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Home » You Won’t BELIEVE How This Scream Meme Shook the Internet Forever! - Crankk.io

You Won’t BELIEVE How This Scream Meme Shook the Internet Forever! - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 \boxed{60^\circ, 120^\circ, 240^\circ, 300^\circ}

📌 Question: Let $ z^4 + z^2 + 1 = 0 $. Express the maximum imaginary part of a root as $ \sin \theta $, where $ \theta \in [0^\circ, 360^\circ] $, reflecting harmonic analysis in linguistic frequency models.

📌 Solution: Factor $ z^4 + z^2 + 1 = (z^2 + z + 1)(z^2 - z + 1) = 0 $. Solving $ z^2 + z + 1 = 0 $ gives roots $ z = \frac{-1 \pm i\sqrt{3}}{2} $, with imaginary parts $ \pm \frac{\sqrt{3}}{2} $. Solving $ z^2 - z + 1 = 0 $ gives $ z = \frac{1 \pm i\sqrt{3}}{2} $, with imaginary parts $ \pm \frac{\sqrt{3}}{2} $. The maximum imaginary part is $ \frac{\sqrt{3}}{2} = \sin 60^\circ $.

📌 \boxed{60^\circ}

📌 Question: If $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ are unit vectors in $ \mathbb{R}^3 $ with $ \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{a} = \frac{1}{2} $, find the maximum value of $ \|\mathbf{a} + \mathbf{b} + \mathbf{c}\| $, modeling directional consistency in linguistic feature spaces.

📌 Solution: Compute $ \|\mathbf{a} + \mathbf{b} + \mathbf{c}\|^2 = 3 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 3 + 2\left(\frac{1}{2} + \frac{1}{2} + \frac{1}{2}\right) = 3 + 3 = 6 $. Thus, the norm is $ \sqrt{6} $.

📌 Question: Determine the minimum value of $ \frac{\sin^2 x + 4}{\sin x} $ for $ x \in (0^\circ, 180^\circ) $, analogous to optimizing resource allocation in predictive environmental models.

📌 Solution: Rewrite $ f(x) = \frac{\sin^2 x + 4}{\sin x} = \sin x + \frac{4}{\sin x} $. Let $ y = \sin x \in (0, 1] $. The function becomes $ f(y) = y + \frac{4}{y} $. The derivative $ f'(y) = 1 - \frac{4}{y^2} $ has critical point at $ y = 2 $, but $ y \leq 1 $. Analyze endpoints: as $ y \to 0^+ $, $ f(y) \to \infty $; at $ y = 1 $, $ f(1) = 1 + 4 = 5 $. The minimum is $ 5 $.

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