Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=7 \cos 5 \theta\) (five-leaved rose)

Step-by-Step Solution

Answer

The graph is a five-petaled rose symmetric about the polar axis.

Step 1: Identify the Polar Equation

The given polar equation is \( r = 7 \cos 5\theta \). This type of equation is a rose curve where \( r \) is expressed in terms of \( \cos \theta \) with a positive integer multiplied by \( \theta \).

Step 2: Determine the Number of Petals

For a polar equation in the form of \( r = a \cos n\theta \), if \( n \) is odd, the graph will have \( n \) petals. Since \( n = 5 \) in the given equation, the graph will have 5 petals.

Step 3: Analyze Symmetry

Since the equation involves \( \cos \theta \), the graph is symmetric about the polar axis (the horizontal axis in polar coordinates). To verify, replace \( \theta \) with \(-\theta\) and observe that \( \cos(-5\theta) = \cos(5\theta) \), confirming symmetry about the line \( \theta = 0 \).

Step 4: Plot Key Points

To sketch the graph, choose key values of \( \theta \) such as \( 0, \frac{\pi}{5}, \frac{2\pi}{5}, \ldots \) and calculate corresponding values of \( r \). Note the maximum \( r \) value is 7, obtained when \( \cos 5\theta = 1 \). This occurs at \( \theta = \frac{k\pi}{5} \) where \( k \) is an integer.

Step 5: Sketch the Graph

Begin drawing the petals by plotting the calculated points and connecting them smoothly, ensuring each petal extends from the center. There will be 5 evenly spaced petals. The completed graph should resemble a symmetric five-leaved flower.

Key Concepts

Rose Curve

A rose curve is a type of polar graph resembling a flower with petals. These curves can be described in polar coordinates using equations of the form \( r = a \cos n\theta \) or \( r = a \sin n\theta \), where \( a \) and \( n \) are positive constants. The values of \( a \) and \( n \) primarily determine the shape and number of petals in the graph.

• When \( n \) is an odd integer, the curve will have exactly \( n \) petals. Conversely, if \( n \) is even, the graph will exhibit \( 2n \) petals.
• The amplitude \( a \) defines the length of each petal; in the equation \( r = 7 \cos 5\theta \), each petal reaches a maximum radius of 7.

Rose curves can make fascinating patterns, often appearing as decorative designs, and illustrate the combination of periodic functions and geometry.

Graph Symmetry

Symmetry plays a significant role in understanding the properties of polar curves. In polar equations, symmetry can assist us in graph sketching by reducing the amount of point plotting we need. The symmetry of a curve depends on whether it involves \( \, \cos \, \theta \, \, \) or \( \, \sin \, \theta \,\).

• Equations involving \( \, \cos \, \theta \,\) are generally symmetric about the polar axis (the horizontal line). This means the curve looks the same above and below this axis.
• Those involving \( \, \sin \, \theta \,\) exhibit symmetry about the line \( \theta = \frac{\pi}{2} \), the vertical line in polar coordinates.
• To check for symmetry in an equation like \( r = 7 \cos 5\theta \), one verifies that replacing \( \theta \) with \(-\theta\) yields the original equation: \( \cos(-5\theta) = \cos(5\theta) \).

This confirmation cements its symmetry about the horizontal polar axis, making plotting both easier and faster.

Polar Equation

A polar equation is a mathematical expression that defines a curve on a polar coordinate system. Unlike Cartesian coordinates, which use \( x \) and \( y \) to denote positions, polar coordinates use \( r \) (the radial distance from the origin) and \( \theta \) (the angle from the positive \( x \)-axis). This system is particularly useful for modeling circular and spiral patterns.

• Given an equation like \( r = 7 \cos 5\theta \), \( r \) tells us how far from the origin the curve is at any angle \( \theta \).
• The angle component \( \theta \) rotates around the origin, generating the curve's shape as it changes.
• Polar equations can produce intricate patterns quickly, primarily when containing trigonometric functions like \( \, \cos \, \) and \( \, \sin\,\).

When interpreting these equations, slight variations in \( \theta \) can lead to distinctive and beautiful mathematical graphs.

Sketching Graphs

Sketching graphs of polar equations involves understanding the behavior of \( r \) as \( \theta \) varies. This strategy helps visualize complex curves without plotting each decimal point manually. Here's a step-by-step approach:

• Begin by identifying key angles for \( \theta \). For a rose curve like \( r = 7 \cos 5\theta \), it is wise to select multiples of \( \frac{\pi}{5} \) because \( 5\theta \) simplifies nicely to integer multiples of \( \pi \).
• Calculate \( r \) at these angles to find the radial distances. The maximum \( r = 7 \) occurs when \( \cos 5\theta = 1 \).
• Plot these polar coordinates on the graph, starting from the origin. Connect the points smoothly to form each petal.
• Ensure each petal's symmetry by reflecting the plotted angles across the relevant axes.

Ultimately, sketching a rose curve results in a symmetrical and balanced graph, visualizing the intriguing patterns of polar coordinates.