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Home » y = 2\left(\frac43\right) + 3 = \frac83 + 3 = \frac83 + \frac93 = \frac173 - Groen Casting

y = 2\left(\frac43\right) + 3 = \frac83 + 3 = \frac83 + \frac93 = \frac173 - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: A materials scientist is analyzing the stiffness of a composite material, modeled by the expression

📌 Find the minimum value of this expression for $ x \in \left(0, \frac{\pi}{2}\right) $.

📌 Recall identities:

📌 \sec^2 x = 1 + \tan^2 x,\quad \csc^2 x = 1 + \cot^2 x

📌 Now, note that:

📌 \tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}

📌 But it’s simpler to minimize directly:

📌 Let $ s = \sin x + \cos x $, $ p = \sin x \cos x $. Then:

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