MathCalculus

Find the Points of Horizontal Tangency to the Polar Curve r = 2 csc θ + 3

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Question

Find the points of horizontal tangency, if any, to the polar curve

r=2csc⁡θ+3.r=2\csc\theta+3.

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Evidence boundary

The polar equation and task wording are taken from the cited National University of Kaohsiung examination. That source gives no restricted θ interval, so the answer lists all angle representatives modulo 2π and the two distinct Cartesian points. Both candidates are checked against dx/dθ ≠ 0.