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his eye is on the sparrow lyrics - Londonproperty

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 \frac{1}{\sin \theta \cos \theta} = 4 \implies \sin \theta \cos \theta = \frac{1}{4}.

📌 Using $ \sin 2\theta = 2\sin \theta \cos \theta $, we get:

📌 \sin 2\theta = \frac{1}{2}.

📌 Solutions for $ 2\theta \in [0^\circ, 720^\circ] $ are $ 2\theta = 30^\circ, 150^\circ, 390^\circ, 510^\circ $, so $ \theta = 15^\circ, 75^\circ, 195^\circ, 255^\circ $.

📌 Sum: $ 15 + 75 + 195 + 255 = 540^\circ $.

📌 \boxed{540^\circ}

📌 Question: A plant biologist studies the growth vector $ \vec{OG} $ of a crop, defined by $ \vec{OG} = m\vec{OA} + n\vec{OB} $, where $ \|\vec{OA}\| = 3 $, $ \|\vec{OB}\| = 4 $, and $ \vec{OA} \cdot \vec{OB} = 0 $. If $ \|\vec{OG}\| = 5 $ and $ \vec{OG} $ is perpendicular to $ \vec{OA} + \vec{OB} $, find $ (m, n) $.

📌 Solution: Since $ \vec{OA} \cdot \vec{OB} = 0 $, they are perpendicular. Let $ \vec{OA} = (3, 0) $, $ \vec{OB} = (0, 4) $. Then $ \vec{OG} = (3m, 4n) $.

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