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Chapter 6: Problem 75

Explain why the graph of \(r^{2}=\cos ^{2} \theta\) and the graph of \(r=\cos \theta\) are not the same.

Short Answer

Expert verified
The graphs of \(r^{2}=\cos ^{2}\theta\) and \(r=\cos \theta\) are not the same because the former includes mirrored locations of the latter in the origin due to the squaring operation, while the latter only includes locations in the first and fourth quadrants.

Step by step solution

01

Understand Polar Coordinates

First, a quick overview of polar coordinates. Any point in a 2D space can be specified by two values: the distance, r, of the point from the origin and the angle, 胃, the line segment from the origin to the point makes with the positive x-axis.
02

Graphing the Equation \(r=\cos \theta\)

To start, graph the equation \(r=\cos \theta\). In polar coordinates, \(r=\cos \theta\) creates a circle of radius 0.5 centered at (0.5, 0) - an illustration of unit circle shifted halfway along positive x-axis.
03

Graphing the Equation \(r^{2}=\cos ^{2} \theta\)

Next, graph the equation \(r^{2}=\cos ^{2} \theta\). Since this equation represents the square of distances from the origin, it includes both the positive and negative restrictions of \(r=\cos\theta\) and creates two circles symmetric about the origin.
04

Comparing Both Graphs

Finally, compare the two resulting graphs. The squaring operation in the second equation leads to the inclusion of negative values for r. This means the second equation includes mirrored locations of the first equation鈥檚 graph in relation to the origin, resulting in different graphs.

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