Graphing function with polar coordinate

Question

I am studying polar coordinates and I am not understanding what's the practical method for graphing this relation:

$$r = \frac{1}{2} + \sin \theta, \text{for } 0 < \theta < 2\pi$$.

I plotted using Wolfram Alpha, but I don't understand what happened in the interval $\pi < \theta < {3\pi}/{2}$ (I mean, where did the inner loop come from?).

Answer 1

Calculate x and y in a table separately for

$$ x = (1/2 + \sin \theta) \cos \theta, $$ $$ y = (1/2 + \sin \theta)\sin \theta. $$

You notice that not always we have a single $r$ for a given $ \theta$ for the entire domain. In this case double angle $\theta $ gives rise to another $r$ after further rotation.

Likewise $ r = \sin 2\theta $ gives rise to four leaves in the plot.