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Home » Question: Find the minimum value of $(\csc x + \cot x)^2 + (\sec x - an x)^2 - Groen Casting

Question: Find the minimum value of $(\csc x + \cot x)^2 + (\sec x - an x)^2 - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: The maximum imaginary part of a root of $ z^5 + z^3 + z + 1 = 0 $ can be expressed as $ \sin \theta $. Find $ \theta $ in degrees.

📌 Solution: Factor the polynomial:

📌 z^5 + z^3 + z + 1 = z^3(z^2 + 1) + 1(z + 1).

📌 Group terms:

📌 Roots: $ z^3 = -1 \implies z = e^{i\pi/3 + 2k\pi/3} $, and $ z^2 = -1 \implies z = e^{i\pi/2}, e^{i3\pi/2} $.

📌 The roots with maximum imaginary part are $ e^{i\pi/3} $ and $ e^{

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