53–56. Simple curves Tabulate and plot enough points to sketch a graph of the following equations.


r = 1 - cos θ

Verified step by step guidance

1. Recognize that the given equation \(r = 1 - \cos \theta\) is in polar coordinates, where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
2. Create a table of values by choosing several values of \(\theta\) between \$0\( and \)2\pi\( (for example, \)0\(, \)\frac{\pi}{6}\(, \)\frac{\pi}{4}\(, \)\frac{\pi}{2}\(, \)\pi\(, \)\frac{3\pi}{2}\(, \)2\pi\(). For each \)\theta\(, calculate the corresponding \)r\( using the formula \)r = 1 - \cos \theta$.
3. Convert each polar coordinate \((r, \theta)\) into Cartesian coordinates \((x, y)\) using the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). This will help in plotting the points on the Cartesian plane.
4. Plot the points \((x, y)\) on the Cartesian plane. Since \(r\) depends on \(\theta\), the points will trace out the curve as \(\theta\) varies from \$0\( to \)2\pi$.
5. Connect the plotted points smoothly to sketch the graph of the curve. Notice the shape formed by the curve \(r = 1 - \cos \theta\) is a cardioid, a heart-shaped curve, which is symmetric about the horizontal axis.

Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). Here, r is the distance from the origin, and θ is the angle from the positive x-axis. Understanding this system is essential for plotting and interpreting curves defined by r as a function of θ.

Graphing Polar Equations

Graphing polar equations involves calculating values of r for various θ values, then plotting these points in polar form. By tabulating points for θ in a suitable range (usually 0 to 2π), you can sketch the curve's shape. This process helps visualize the curve defined by r = 1 - cos θ.

Properties of the Cardioid

The equation r = 1 - cos θ describes a cardioid, a heart-shaped curve in polar coordinates. Recognizing this helps anticipate the curve's shape and symmetry. The cardioid has a cusp at the origin and is symmetric about the polar axis, which aids in sketching and understanding its behavior.