KNOTT-2.1 — Exact Radical Sines, Cosines, Tangents & Spreads of 18°, 36°, 54°, 72°

On the unit circle, cos 36° = sin 54° = Φ/2 and sin 18° = cos 72° = φ/2 differ by exactly 1/2 and multiply to 1/4, while the pentagonal spreads s(36°) = (5 − √5)/8 and s(72°) = (5 + √5)/8 form the quadratic roots of Wildberger’s five-fold spread factor 16 s² − 20 s + 5 = 0.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-2.1
Curated Source	Dr. Ron Knott, Exact Trig Values for Simple Angles (#3654)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.trigonometry.construct_pentagonal_trig_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/wildberger, geometor/euclid, geometor/divine

Constructive Synthesis & Exact Proof

On simpleTrig.html (KnottSimpleTrig), Dr. Ron Knott derives the closed-form quadratic radicals for every multiple of \(18^\circ\) directly from the golden isosceles triangle—linking Claudius Ptolemy’s chord table to Norman J. Wildberger’s Rational Trigonometry spreads (Wildberger2005).

• Unit-Circle Pentagonal Rays: On unit circle (O, U) bounded by orthogonal axes OU and OV, the four acute pentagonal rays OP18, OP36, OP54, and OP72 are equal unit radii. Dropping altitudes P36C36 (forming right triangle OC36P36) and P72C72 onto OU projects C36 at \(x = \cos 36^\circ = \Phi/2\) and C72 at \(x = \cos 72^\circ = \sin 18^\circ = \phi/2\).

• Rational Difference, Product & Five-Fold Spread Factor: Because \(\Phi - \phi = 1\) and \(\Phi\phi = 1\), segment C72C36 has exact rational length \(\cos 36^\circ - \cos 72^\circ = 1/2\) while \(\cos 36^\circ \cdot \cos 72^\circ = 1/4\). Squaring the ordinates eliminates the nested square roots \(\sqrt{10 \pm 2\sqrt{5}}\) completely, leaving pure \(\mathbb{Q}(\sqrt{5})\) spreads \(s(18^\circ) = (3 - \sqrt{5})/8\), \(s(36^\circ) = (5 - \sqrt{5})/8\), \(s(54^\circ) = (5 + \sqrt{5})/8\), and \(s(72^\circ) = (5 + \sqrt{5})/8\), where \(s(36^\circ)\) and \(s(72^\circ)\) are the exact roots of \(16 s^2 - 20 s + 5 = 0\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
cos 36	\(\frac{1}{4} + \frac{\sqrt{5}}{4}\)
sin 18	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)
spread 18	\(\frac{3}{8} - \frac{\sqrt{5}}{8}\)
spread 36	\(\frac{5}{8} - \frac{\sqrt{5}}{8}\)
spread 54	\(\frac{\sqrt{5}}{8} + \frac{3}{8}\)
spread 72	\(\frac{\sqrt{5}}{8} + \frac{5}{8}\)
projection span C72 C36	\(\frac{1}{2}\)