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The case of these triangles is special and close to the Pythagorean theorem, namely $c^2=a^2+b^2\pm ab$. This is a particular case of trigonometric relations and the law of cosines, and remains quite easy without it, using similar triangles and a few tricks.

So my question is: is it known when this result was studied/written for the first time and was it long before the study of trigonometry? I think it would be strange if it has not been found before understanding the general case. This question was first asked on Math Stack Exchange.

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  • $\begingroup$ Your question is unclear if one does not read the MathSE version. Please make it clear. $\endgroup$
    – peterh
    Dec 13 '20 at 18:10
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Geometric analogs of the law of cosines are the propositions II.12 and II.13 of Euclid's Elements (c. 300 BC) for obtuse and acute angles, respectively. Here is Euclid's formulation for the acute case:

"In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle."

Geometric constructions needed to establish numerical values for the cosines of $60^\circ$, $120^\circ$ and some other angles, e.g. $36^\circ$ and $72^\circ$, also occur in the Elements, when Euclid is inscribing regular triangles, pentagons and hexagons into a circle. Ptolemy in Almagest (c. 150 AD) uses such constructions and additional geometric theorems to derive an entire table of chord values for such angles and their subdivisions (chord trigonometry is equivalent to the modern one). See Aaboe, Episodes From the Early History of Astronomy for a modern exposition of his methods.

The modern formulation of the law of cosines is usually attributed to Al-Kashi (c. 1400 AD), see An Overview of Mathematical Contributions of Ghiyath al-Din Jamshid Al-Kashi by Azarian, he also refined Ptolemy's tables and expressed them in terms of sines and cosines. In Europe the law was popularized by Viete. None of these authors singled out $60^\circ$ or $120^\circ$ triangles for special treatment.

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