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To graph a polar equation like \(r = \cos \theta\), it's essential to understand how to plot points in polar coordinates. Polar coordinates represent points in a plane using a radius and an angle. To begin, choose several values for the angle \(\theta\) between \(0\) and \(2\pi\). Calculate the corresponding radius \(r\) for each angle. For example:

• \(\theta = 0^\circ \rightarrow r = \cos(0^\circ) = 1\)
• \(\theta = 90^\circ \rightarrow r = \cos(90^\circ) = 0\)
• \(\theta = 180^\circ \rightarrow r = \cos(180^\circ) = -1\)

Now that you have (\(r, \theta\)) pairs, plot these points on polar graph paper. The paper is divided into circles (for \(r\) values) and lines (for \(\theta\) values), making it easier to locate each point accurately. Repeat this process for multiple points to accurately plot the curve. Connecting these points smoothly should reveal the shape of the graph.

In polar coordinates, each point is defined by an angle \(\theta\) and a radius \(r\). The angle \(\theta\) is measured from the positive x-axis, counterclockwise. The radius \(r\) is the distance from the origin to the point. When graphing \(r = \cos \theta\), you are essentially plotting how the radius changes with the angle:

• For \(\theta = 0^\circ\), \(r = 1\). This means 1 unit away from the origin along the positive x-axis.
• For \(\theta = 90^\circ\), \(r = 0\). This means the point is at the origin.
• For \(\theta = 180^\circ\), \(r = -1\). A negative radius flips the point across the origin.

By plotting these key points and others between 0 and \(2\pi\) radians (or 0° to 360°), you map out the curve more accurately. The relationship between the angle and radius in this equation generates a circle when plotted.

After manually plotting the polar equation \(r = \cos \theta\), a graphing calculator can be a useful tool to verify your work. Most graphing calculators allow you to input polar equations directly:

• Select the polar coordinates mode on the calculator.
• Input the equation \(r = \cos \theta\).
• Set the range for \(\theta\) from \(0\) to \(2\pi\).

The graphing calculator will produce a visual representation of the equation. Use this to cross-check the manually plotted points and the overall shape of the graph. This verification step ensures your understanding and accuracy in plotting polar equations. It additionally helps in recognizing the familiar shapes that different polar equations produce.