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Home » From Ordinary to *Shiny* Charcadet: Calling All Creatives Who Can’t Look Away! - Parker Core Knowledge

From Ordinary to Shiny Charcadet: Calling All Creatives Who Can’t Look Away! - Parker Core Knowledge

📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta.

📌 Substitute into $ f(\theta) $:

📌 f(\theta) = (1 - 2\sin^2 \theta \cos^2 \theta) - 4\sin^2 \theta \cos^2 \theta = 1 - 6\sin^2 \theta \cos^2 \theta.

📌 Now use the identity $ \sin(2\theta) = 2\sin\theta \cos\theta $, so:

📌 \sin^2(2\theta) = 4\sin^2\theta \cos^2\theta \Rightarrow \sin^2\theta \cos^2\theta = \frac{1}{4} \sin^2(2\theta).

📌 f(\theta) = 1 - 6 \cdot \frac{1}{4} \sin^2(2\theta) = 1 - \frac{3}{2} \sin^2(2\theta).

📌 The maximum value of $ \sin^2(2\theta) $ is 1, so the minimum value of $ f(\theta) $ is:

📌 - \frac{3}{2}(1) = -\frac{1}{2}.

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