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

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Recall the three common tests for symmetry in polar equations: symmetry about the polar axis (x-axis), symmetry about the line \( \theta = \frac{\pi}{2} \) (y-axis), and symmetry about the pole (origin).

To test for symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation \( r = 1 - \sin \theta \) and check if the equation remains unchanged.

To test for symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged.

To test for symmetry about the pole, replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \) and check if the equation remains unchanged.

After determining the symmetries, plot points for various values of \( \theta \) (for example, \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi \), etc.) by calculating corresponding \( r \) values using \( r = 1 - \sin \theta \), then sketch the graph accordingly.

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주요 개념

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Polar Coordinates and Equations

Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar equations express r as a function of θ, describing curves in the plane. Understanding how to interpret and plot these equations is essential for graphing polar curves.

Symmetry Tests in Polar Graphs

Symmetry in polar graphs can be tested about the polar axis, the line θ = π/2, and the pole (origin). These tests involve substituting θ with -θ, π - θ, or replacing r with -r to check if the equation remains unchanged, helping to simplify graphing and understand the curve's shape.

Graphing Polar Equations Involving Sine

Polar equations with sine functions, like r = 1 − sin θ, often produce limaçon or cardioid shapes. Recognizing the effect of the sine term on the radius as θ varies helps in sketching the curve accurately, noting key points such as maximum and minimum r values.