91Ó°ÊÓ

The equation given is in polar coordinates, where each point on the curve is defined by a radius \( r \) and an angle \( \theta \). In this case, \( r = 1 + \sin \theta \).

To determine symmetry, we check three types:- **Symmetry about the x-axis (horizontal axis):** Substitute \( \theta \) with \(-\theta\) and see if the equation remains unchanged.- **Symmetry about the y-axis (vertical line \(\theta = \frac{\pi}{2}\)):** Replace \( \theta \) with \(\pi - \theta\).- **Symmetry about the origin:** Apply \( r \) to \(-r \).For \( r = 1 + \sin \theta \):- Substituting \( \theta \) with \(-\theta\) gives \( r = 1 - \sin \theta \), which is not the same, so no x-axis symmetry.- Substituting \( \theta \) with \(\pi - \theta\) results in \( r = 1 + \sin \theta \), unchanged, so there is symmetry about the y-axis (vertical line at \(\theta = \frac{\pi}{2}\)).- Using \(-r\), we see it does not hold true for origin symmetry.

To sketch \( r = 1 + \sin \theta \):- **Set \( \theta = 0 \):** \( r = 1 + \sin 0 = 1 \).- **Set \( \theta = \frac{\pi}{2} \):** \( r = 1 + \sin \frac{\pi}{2} = 2 \).- **Set \( \theta = \pi \):** \( r = 1 + \sin \pi = 1 \).- **Set \( \theta = \frac{3\pi}{2} \):** \( r = 1 + \sin \frac{3\pi}{2} = 0 \).- Plot these values and others between them to sketch the curve. The resulting shape is a cardioid centered at (1,0) with its peak point at radius 2 when \(\theta = \frac{\pi}{2}\).

The shape is confirmed as a cardioid, which can be visualized by the points plotted from the sketch, verifying the symmetry about the y-axis as expected. The curve touches the origin and then loops around with a maximum radial length of 2 at \(\theta = \frac{\pi}{2}\).