In Exercises 64–70, graph each polar equation. Be sure to test for symmetry. r = 2 + 2 sin θ

Verified step by step guidance

1. Identify the type of polar equation. The given equation $r=2+2\sin \theta$ is a limaçon with an inner loop because the coefficients of $r$ and $\sin \theta$ are equal.
2. Test for symmetry. Substitute $\theta$ with $-\theta$ to check for symmetry about the polar axis. The equation becomes $r=2+2\sin (-\theta )=2-2\sin \theta$, which is not the same as the original equation, so there is no symmetry about the polar axis.
3. Test for symmetry about the line $\theta =\frac{\pi }{2}$. Substitute $\theta$ with $\pi -\theta$. The equation becomes $r=2+2\sin (\pi -\theta )=2+2\sin \theta$, which is the same as the original equation, indicating symmetry about the line $\theta =\frac{\pi }{2}$.
4. Test for symmetry about the pole. Substitute $r$ with $-r$ and $\theta$ with $\theta +\pi$. The equation becomes $-r=2+2\sin (\theta +\pi )=2-2\sin \theta$, which is not the same as the original equation, so there is no symmetry about the pole.
5. Plot the graph. Start by plotting key points for $\theta =0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$ and connect them smoothly, considering the symmetry about the line $\theta =\frac{\pi }{2}$. The graph will have an inner loop.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the pole) and an angle from a reference direction. In the equation r = 2 + 2 sin θ, 'r' denotes the radius or distance from the origin, while 'θ' is the angle measured from the positive x-axis. Understanding how to convert between polar and Cartesian coordinates is essential for graphing polar equations.

Symmetry in Polar Graphs

Symmetry in polar graphs can be determined by analyzing the equation with respect to specific angles. A graph is symmetric about the polar axis if replacing θ with -θ yields the same equation, and symmetric about the line θ = π/2 if replacing r with -r gives the same result. Testing for symmetry helps in sketching the graph accurately and understanding its properties.

Graphing Polar Equations

Graphing polar equations involves plotting points based on the values of 'r' for various angles 'θ'. The shape of the graph can vary significantly depending on the equation's form. For r = 2 + 2 sin θ, the graph will exhibit a specific shape, which can be identified by calculating values of 'r' at key angles and observing the overall pattern, including any loops or intersections.