F.53 Geometric Half-Angle Trigonometry — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.53 (Geometric Half-Angle Trigonometry) (SeeligNotebooks).

Folio 0346 (Page 1 of 1): Unit Semicircle Proof of Half-Angle Identities

[Folio 0346]

Folio 0346 (original scan 0346.jpg) from resources/seelig/53_trig.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0346 (0346.jpg, Sheet 1 of 1):

Constructs right triangle \(AHP\) inside a unit semicircle (\(R=1, |AH|=1+\cos\theta, |PH|=\sin\theta, |AP|=2\cos(\theta/2)\)) to prove \(\cos^2(\theta/2) = \frac{1}{2}(1+\cos\theta)\) and \(\sin^2(\theta/2) = \frac{1}{2}(1-\cos\theta)\).

\[(2 x)^2 = \sin^2\theta + (1 + \cos\theta)^2 = 2 + 2\cos\theta \implies \cos^2\frac{\theta}{2} = \frac{1 + \cos\theta}{2}\]

Editorial & Symbolic Notes

Archival scan 0346.jpg from Fred W. Seelig’s notebook dossier on F.53 Geometric Half-Angle Trigonometry (Page 1 of 1). Companion OpenDocument worksheet(s): Trig Identities, General Reference. Classical Euclidean foundations: I.47, III.20, III.31.