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Accessoirescom’s Best Kept Secrets: Upgrade Your Look Like a Pro! - American Beagle Club

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 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) $.

📌 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)

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\[ = x^2 - x - 20. \]
The expanded form is \(\boxed{x^2 - x - 20}\).
Compute the square of \( (3x - 2) \).
Use the formula \((a - b)^2 = a^2 - 2ab + b^2\):
\[(3x - 2)^2 = (3x)^2 - 2(3x)(2) + (2)^2.\]
\[ = 9x^2 - 12x + 4. \]
The square of \( (3x - 2) \) is \(\boxed{9x^2 - 12x + 4}\).**Question:
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