Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=2 \cos \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 2 \cos \theta\) is a circle with a diameter of 2 located on the x-axis from the origin to 2 in the positive direction.

Step 1: Understand the effects of cosine function on the radius

In polar coordinates, \(r = 2 \cos \theta\) means that the radius is twice the cosine of angle \(\theta\). As \(\theta\) increases from 0 to \(2\pi\), the cosine function oscillates from 1 to -1. Therefore, the radius \(r\) will oscillate from 2 to -2. We must note the negative radius implies that the point is in the opposite direction.

Step 2: Sketch the graph

We plot a few important points. When \(\theta = 0\), \(r = 2 \times 1 = 2\). When \(\theta = \pi/2\), \(r = 2 \times 0 = 0\). When \(\theta = \pi\), \(r = 2 \times -1 = -2\), means the point will be at 2 in the opposite direction. Lastly, when \(\theta = (3\pi)/2\), \(r = 0\). Connecting these key points, the graph will form a circle with a diameter of 2 located on the x-axis from the origin to 2 in the positive direction.

Step 3: Verify the graph

We use a graphing utility to sketch the graph of the given polar equation. It should generate a graph that matches the sketch we have drawn in the previous step. This will confirm that the approach is correct.

Key Concepts

Cosine Function in Polar Coordinates

In the polar coordinate system, the cosine function has a unique representation that directly affects the radius of a point from the origin. Normally, the cosine function oscillates between 1 and -1 as its angle varies. However, when we express it as a polar equation, such as \(r = 2 \cos \theta\), we are defining the distance, \(r\), from the origin to a point at an angle \(\theta\) from the positive x-axis.

As the angle \(\theta\) increases, the cosine value determines if the point is moving closer to or farther from the origin within a circular path. Specifically, when \(\theta = 0\), cosine is at its maximum, which means the point starts at the furthest distance on the x-axis. Thus, picturing the graph of \(r = 2 \cos \theta\) involves a series of points that are two times the cosine of the angle away from the origin, ultimately tracing out a familiar shape - a circle, in this particular instance.

Negative Radius in Polar Coordinates

A negative radius in polar coordinates may seem counterintuitive at first. In a Cartesian system, negative values simply indicate direction along an axis, but in polar coordinates, a negative radius indicates that the point is to be plotted in the direction opposite the angle \(\theta\).

For the equation \(r = 2 \cos \theta\), as \(\theta\) crosses \(\pi/2\) and approaches \(\pi\), the cosine function delivers negative values, and these results suggest plotting points in the opposite direction of \(\theta\). This concept is crucial in understanding the symmetry and the full structure of the graphs created by polar equations. It's these negative radii that allow for the complete formation of the circular shapes when the angle spans from 0 to \(2\pi\).

Graphing Polar Equations

Graphing polar equations is a process that involves plotting points which are defined by a distance from the origin and an angle with respect to the positive x-axis. To graph \(r = 2 \cos \theta\), one begins by calculating the radius for several key angles (typically \(0\), \(\pi/2\), \(\pi\), and \(3\pi/2\)) and then plotting these points in their corresponding direction.

It's vital to include the variations caused by the cosine function. Since cosine is highest at \(0\) and \(2\pi\), these points lie on the maximum radius on the x-axis. At \(\pi/2\) and \(3\pi/2\), the cosine function is at 0, so these points will be at the origin. Finally, the negative radii that occur when \(\theta > \pi/2\) should be mirrored across the origin. By connecting all these points smoothly, one can trace the path of the graph, which in the case of \(r = 2 \cos \theta\) results in a circle.

Verifying Graphs with Graphing Utility

After sketching a polar graph by hand, it's wise to verify its accuracy. This is where a graphing utility becomes invaluable. A graphing utility can plot polar equations precisely, without the errors that can occur when sketching manually. By inputting the equation \(r = 2 \cos \theta\) into such a tool, students can confirm that their hand-drawn graphs are correct.

If discrepancies are found between the two graphs, one can re-evaluate the step-by-step plotting process or confirm the settings and inputs in the graphing utility. Verifying graphs with technology ensures a clearer understanding of the concepts and prevents the reinforcement of any misconceptions. It’s a powerful method for learning graphing skills in polar coordinates.