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The R-formula — two waves collapsing into one

A seven-scene guided mini-lesson: a sin θ + b cos θ is secretly ONE wave, R sin(θ + α). Prev / Next drives the story (← → keys work too) — or press ▶ Play all.
compound angle:  R sin(θ + α) = (R cos α) sin θ + (R sin α) cos θ
match coefficients:  R cos α = 3  and  R sin α = 4 — the two legs
square & add:  R² = 3² + 4² = 25  →  R = 5 (Pythagoras)
divide:  tan α = 43  →  α ≈ 53.1° (tan⁻¹ of b over a)