57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = 2 - 2 sin θ  b

Verified step by step guidance

1. Identify the given polar equation: \(r = 2 - 2 \sin \theta\). This represents a polar curve where \(r\) depends on the angle \(\theta\).
2. Recall that \(r\) is the distance from the origin to a point on the curve, and \(\theta\) is the angle measured from the positive x-axis. To graph, you will plot points for various values of \(\theta\) between \(0\) and \(2\pi\).
3. Create a table of values by choosing several \(\theta\) values (for example, \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\pi\), etc.) and compute the corresponding \(r\) values using the equation \(r = 2 - 2 \sin \theta\).
4. Plot each point in polar coordinates by moving \(r\) units from the origin at the angle \(\theta\). Connect these points smoothly to reveal the shape of the curve.
5. Use a graphing utility to input the equation \(r = 2 - 2 \sin \theta\) and compare the plotted points and shape with your manual graph to verify accuracy.

Key Concepts

Polar Coordinates and Polar Equations

Polar coordinates represent points in the plane using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). Understanding how to interpret and plot equations given in polar form, like r = 2 - 2 sin θ, is essential for graphing polar curves.

Graphing Polar Curves

Graphing polar curves involves plotting points for various values of θ and connecting them smoothly. Recognizing common shapes such as cardioids, limacons, and circles helps in sketching the curve accurately before verifying with a graphing utility.

Use of Graphing Utilities

Graphing utilities, such as graphing calculators or software, allow for precise visualization of polar curves. They help confirm the shape and key features of the graph, ensuring accuracy and aiding in understanding complex polar equations.