1. **State the problem:** Identify the symmetries of the polar curve $$r = \sin\left(\frac{\theta}{2}\right)$$ and then sketch it. 2. **Recall symmetry types for polar curves:** - Symmetry about the polar axis (horizontal axis) means replacing $$\theta$$ by $$-\theta$$ leaves $$r$$ unchanged. - Symmetry about the line $$\theta = \frac{\pi}{2}$$ means replacing $$\theta$$ by $$\pi - \theta$$ leaves $$r$$ unchanged. - Symmetry about the pole means replacing $$r$$ by $$-r$$ when $$\theta$$ is replaced by $$\theta + \pi$$ leaves the curve unchanged. 3. **Test symmetry about the polar axis:** Calculate $$r(-\theta) = \sin\left(\frac{-\theta}{2}\right) = -\sin\left(\frac{\theta}{2}\right) = -r(\theta)$$. Because $$r(-\theta) = -r(\theta)$$, the curve is symmetric with respect to the pole, not the polar axis. 4. **Test symmetry about the line $$\theta = \frac{\pi}{2}$$:** Calculate $$r(\pi - \theta) = \sin\left(\frac{\pi - \theta}{2}\right) = \sin\left(\frac{\pi}{2} - \frac{\theta}{2}\right) = \cos\left(\frac{\theta}{2}\right)$$. Since $$r(\pi - \theta) \ne r(\theta)$$, no symmetry about $$\theta = \frac{\pi}{2}$$. 5. **Test symmetry about the pole:** Calculate $$r(\theta + \pi) = \sin\left(\frac{\theta + \pi}{2}\right) = \sin\left(\frac{\theta}{2} + \frac{\pi}{2}\right) = \cos\left(\frac{\theta}{2}\right)$$. Note $$r(\theta + \pi) \ne -r(\theta)$$, so this is not exactly symmetric about the pole either. 6. **Conclusion on symmetry:** The curve shows symmetry originating from the periodicity and sine function structure, producing a petal-like shape, but no classical symmetry about polar axis, line $$\theta=\frac{\pi}{2}$$, or pole. 7. **Sketching:** The curve $$r=\sin\left(\frac{\theta}{2}\right)$$ produces 4 petals because the argument of sine is halved and covers $$\theta$$ from $$0$$ to $$4\pi$$ to complete the pattern. It is centered at the pole and extends symmetrically in a petal pattern. **Final answer:** The curve $$r=\sin\left(\frac{\theta}{2}\right)$$ produces a 4-petal rose symmetric about the pole but does not exhibit symmetry about the polar axis or $$\theta=\frac{\pi}{2}$$ line individually.