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Home » 13-Year-Old Geniusality: What This Teen Series Has Landed on Every Top List! - Midis

13-Year-Old Geniusality: What This Teen Series Has Landed on Every Top List! - Midis

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 Question: A historian of science analyzes an ancient diagram of celestial orbits, where the minimum value of the expression $ |2\sin x + 3\cos x + 4| $ over real $ x $ reflects a key symmetry in Ptolemaic model predictions. Find this minimum value.

📌 Solution: We aim to minimize:

📌 |2\sin x + 3\cos x + 4|

📌 First, consider the expression inside the absolute value: $ 2\sin x + 3\cos x $. This is a linear combination of sine and cosine. We write it in amplitude-phase form:

📌 2\sin x + 3\cos x = R\sin(x + \phi)

📌 where $ R = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $.

📌 2\sin x + 3\cos x = \sqrt{13} \sin(x + \phi)

📌 The range of $ \sqrt{13} \sin(x + \phi) $ is $ [-\sqrt{13}, \sqrt{13}] $.

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