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Start by evaluating the function \(r = \sin \theta\) at key angles (e.g., multiples of \(\pi/2\)): - \(\theta = 0\): \(r = \sin (0) = 0\) - \(\theta = \frac{\pi}{2}\): \(r = \sin \left(\frac{\pi}{2}\right) = 1\) - \(\theta = \pi\): \(r = \sin (\pi) = 0\) - \(\theta = \frac{3\pi}{2}\): \(r = \sin \left(\frac{3\pi}{2}\right) = -1\) - \(\theta = 2\pi\): \(r = \sin (2\pi) = 0\) Based on these values, we can see that the curve starts at the origin, moves away from the origin as \(\theta\) increases, and then moves back toward the origin. This suggests that the curve has a loop-like structure.

Symmetry and periodic behavior are essential when analyzing polar curves. Begin by noting that the sine function is periodic with a period of \(2\pi\). Therefore, the curve will repeat itself over each interval of \(2\pi\). In addition, we can check for symmetry by evaluating \(\sin (-\theta)\). Since \(\sin (-\theta) = - \sin \theta\), the polar curve exhibits symmetry across the origin; that is, reflecting the curve over the origin yields the same curve.

Based on our analysis, we can sketch the curve in the following manner: 1. Plot the points corresponding to the important angles we found in Step 1, such as the origin (0,0), (1, \(\pi/2\)), (0, \(\pi\)), (-1, 3\(\pi/2\)), and (0, 2\(\pi\)). 2. Using these points as a guide, draw a loop that extends from the origin outward, with the distance from the origin increasing as we move from \(\theta = 0\) to \(\frac{\pi}{2}\) and then decreasing again as we move from \(\frac{\pi}{2}\) to \(\pi\). 3. Reflect this loop across the origin, as we determined the curve has symmetry across the origin. 4. Finally, since the curve is periodic with a period of 2\(\pi\), we can draw additional instances of this looped curve for any integer multiples of \(2\pi\). By following these steps, we can sketch the polar curve \(r = \sin \theta\), which appears as a series of symmetric loops revolving around the origin.