Considering multiple coordinates ensures the graph is three-dimensional, allowing $$r = 2\cos\theta$$ to form a sphere instead of a circle.

Multiple coordinate sets are needed to plot discontinuities, making $$r = 2\cos\theta$$ a series of disconnected points rather than a continuous curve.

Multiple coordinate sets for a point account for periodicity, causing $$r = 2\cos\theta$$ to form a complete circle rather than just an arc, as each point has infinite representations.

Considering multiple coordinates reveals that $$r = 2\cos\theta$$ actually forms a spiral, not a closed shape, due to the infinite nature of angle measures.