Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 polar graph of \(r=7\cos 5\theta\) is a five-leaved rose, symmetric about the polar axis.

Step 1: Understand the Equation

The given equation is a polar equation, which is of the form \(r = a \cos(n \theta)\). This is a polar rose equation, with \(a = 7\) and \(n = 5\). The coefficient \(n\) tells us that this will be a five-leaved rose, as it is odd.

Step 2: Determine Symmetry

The equation \(r = 7 \cos 5 \theta\) indicates symmetry about the polar axis (x-axis in Cartesian coordinates), because cosine functions are symmetric about the x-axis. For polar graphs, if \(r = \cos(n \theta)\), symmetry about the line \(\theta = 0\) is expected.

Step 3: Plot the Key Points

To plot the graph, we find key angles where the function has notable values. Typically, for \(r = a \cos(n \theta)\), start with \(\theta = 0\). Compute \(r\) at critical points like \(\theta = 0, \frac{\pi}{5}, \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{4\pi}{5}, \frac{\pi}{2}, \pi\) etc., and record the radius at these points. For example, \(r = 7 \cos(0) = 7\) at \(\theta = 0\).

Step 4: Sketch the Graph

Using the key points calculated, sketch the polar graph. Draw the radial lines corresponding to each calculated \(\theta\) and mark points at the respective \(r\) values. Join these points smoothly to form the petals of the rose.

Step 5: Verify the Graph

Check if the graph conforms to expectations: it should have 5 symmetric petals due to \(n = 5\) being odd. Examine if the symmetry about the polar axis is visually clear, confirming the theoretical symmetry determined earlier.

Key Concepts

Polar Graph Symmetry

In polar coordinates, symmetry is a key principle that simplifies plotting and understanding graphs. For the polar graph given by the equation \( r = 7 \cos 5 \theta \), we're looking at specific symmetry types:

• Polar Axis Symmetry: If you draw a line along the polar axis (the equivalent of the x-axis), the polar graph will appear mirror-symmetric.
• Line Symmetry: For cosine functions like this one, symmetry around the line \( \theta = 0 \) is a critical property. It effectively means that the graph will look the same on either side of this line.

Visualizing these symmetries helps in predicting how the polar graph should look, aiding in the accurate sketching of the graph. Remember that using symmetry ensures that if you accurately plot part of the graph, you can reflect it to get the rest, simplifying the work needed.

Five-Leaved Rose

The term "five-leaved rose" refers to a specific type of polar graph that creates a flower-like shape with five distinct petals. Polar equations of the form \( r = a \cos (n \theta) \) or \( r = a \sin (n \theta) \) can depict roses. Here, \( n \) represents the number of petals for odd values. For our equation, \( n = 5 \) leads to five petals.

• Consider this: when \( n \) is odd, the number of petals is exactly \( n \).
• The factor \( a \), which is 7 in our case, determines the length of each petal from the origin.

Understanding these properties helps in predicting and sketching the rose's appearance. The petals are evenly spaced out around the origin, displaying a beautiful radial pattern.

Plotting Polar Graphs

Plotting polar graphs involves calculating key points and understanding symmetry. Here's a step-by-step guide as demonstrated in the solution:

• Identify Critical Angles: Angles such as \( \theta = 0, \frac{\pi}{5}, \frac{2\pi}{5}, ... \) give us notable values for \( r \).
• Calculate Radius: Substitute these angles into our equation \( r = 7 \cos 5 \theta \) to find corresponding radius values.
• Sketch the Graph: Draw radial lines at each critical angle and plot points at the calculated radius on these lines. The symmetrical nature of the graph means we can reproduce this pattern around the origin to sketch the entire rose.

This straightforward process uses both the mathematical computations and visual symmetry to complete the graph.

Cosine Symmetry

Cosine functions, like \( \cos 5 \theta \) in our equation, hold a special symmetrical property. Here's how it applies:

• Symmetrical About the X-axis: Cosine graphs in Cartesian coordinates are symmetric about the x-axis—this feature translates to symmetry about the polar axis in polar coordinates.
• Repetitive Behavior: The periodic nature of cosine means that values repeat over regular intervals, contributing to a balanced, predictable graph.

These features make plotting graphs with cosine components easier, as you can anticipate how parts of the graph mirror each other along certain axes or lines. This ability to predict behavior simplifies the plotting of complex polar equations, such as our five-leaved rose.