Problem 26 - Precalculus

Question

Graph the polar equations. \(r=2 \cos 5 \theta\) (five-leafed rose)

Step-by-Step Solution

Answer

The graph is a five-leafed rose with leaves extending 2 units from the origin.

Step 1: Understand the Polar Equation

The given polar equation is \(r = 2\cos 5\theta\). This type of polar equation represents a rose curve. The coefficient of \(\theta\) inside the cosine function determines the number of petals. In this case, \(5\theta\) suggests a five-leafed rose.

Step 2: Determine the Number of Leaves

For the equation \(r = a\cos(n\theta)\), if \(n\) is odd, the number of leaves is exactly \(n\). Here, \(n = 5\), which is odd, so the rose has 5 leaves.

Step 3: Determine the Length of Leaves

The coefficient of the cosine function, which is \(2\), determines the length of each petal. Therefore, each leaf will have a maximum length of 2 units from the origin.

Step 4: Sketch the Graph

To graph \(r = 2\cos 5\theta\), note that since the maximum value of \(\cos(5\theta)\) is 1, the radius \(r\) will range from 0 to 2. For different values of \(\theta\), calculate \(r\) to find points on the curve. For example, at \(\theta = 0\), \(r = 2\), which is one of the leaf tips.

Step 5: Plotting Points and Drawing the Curve

Continue plotting other points by choosing various \(\theta\) values (e.g., 0, \(\frac{\pi}{10}\), \(\frac{2\pi}{10}\), and so on). Note the symmetry and periodic structure in their arrangement. Join these points smoothly to form five lobed petals symmetrically around the origin.

Step 6: Check for Symmetry and Completeness

Ensure the graph is symmetric about both the x-axis and y-axis. Verify that each petal is equally spaced, forming a pattern where each leaf occurs 72 degrees apart \((360/5)\). This confirms the completeness of the graph plotting.

Key Concepts

Understanding Graphing Polar Curves

When you're graphing polar curves, you're essentially drawing the path traced by a point that moves in the plane according to a polar equation. This method uses polar coordinates, where each point on the plane is determined by a distance from a reference point (radius) and an angle from a reference direction. The main difference from Cartesian coordinates is that here you describe locations using a radius and angle, which is very useful for capturing circular or angular forms.

A polar equation can be of the form \( r = f(\theta) \), where \( r \) is the distance from the pole (usually the origin) and \( \theta \) is the angle measured from the positive x-axis. When sketching these curves, it's crucial to know:

• How the radius \( r \) changes as the angle \( \theta \) changes.
• The range of \( \theta \) values that provide a complete understanding of the graph.
• The symmetry within the polar graph which simplifies plotting points.

For example, preparing to graph the curve described by a polar equation like \( r = 2 \cos 5\theta \) involves understanding how \( r \) changes as you input different values of \( \theta \), indicating where the petals will locate and how they stretch around the pole.

Exploring Rose Curves

Rose curves are a special type of polar graphs that look like petalled flowers. These curves have equations of the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \), where \( a \) determines the petal length and \( n \) dictates the pinwheel-like pattern's petal count. If \( n \) is odd, you'll get \( n \) petals; if \( n \) is even, you will see \( 2n \) petals.

In the example of \( r = 2 \cos 5\theta \), the formula tells us:

• The multiplier of \( \theta \), which is 5, means this is a five-leafed rose.
• The coefficient outside of \( \cos \), which is 2, defines the length of each petal. Hence, these petals extend 2 units from the center.

Each petal forms at specific angles and reaches its maximum at those peak positions of \( \theta \), fully portraying that enchanting symmetrical flower shape. These beautifully systematic rose curves provide a quick, artistic representation of mathematical symmetry and periodicity.

Discovering Symmetry in Polar Graphs

Symmetry in polar graphs makes them especially intriguing and beautiful. Identifying this balance helps to simplify the graphing process, as you often only need to plot part of the graph before mirroring it to complete the picture.

Polar graphs can exhibit several types of symmetry:

• Symmetry about the polar axis: If replacing \( \theta \) with \(-\theta\) yields the same equation, the graph is symmetric with respect to the polar (or horizontal) axis.
• Symmetry about the line \( \theta = \frac{\pi}{2} \): If replacing \( \theta \) with \( \pi - \theta \) results in an identical equation, it shows symmetry regarding the vertical line through the pole.
• Origin symmetry: The graph is origin-symmetric if \( r, \theta \) and \(-r, \theta + \pi \) both satisfy the equation.

For the curve \( r = 2 \cos 5\theta \), observe that this symmetry ensures we have a neat radial layout for the petals, conveniently arrayed every 72 degrees (since 360° divided by 5 petals equals 72° per petal). These symmetrical properties are key to elegant and accurate plotting of polar curves.