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edward guinness 1st earl of iveagh - NBX Soluciones

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: Find the minimum value of $ (\cos x + \sec x)^2 + (\sin x + \csc x)^2 $.

📌 Solution: Expand the expression: $ \cos^2x + 2 + \sec^2x + \sin^2x + 2 + \csc^2x $. Simplify using $ \cos^2x + \sin^2x = 1 $ and $ \sec^2x = 1 + an^2x $, $ \csc^2x = 1 + \cot^2x $: $ 1 + 2 + 1 + an^2x + 1 + \cot^2x + 2 = 7 + an^2x + \cot^2x $. Let $ t = an^2x $, so $ \cot^2x = rac{1}{t} $. The expression becomes $ 7 + t + rac{1}{t} $. By AM-GM, $ t + rac{1}{t} \geq 2 $, so the minimum is $ 7 + 2 = 9 $. Thus, the minimum value is $ oxed{9} $.

📌 Question: Among all roots of $ z^6 + z^4 + z^2 + 1 = 0 $, the maximum imaginary part can be expressed as $ \sin heta $, where $ 0 < heta < rac{\pi}{2} $. Find $ heta $.

📌 Solution: Factor the equation: $ z^6 + z^4 + z^2 + 1 = rac{z^8 - 1}{z^2 - 1} $ (for $ z^2

📌 eq 1 $). The roots are the 8th roots of unity except $ z = \pm 1 $. The roots with positive imaginary parts are $ e^{i\pi/8}, e^{i3\pi/8} $. The maximum imaginary part is $ \sin(3\pi/8) $. Thus, $

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