Problem 41 - Precalculus with Limits

Question

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r= 6\ \cos\ 3\theta\)

Step-by-Step Solution

Answer

The polar function \(r = 6\cos3\theta\) represents a rose with 6 petals of length 6, centered at the origin, symmetric about the x-axis and spanning all quadrants in the polar plane.

Step 1: Identify the Form of the Equation

The equation \(r = 6\cos3\theta\) represents a 'rose' polar graph. In general, if the equation has form \(r = a\cos n\theta\) or \(r = a\sin n\theta\), then it's a rose polar graph, which has \(n\) petals if \(n\) is odd, and \(2n\) petals if \(n\) is even.

Step 2: Determine the Symmetry

As the equation contains a \(\cos\) term, we can conclude that the graph is symmetric about the x-axis.

Step 3: Find the Zeros of the Polar Function

The zeros are the values of \(\theta\) that make \(r = 0\). Setting up \(6\cos3\theta = 0\) and solving \(\cos3\theta = 0\) gives the solutions \(\theta = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6}\). These zeros divide the x-axis into six distinct zones, which correspond to the six 'petals' of our rose.

Step 4: Discover the Maximum \(r\)-values

The maximum value of \(r\) is simply the absolute value of the coefficient of the \(\cos\) term, which is 6.

Step 5: Plot the Curve

Given these points, we can plot the graph of the polar equation. The six petals are equally spaced around the origin, with the first petal centered above the origin, along the positive x-axis.

Key Concepts

Rose Curve

A rose curve is a type of polar graph characterized by a petal-like pattern that is both visually attractive and mathematically intriguing. These curves are defined by a polar equation of the form \( r = a \cos n\theta \) or \( r = a \sin n\theta \). In these equations:

• \(a\) represents the length of each petal.
• \(n\) determines the number of petals: if \(n\) is odd, the rose will have \(n\) petals, and if it is even, \(2n\) petals.

For example, with the equation \( r = 6 \cos 3\theta \), we have a rose curve with 3 petals. Each petal extends to a maximum radius of 6. The number of petals and their arrangement are heavily influenced by the cosine (or sine) function's periodic nature and symmetry properties.

Symmetry in Polar Graphs

Symmetry in polar graphs refers to the graph's reflective balance around a particular axis or point. Certain elements within the polar equation help determine this symmetry. For instance, the presence of a \( \cos \) function suggests symmetry about the x-axis.
The graph of the equation \( r = 6 \cos 3\theta \) is symmetric about the x-axis due to the cosine term. This means if we were to fold the graph along the x-axis, both halves would overlap perfectly.
Symmetry simplifies the sketching of polar curves: you can accurately predict the plot's shape and behavior without calculating each point manually. This feature is particularly useful for complex equations, allowing for faster and correct visualization.

Polar Equations

Polar equations describe the position of points in the plane using a distance from a fixed point (the origin) and an angle from a fixed direction (typically the positive x-axis). These equations take the form \( r = f(\theta) \), where \( r \) is the radius and \( \theta \) is the angle.
For example, in the equation \( r = 6 \cos 3\theta \), \( \theta \) ranges from 0 to \( 2\pi \).

• As \( \theta \) changes, \( r \) gives the distance from the origin for each corresponding angle.
• This relationship depends on trigonometric functions like sine and cosine, defining how radii lengthen or shorten.

Through polar equations, we can explore diverse graph types beyond simple Cartesian plots. Polar equations offer unique insights and visually engaging results, like rose curves, that showcase nature's symmetry.

Maximum r-values in Polar Graphs

The maximum \( r \)-value in a polar graph signifies the farthest distance a point reaches from the origin. Determining this value is crucial for understanding the graph's boundary and extent as you plot it.
In a polar equation such as \( r = 6 \cos 3\theta \), the maximum \( r \)-value is determined by the coefficient of the cosine term. Here, the coefficient is 6, indicating that each petal will extend out to a radius of 6.
Maximum \( r \)-values offer a direct measure to visualize and predict the graph's overall size and shape. These values also highlight points where the graph shows peaks or directional extension, giving a clear, concise metric for comparing different polar graphs.