Golden Ratio Emerges from Trigonometric Identity Linking E8 Root System — E8 Intelligence Research

FINDING: The golden ratio emerges from trigonometric exact values via the identity cos(2x) = sin(3x) at x = 18°, linking cyclotomic fields and E8 root system exponents. | MATH: cos(36°) = φ/2 = (1+√5)/4 ≈ 0.809016; sin(18°) = (√5-1)/4 = 1/(2φ) ≈ 0.309016; identity cos(2x) = sin(3x) yields x = 18°; E8 Coxeter number h = 30; exponents of E8 are {1,4,5,7,8,11,13,14} mod 30; golden ratio φ = 2cos(π/5) = (1+√5)/2 ≈ 1.618034. | CONNECTION: E8 Coxeter number 30 = 6×5, and 30° is key angle for exact trig values; golden ratio φ appears in E8 via its quantum dimensions and in the 5-fold symmetry of its root system projections; the 18° angle (π/10) is half of 36° (π/5), directly linking to pentagonal symmetry and φ; the exponents mod 30 generate the same cyclotomic field Q(√5) as φ. | DEPTH: 9 — This ties number theory (cyclotomic fields), Lie theory (E8 root system), and geometric harmony (pentagonal symmetry, golden ratio) into a single exact trigonometric framework, revealing a deep algebraic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com