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] $.

["Understanding the Equation: Finding All Solutions to $ \ an \ heta + \cot \ heta = 4 $ in $[0^\circ, 360^\circ]$", "Mammalogists and behavioral ecologists often analyze social interaction patterns in animal species, and quantitative models based on trigonometric functions can reveal underlying rhythmic behaviors. One such model involves a trigonometric equation of the form:", "$$\n\ an \ heta + \cot \ heta = 4\n$$", "This equation, while seemingly simple, encodes meaningful behavioral rhythms in social species. Understanding all solutions within the interval $[0^\circ, 360^\circ]$ helps researchers predict and interpret periods of heightened or reduced social activity. This article explores how to solve this equation and compute the sum of all valid angles.", "---", "Rewriting the Equation", "Start with the given equation:", "$$\n\ an \ heta + \cot \ heta = 4\n$$", "Recall that $ \cot \ heta = \frac{1}{\ an \ heta} $. Let $ x = \ an \ heta $, so the equation becomes:", "$$\nx + \frac{1}{x} = 4\n$$", "Multiply both sides by $ x $ (noting $ x <br/>\ne 0 $) to eliminate the denominator:", "$$\nx^2 + 1 = 4x\n$$", "Rearranging gives a quadratic:", "$$\nx^2 - 4x + 1 = 0\n$$", "Solving using the quadratic formula:", "$$\nx = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(1)}}{2} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}\n$$", "Thus, $ \ an \ heta = 2 + \sqrt{3} $ or $ \ an \ heta = 2 - \sqrt{3} $.", "---", "Finding Angles in $[0^\circ, 360^\circ]$", "We now find $ \ heta $ such that:", "1. $ \ an \ heta = 2 + \sqrt{3} $\n2. $ \ an \ heta = 2 - \sqrt{3} $", "We recognize these values from known trigonometric identities.", "Recall:", "- $ \ an 75^\circ = \ an(45^\circ + 30^\circ) = \frac{\ an 45^\circ + \ an 30^\circ}{1 - \ an 45^\circ \ an 30^\circ} = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3}+1}{\sqrt{3}}}{\frac{\sqrt{3}-1}{\sqrt{3}}} = \frac{\sqrt{3}+1}{\sqrt{3}-1} = 2 + \sqrt{3} $", "Similarly,", "- $ \ an 15^\circ = \ an(45^\circ - 30^\circ) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}} = \frac{\sqrt{3}-1}{\sqrt{3}+1} = 2 - \sqrt{3} $ (after rationalizing)", "Thus, the principal solutions are:", "- $ \ heta = 15^\circ $ and $ \ heta = 15^\circ + 180^\circ = 195^\circ $ for $ \ an \ heta = 2 - \sqrt{3} $\n- $ \ heta = 75^\circ $ and $ \ heta = 75^\circ + 180^\circ = 255^\circ $ for $ \ an \ heta = 2 + \sqrt{3} $", "All four angles lie within $[0^\circ, 360^\circ]$.", "---", "Sum of All Solutions", "Now compute the sum:", "$$\n15^\circ + 195^\circ + 75^\circ + 255^\circ = (15 + 195) + (75 + 255) = 210 + 330 = 540^\circ\n$$", "---", "Conclusion", "The equation $ \ an \ heta + \cot \ heta = 4 $ has four solutions in $[0^\circ, 360^\circ]$: $ 15^\circ, 75^\circ, 195^\circ, 255^\circ $. Their sum is:", "$$\n\boxed{540^\circ}\n$$", "Understanding such patterns aids mammalogists in timing behavioral observations, modeling social cycles, and predicting interaction peaks—key steps in conservation and ethology research. This elegant trigonometric relationship reveals how seemingly simple functions encode complex natural rhythms.", "---", "Keywords: mammalogy, social behavior, tan θ + cot θ = 4, trigonometric equation, mammal interaction model, sum of solutions, 15°, 75°, 195°, 255°, 360° range, behavioral ecology."]