Problem 44 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=2-4 \cos 2 \theta$$

Step-by-Step Solution

Answer

The equation \(r=2-4 \cos 2 \theta\) does not exhibit symmetry about the x-axis, y-axis or origin. \nThe graph of this equation needs to be plotted manually by substituting values for \(\theta\) and calculating the corresponding \(r\) values.

Step 1: Test for Symmetry about the x-axis

Substitute \(-\theta\) for \(\theta\) in the equation, if the equation remains unchanged, then the graph is symmetric about the x-axis. However, when replacing \(\theta\) with \(-\theta\) in the equation \(r = 2-4 \cos 2 \theta\), we get \(r = 2 - 4 \cos (-2 \theta)\), which is not equivalent to the original equation. Therefore, this graph is not symmetric about the x-axis.

Step 2: Test for Symmetry about the y-axis

Replace \(-\theta\) for \(\theta\) and \(-r\) for \(r\), if the equation is still the same, then the graph is symmetric about the y-axis. Here, replacing \(\theta\) with \(-\theta\) and \(r\) with \(-r\), we obtain \(-r = 2 - 4 \cos (-2 \theta)\), which, in expanded form, would be \(-r = 2 - 4 \cos 2 \theta\). This is not equivalent to the original equation, hence, the given polar equation is not symmetric about the y-axis.

Step 3: Test for Origin Symmetry

To test for origin symmetry, substitute \(\theta + \pi\) for \(\theta\) and \(-r\) for \(r\). If the equation remains the same, then the graph is symmetric about the origin. Here, replacing \(\theta\) with \(\theta + \pi\) and \(r\) with \(-r\) in the equation, we get \(-r = 2 - 4 \cos 2 (\theta + \pi)\), which simplifies to \(-r = 2 + 4 \cos (2 \theta)\). This is also not equivalent to the original equation. Therefore, the polar graph of the equation \(r = 2-4 \cos 2 \theta\) is not symmetric with respect to the origin.

Step 4: Graph the Polar Equation

The polar coordinate system is used to plot points in the form \( (r,\theta) \) - one coordinate represents the radius and the other the angle in standard position. It's common to start with common angles (i.e., \(\theta = 0\), \(\pi/4\), \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\)) and compute the corresponding \(r\). Although, the shape of the graph might require additional points to be plotted. After plotting these points in the polar coordinate system, the points are connected smoothly to form the graph. Given that this is not a simple polar equation (like a circle, line, etc.) whose shape we can anticipate - we have to rely on plotting sufficient points to capture the shape.