F.53 Geometric Half-Angle Trigonometry

Unit Semicircle Chord Proof of cos^2(theta/2) and sin^2(theta/2)

In a unit semicircle of radius R = 1 and diameter AB = 2, a point P at central angle theta has altitude sin(theta), base projections 1 + cos(theta) and 1 - cos(theta), and chord |AP| = 2 cos(theta/2) = sqrt(2 + 2 cos(theta)), giving cos^2(theta/2) = (1 + cos(theta))/2 and sin^2(theta/2) = (1 - cos(theta))/2.

Interactive GEOMETOR construction for F.53 (Geometric Half-Angle Trigonometry). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A : Point P : Point ===== c_unit = circle(O, A) l_oa = line(O, A) B = intersection(l_oa, c_unit) H = foot(P, l_oa) s_ap = segment(A, P) s_bp = segment(B, P)

Mathematical Synthesis

In this elegant single-folio geometric proof (SeeligNotebooks), Seelig derives the half-angle identities for \(\cos^2(\theta/2)\) and \(\sin^2(\theta/2)\) directly from a unit semicircle (\(R = 1\), diameter \(|AB| = 2\)) without algebraic angle-addition expansion. Dropping the altitude \(|PH| = \sin\theta\) from P to diameter A B divides the diameter into \(|AH| = 1 + \cos\theta\) and \(|HB| = 1 - \cos\theta\).

By the inscribed angle theorem (Euclid Elements III.20), the angle at A is \(\theta/2\), so the chord sap has length \(2x = 2\cos(\theta/2)\). Applying the Pythagorean theorem (I.47) to right triangle \(\triangle AHP\) gives \((2x)^2 = \sin^2\theta + (1 + \cos\theta)^2 = 2 + 2\cos\theta\), whence \(x^2 = \cos^2\frac{\theta}{2} = \frac{1}{2} + \frac{1}{2}\cos\theta\) and \(\sin^2\frac{\theta}{2} = \frac{1}{2} - \frac{1}{2}\cos\theta\).

Key Algebraic Identities

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

\[\sin^2\frac{\theta}{2} = 1 - \cos^2\frac{\theta}{2} = \frac{1 - \cos\theta}{2}, \qquad \tan\frac{\theta}{2} = \frac{\sin\theta}{1 + \cos\theta}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.53 Geometric Half-Angle Trigonometry — Folios (Folios 0346–0346, 1 scanned manuscript page in resources/seelig/53_trig)
Euclid’s Elements	I.47, III.20, III.31 (EuclidElements)
GEOMETOR Elements (G-Index)	G.7 plane surface, G.11 rectilineal figures
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Trig Identities, General Reference (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.10 Circles and Geometric Mean