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Home » Pinewood Derby Car Designs That Look Like They’re Straight Out of a Fast Lane! - Abbey Badges

Pinewood Derby Car Designs That Look Like They’re Straight Out of a Fast Lane! - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Question: Find all angles $ z \in [0^\circ, 360^\circ] $ such that $ \sin(2z) + \sqrt{3} \cos(2z) = 1 $, relevant to aligning phase shifts in a glaciological time-series model of basal sliding.

📌 \sin(2z) + \sqrt{3} \cos(2z) = 1

📌 We rewrite the left-hand side as a single sine function. Let:

📌 R \sin(2z + \alpha) = \sin(2z) + \sqrt{3} \cos(2z)

📌 R \left( \sin(2z)\cos\alpha + \cos(2z)\sin\alpha \right) = R \cos\alpha \sin(2z) + R \sin\alpha \cos(2z)

📌 Matching coefficients:

📌 R \cos\alpha = 1, \quad R \sin\alpha = \sqrt{3}

📌 R^2 = 1^2 + (\sqrt{3})^2 = 4 \Rightarrow R = 2

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