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Home » Now check which satisfy the original equation \(\sin z + \cos z = 1\): - Dachbleche24

Now check which satisfy the original equation \(\sin z + \cos z = 1\): - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Using the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\), the expression becomes:

📌 + 2 + \sec^2 \theta + 2 + \csc^2 \theta = 5 + \sec^2 \theta + \csc^2 \theta

📌 Recall the identities \(\sec^2 \theta = 1 + \tan^2 \theta\) and \(\csc^2 \theta = 1 + \cot^2 \theta\), so:

📌 + \sec^2 \theta + \csc^2 \theta = 7 + \tan^2 \theta + \cot^2 \theta

📌 Using the AM-GM inequality, \(\tan^2 \theta + \cot^2 \theta \geq 2\), hence:

📌 + \tan^2 \theta + \cot^2 \theta \geq 7 + 2 = 9

📌 The minimum value is achieved when \(\tan^2 \theta = \cot^2 \theta = 1\), i.e., \(\theta = \frac{\pi}{4} + n\pi\). Therefore, the minimum value is:

📌 Question:** An entomologist specializing in insect ecology is studying the movement patterns of a certain species of insect, modeled by the equation \((x - 3)^2 + (y - 4)^2 = 25\). Determine the number of integer coordinate points \((x, y)\) that lie on this circle.

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