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Home » \[ 2 = \fraca + b - 102 \] - Crosslake

\[ 2 = \fraca + b - 102 \] - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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) $.

📌 The angular frequencies are $ \frac{3\pi}{7} $ and $ \frac{4\pi}{7} $, so periods are $ \frac{2\pi}{3\pi/7} = \frac{14}{3} $, and $ \frac{2\pi}{4\pi/7} = \frac{7}{2} $. The LCM of $ \frac{14}{3} $ and $ \frac{7}{2} $ is found by expressing as rational multiples.

📌 Let $ t \mapsto \theta = \frac{\pi}{7} t $, so $ t \in [0,7) \Rightarrow \theta \in [0,7\pi) $. Then:

📌 S = \left| \sin(3\theta) + \cos(4\theta) \right|, \quad \theta \in [0, 7\pi)

📌 We seek the number of solutions to $ \left| \sin(3\theta) + \cos(4\theta) \right| = 1 $ in $ \theta \in [0, 7\pi) $, then map back to $ t $.

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