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Home » A) A tone and salivation - Silent Sales Machine

A) A tone and salivation - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 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 $.

📌 The function $ \sin(3\theta) + \cos(4\theta) $ is smooth and periodic. Its derivative is $ 3\cos(3\theta) - 4\sin(4\theta) $, which is non-zero almost everywhere, implying the function is strictly increasing or decreasing in intervals, but due to high frequency, we estimate number of times it crosses $ \pm1 $.

📌 Instead, count the number of times $ \left| \sin(3\theta) + \cos(4\theta) \right| = 1 $, i.e., when $ \sin(3\theta) + \cos(4\theta) = \pm1 $.

📌 The function $ f(\theta) = \sin(3\theta) + \cos(4\theta) $ is analytic and not periodic with simple period, but we can consider how many times it attains values $ \pm1 $.

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