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Home » Spanish Coloring: The Easy Way to Create Vibrant Art Like a Local! - Crosslake

Spanish Coloring: The Easy Way to Create Vibrant Art Like a Local! - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Using the identity $ \sin(2x) = 2\sin x \cos x $, we have:

📌 \sin x \cos x = \frac{1}{2} \sin 2x \Rightarrow \sin^2 x \cos^2 x = \frac{1}{4} \sin^2 2x

📌 f(x) = \frac{1}{\frac{1}{4} \sin^2 2x} + 1 = \frac{4}{\sin^2 2x} + 1

📌 Since $ 0 < x < \frac{\pi}{2} $, then $ 0 < 2x < \pi $, and $ \sin 2x > 0 $. The minimum of $ f(x) $ occurs when $ \sin^2 2x $ is maximized, i.e., when $ \sin^2 2x = 1 $. This happens when $ 2x = \frac{\pi}{2} \Rightarrow x = \frac{\pi}{4} $.

📌 f\left(\frac{\pi}{4}\right) = \frac{4}{1} + 1 = 5

📌 Thus, the minimum value is $ \boxed{5} $.

📌 Question: A mammalogist observes that the social cohesion index $ S $ of a primate group over time is modeled by $ S = \left| \sin\left(\frac{3\pi}{7} t\right) + \cos\left(\frac{4\pi}{7} t\right) \right| $, where $ t $ is time in days. How many times in one full cycle of $ S $ (i.e., over $ t \in [0, 7) $) does $ S = 1 $?

📌 Solution: The function $ S(t) = \left| \sin\left(\frac{3\pi}{7} t\right) + \cos\left(\frac{4\pi}{7} t\right) \right| $ has period $ T $, where both components repeat. The fundamental period is the least common multiple of the periods of $ \sin\left(\frac{3\pi}{7} t\right) $ and $ \cos\left(\frac{4\pi}{7} t\right) $.

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