In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − sin θ

Verified step by step guidance

1. Identify the types of symmetry to test for polar equations: symmetry with respect to the polar axis (x-axis), the line $\theta =\frac{\pi }{2}$ (y-axis), and the pole (origin).
2. Test for symmetry with respect to the polar axis by replacing $\theta$ with $-\theta$ in the equation $r=1-\sin \theta$. Check if the resulting equation is equivalent to the original.
3. Test for symmetry with respect to the line $\theta =\frac{\pi }{2}$ by replacing $\theta$ with $\pi -\theta$ in the equation $r=1-\sin \theta$. Check if the resulting equation is equivalent to the original.
4. Test for symmetry with respect to the pole by replacing $r$ with $-r$ and $\theta$ with $\theta +\pi$ in the equation $r=1-\sin \theta$. Check if the resulting equation is equivalent to the original.
5. After testing for symmetry, plot the graph of the polar equation $r=1-\sin \theta$ by calculating $r$ for various values of $\theta$ and plotting the corresponding points in polar coordinates.

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 polar equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. 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 tested by substituting specific values into the polar equation. A graph is symmetric about the polar axis if replacing θ with -θ yields the same equation, and it is symmetric about the line θ = π/2 if replacing r with -r gives the same result. Recognizing these symmetries helps in sketching the graph accurately.

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. For the equation r = 1 - sin θ, understanding how to evaluate 'r' at key angles (like 0, π/2, π, and 3π/2) will help in accurately sketching the graph and identifying its features.