Problem 21 - Precalculus

Question

Graph the polar equations. $$r^{2}=\cos 4 \theta$$

Step-by-Step Solution

Answer

The graph is a rose curve with 8 petals, symmetric about the axes and the origin.

Step 1: Identify Symmetry

Polar equations of the form \(r^2 = \cos(k\theta)\) exhibit rose curve patterns when \(k\) is an even integer. This equation has symmetry about the x-axis, y-axis, and origin due to its cosine term.

Step 2: Determine Number of Petals

For the equation \(r^2 = \cos 4\theta\), the number of petals can be determined by the coefficient of \(\theta\). Since \(k = 4\) is even, the number of petals is \(2k\), which makes 8 petals.

Step 3: Set up for Graphing

In polar coordinates, it's helpful to start by determining the range of \(\theta\) values. The cosine function \(\cos 4\theta\) completes a full cycle as \(\theta\) varies from \(0\) to \(\pi\). Consider \(\theta\) values between \(0\) and \(\pi\).

Step 4: Test Specific Values

Select intervals to plug into the equation \(r^2 = \cos 4\theta\). For instance, \(\theta = 0, \frac{\pi}{8}, \frac{\pi}{4}, \ldots, \frac{7\pi}{8}\), and compute \(r^2\).

Step 5: Solve for \(r\)

For each chosen \(\theta\), solve \(r^2 = \cos 4\theta\) for \(r\). Remember that \(r = \pm\sqrt{\cos 4\theta}\). This accounts for the reflection of each petal across the origin.

Step 6: Plot the Points

Plot the points derived from \(r = \pm\sqrt{\cos 4\theta}\). Connect these points smoothly, ensuring symmetry to illustrate the rose with 8 petals.

Step 7: Sketch the Graph

With the points from previous steps, sketch the graph. Ensure to show all 8 petals distributed evenly around the origin with equal spacing.

Key Concepts

Symmetry in Polar Graphs

When working with polar graphs, identifying symmetry is key. Polar equations can have symmetries about the x-axis, y-axis, or the origin. This makes graphing them much simpler. In our case, the equation \(r^2 = \cos 4\theta\) has symmetry in all these axes because of the cosine term.

Here are some basics to understand symmetry in polar graphs:

• X-axis Symmetry: If replacing \(\theta\) with \(-\theta\) gives an equivalent equation, the graph is symmetric about the x-axis.
• Y-axis Symmetry: If \(r\) may be replaced with \(-r\), the graph is symmetric about the y-axis.
• Origin Symmetry: If both \(r\) and \(\theta\) can be replaced by \(-r\) and \(-\theta\) respectively, there's symmetry about the origin.

Checking symmetry before graphing can reduce the amount of calculations needed, as symmetric parts can just be mirrored across their respective axes.

Rose Curves

Rose curves are a fascinating type of polar graph. They feature petal-like patterns, making them visually striking and methodologically interesting. Understanding how to determine their number of petals is vital for sketching them correctly.

Rose curves follow the general form \(r = \cos(k\theta)\) or \(r = \sin(k\theta)\). The key to understanding their structure lies in the integer \(k\):

• When \(k\) is even, the curve will have \(2k\) petals.
• When \(k\) is odd, the curve will have \(k\) petals.

In our exercise, the equation \(r^2 = \cos 4\theta\) translates into a rose with 8 petals since \(k = 4\), an even number. Rose curves are often symmetric, displaying a rhythmical beauty in mathematics.

Graphing Polar Equations

Graphing equations in polar form is different from the Cartesian method. It's essential to understand how to work with \(r\) and \(\theta\), the two polar variables. Typically, you start by determining how \(\theta\) progresses:

• Here, \(\theta\) runs from \(0\) to \(\pi\) to complete a full cycle for the cosine function.
• At selected intervals, substitute \(\theta\) back into the equation to find \(r\).

When dealing with equations like \(r^2 = \cos 4\theta\), solving for \(r\) involves taking the square root: \(r = \pm \sqrt{\cos 4\theta}\). Always remember to consider both the positive and negative roots for complete graphs.

After getting the values of \(r\), plot these points on the polar coordinate plane. Connect them smoothly, ensuring you reflect the petals over the axes according to symmetry. Upon sketching, the distinct shape of rose curves becomes evident, showing precise, symmetrical petal patterns emerging clearly around the origin.