How do I graph cardioid r=a(1-cos θ)?

To graph the cardioid given by the equation r = a(1 - cos θ), follow these steps:

Understanding the Equation

In this polar equation, r represents the distance from the origin, θ is the angle, and a is a constant that affects the size of the cardioid.

Setting Up the Graph

• Choose a value for a. For example, let’s use a = 1.
• Set up a polar coordinate system, where the angle θ is measured from the positive x-axis.

Calculating Points

Calculate the values of r for different angles θ. Common angles to use are:

• θ = 0°0 radians: r = 1(1 - cos(0)) = 0
• θ = 90°π/2 radians: r = 1(1 - cos(π/2)) = 1
• θ = 180°π radians: r = 1(1 - cos(π)) = 2
• θ = 270°3π/2 radians: r = 1(1 - cos(3π/2)) = 1

Plotting the Points

On your polar graph, plot the calculated points:

• At θ = 0°, plot (0, 0).
• At θ = 90°, plot (1, π/2).
• At θ = 180°, plot (2, π).
• At θ = 270°, plot (1, 3π/2).

Connecting the Dots

Once you have plotted several points, connect them smoothly to form the cardioid shape. The graph will resemble a heart or a rounded shape with a cusp at the origin.

Final Touches

Label your axes and ensure your graph is clear. You can also try different values of a to see how the size of the cardioid changes.