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📅 March 7, 2026 👤 scraface

Mar 07, 2026

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📌 + \sec^2 \theta + \csc^2 \theta = 7 + \tan^2 \theta + \cot^2 \theta

📌 Using the AM-GM inequality, \(\tan^2 \theta + \cot^2 \theta \geq 2\), hence:

📌 + \tan^2 \theta + \cot^2 \theta \geq 7 + 2 = 9

📌 The minimum value is achieved when \(\tan^2 \theta = \cot^2 \theta = 1\), i.e., \(\theta = \frac{\pi}{4} + n\pi\). Therefore, the minimum value is:

📌 Question:** An entomologist specializing in insect ecology is studying the movement patterns of a certain species of insect, modeled by the equation \((x - 3)^2 + (y - 4)^2 = 25\). Determine the number of integer coordinate points \((x, y)\) that lie on this circle.

📌 The given equation represents a circle centered at \((3, 4)\) with radius 5. The equation is:

📌 To find integer solutions, consider \(x = 3 + a\) and \(y = 4 + b\), where \(a^2 + b^2 = 25\). We need integer pairs \((a, b)\) such that:

📌 a^2 + b^2 = 25

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