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📅 May 10, 2026 👤 scraface

May 10, 2026

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📌 Thus, the range of $ f(x) $ is $[-2, 2]$.

📌 \boxed{[-2, 2]}

📌 Question: A mammalogist observes that the social interaction frequency of a species follows $ \theta $ satisfying $ \tan \theta + \cot \theta = 4 $. Find the sum of all solutions $ \theta \in [0^\circ, 360^\circ] $.

📌 Solution: Rewrite $ \tan \theta + \cot \theta = 4 $ as:

📌 \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = 4 \implies \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = 4.

📌 Since $ \sin^2 \theta + \cos^2 \theta = 1 $, this simplifies to:

📌 \frac{1}{\sin \theta \cos \theta} = 4 \implies \sin \theta \cos \theta = \frac{1}{4}.

📌 Using $ \sin 2\theta = 2\sin \theta \cos \theta $, we get:

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