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Home » $ f(x) = x $ for all $ x $ - Simpleprint

$ f(x) = x $ for all $ x $ - Simpleprint

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 \sin \theta = \frac{\sqrt{2}}{2} \quad \Rightarrow \quad \theta = \frac{\pi}{4} \quad \text{or} \quad \theta = \frac{3\pi}{4}

📌 Both are in $ (0, \pi) $, but since the maximum imaginary part is positive and corresponds to $ \frac{\sqrt{2}}{2} $, and the angle is often taken as the acute one in such contexts, but the problem asks for $ \theta $ such that $ \sin \theta = \text{max imag part} $, and $ \theta $ is to be found. However, $ \frac{\sqrt{2}}{2} = \sin \frac{\pi}{4} = \sin \frac{3\pi}{4} $, but $ \frac{3\pi}{4} $ is larger and still in range. But since the root has argument $ \frac{3\pi}{4} $, and the imaginary part is $ \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} $, the value $ \theta $ such that $ \sin \theta = \frac{\sqrt{2}}{2} $ and corresponds to the phase is naturally $ \frac{\pi}{4} $, but the problem says "expressed as $ \sin \theta $", and to find $ \theta $. However, since $ \sin \theta = \frac{\sqrt{2}}{2} $,

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📌 5Question: A drone uses 3/8 of its battery power during a 45-minute flight. How many minutes of flight time remain if the battery is fully charged?

📌 Solution: If the drone uses $ \frac{3}{8} $ of its battery for 45 minutes, then each $ \frac{1}{8} $ of the battery powers $ \frac{45}{3} = 15 $ minutes. Thus, the remaining battery is $ \frac{5}{8} $, which powers $ 5 \times 15 = 75 $ minutes. Therefore, the remaining flight time is $ \boxed{75} $ minutes.

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