
drawn but the actual values can be scaled; thus 6, 8, 10 plays out the same.
Other triplets have been identified: 5, 12, 13 is one as well. But it will look quite
different, and will have a reduced area.
* * *
But other triangular shapes are possible as well. What if I have three pieces of fencing
in the proportions 6, 7, 8. What would be the area of that triangle. Modern mathematics
has an answer for me, and it involves precalculated tables of trigonometric functions
(or, in my actual case, instantly calculated on a pocket calculator).
Fun fact: the Law of Cosines, when published in France, is known as the théorème
d'Al-Kashi, for the Persian mathematician(1380-1429) .
* * *
How to derive these famous laws. For the Law of Cosines, it's complicated.
The essential difficulty is labeling the 'random' triangle correctly. Once this is done,
basic algebra will do it.
Because we have created a right angle triangle, a can be considered a hypotenuse
whose actual length scales the cosine value of the C angle. The height of the triangle
becomes a sine function on that same angle...
The Law of Sines is generally considered self-evident.
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