# Establishment of relation between angles and sides of a triangle

I was looking for an historical approach to learn about the initiation of trigonometry but got stuck in a part where I have been trying to undestand but couldn't till now.

By far, I could learn that trigonometry has basically evolved from calculations regarding chords of a circle of unit radius. And the development of mathematics till then mainly developed due to establishment of relation of sides of two congruent triangles, that is ratio of similar sides of two similar triangles are equal. Perhaps trigonometry evolved from that relation. But how come anvient mathematicians establish relation between angles and sides of a triangle? I have assumed that they have been carrying out experiments through brute measurements of angles and their relation to sides(ratio of sides) but again stumbled onto were there such precise instruments to measure angles at that time or not. I am pretty much puzzled about this.

• Commented Aug 19, 2021 at 15:13
• I think you are asking what motivated the ancients to study triangles, and not how their study of triangles led to trigonometry ideas. If so, then you probably want to read about how the ancients carried out various engineering projects, among other things. I don't know this literature well enough to cite anything without doing a bit of googing (and thus all I'd be doing is mentioning things you could find out from what I've already said) except for one book I know about, which is L. Sprague De Camp's The Ancient Engineers (first appeared in 1960). Commented Aug 19, 2021 at 18:39
• I am basically looking for how the relation between angles and sides of a triangle established in the beginning.
– MSKB
Commented Aug 19, 2021 at 19:49
• Greek mathematicians did not need experiments or measurements, they had theorems of geometry and performed calculations based on them. See How Ptolemy computed chords in the Almagest for an example, but the methods go back to Hipparchus at least. Commented Aug 20, 2021 at 9:14
• More precisely, there was a strict distinction between pure mathematics (they called it geometry) and applied mathematics (for example astronomy). Only in applied mathematics angles were measured with degrees. Commented Aug 20, 2021 at 15:37

Trigonometry was developed for the needs of astronomy. So originally this was not the plane trigonometry that we study at school but spherical trigonometry. The main (and almost only) source that survived to our days from antiquity is the book of Ptolemy, Mathematical synthesis (a.k.a. Almagest, available in English translation). Improvements and developments were slowly made in this business in the Middle age, mainly by Muslim astronomers. It is they, who introduced our main trigonometric functions; Ptolemy had only one: the chord.

Then this knowledge penetrated to the West, and trigonometry acquired more or less modern form in the work of Johannes Muller (Regiomontanus). For example, he proved the cosine rule.

The conjectural picture of "development of mathematics" that you describe is incorrect. First of all, similar triangles did not play important role in this development, since the primary subject was spherical triangles, and there are no similar spherical triangles. Second, mathematicians usually do not discover their theorems by carrying on "brute measurements".

• How did mathematicians discover that for an angle 30° the ratio of perpendicular and hypotenuse is 0.5?
– MSKB
Commented Aug 20, 2021 at 13:55
• @MSKB: this is an easy consequence of Pythagoras theorem if you state is properly: Euclid and other pure mathematicians did not use degrees. "Angle 30deg" is what they would call 1/12 of the circle. The fact you mention is established by inscribing a regular hexagon into a circle. Commented Aug 20, 2021 at 15:32
• 😅😅😅 whatever they called it.....but how did they establish this relation that for 1/12th of a circle the ratio of perpendicular and base is 1/2 and so on? I just want bit of a clue......
– MSKB
Commented Aug 20, 2021 at 19:06