Question

The butterfly curve of Example 8 is enhanced by adding a term: $$r=e^{\sin \theta}-2 \cos 4 \theta+\sin ^{5} \frac{\theta}{12}, \quad \text { for } 0 \leq \theta \leq 24 \pi$$ a. Graph the curve. b. Explain why the new term produces the observed effect. (Source: S. Wagon and E. Packel, Animating Calculus, Freeman, 1994)

Short answer

Answer: The term \(\sin^5\frac{\theta}{12}\) introduces geometric patterns and symmetry in the butterfly curve while preserving its general shape and structure. It causes slight modifications in the radial distance at certain values of \(\theta\), resulting in a visually enhanced and intricate pattern in the graph.

Step-by-step solution

Understanding the polar function

First, let's recall that a polar function is defined in terms of a radial distance \(r\) and the angle \(\theta\). In this case, given the polar equation $$r= e^{\sin \theta} - 2 \cos 4 \theta + \sin^5 \frac{\theta}{12}$$ for \(0 \leq \theta \leq 24 \pi\), the radial distance \(r\) depends on the angle \(\theta\). The coordinates are given by \((r, \theta)\).

Graphing the polar curve

To graph the given polar curve, start by setting the limits for the angle \(\theta\) from \(0\) to \(24\pi\). For each value of \(\theta\) in that range, calculate the corresponding value of \(r\) using the equation and plot the point \((r, \theta)\) in polar coordinates. Note that while graphing the curve, it is helpful to use a graphing tool or software to visualize the curve more accurately.

Understanding the effect of the new term

The enhanced butterfly curve adds the term \(\sin^5\frac{\theta}{12}\) to the original polar equation. This term introduces a symmetry in the curve, seen as a repeated pattern in the graph. The presence of this term locally modifies the radial distance for certain values of \(\theta\). The symmetry is due to the function \(\sin\) and the scaling by a factor of \(\frac{\theta}{12}\) slows down the amplitude changes. The exponent of \(5\) in this term ensures that the effect of the modifications is small and localized.

Explaining the observed effect

The observed effect of the new term \(\sin^5\frac{\theta}{12}\) on the butterfly curve from Example 8 is seen as a slightly modified radial distance at certain values of \(\theta\). It introduces geometric patterns in the curve while preserving the general shape and structure of the original butterfly curve. These patterns can be observed as repeated and symmetrical modifications in the graph of the enhanced curve.

Key concepts

Graphing Polar Functions

Graphing polar functions involves plotting points in a plane using a specific type of coordinate system, known as polar coordinates. Unlike Cartesian coordinates that use \(x\) and \(y\) values, polar coordinates are defined by the radial distance \(r\) from the origin and the angle \(\theta\), measured from the positive x-axis.
To graph a polar function, such as the one given, \(r = e^{\sin \theta} - 2 \cos 4\theta + \sin^5 \frac{\theta}{12}\), you follow these steps:

• Set \(\theta\) to vary within a given interval, here from 0 to 24\(\pi\).
• For each angle \(\theta\), compute the corresponding \(r\) using the equation.
• Plot the points \((r, \theta)\) to visualize the curve.

Graphing polar functions manually can be complex due to the calculations involved, particularly with intricate equations. Therefore, utilizing graphing software can significantly simplify the process and provide accurate visualizations.

Effect of Trigonometric Terms

In polar equations, trigonometric terms play a crucial role in determining the shape and behavior of the graph. These functions create oscillations, influencing the radius \(r\) based on \(\theta\).
The original polar equation includes trigonometric functions such as \(\sin\) and \(\cos\). Each term influences:

• Amplitude: Determines how far outwards or inwards the curve stretches.
• Frequency: Dictates how often the curve repeats its pattern.
• Phase Shift: Alters the starting point of the curve's wave.

Specifically, the term \(sin^5 \frac{\theta}{12}\) in the equation adds a local modification to the radius. Its \(\theta/12\) factor means changes occur gradually over larger intervals. The power of five further reduces drastic variations, inflecting minor adjustments that enhance the intricacy of the curve.

Symmetry in Polar Graphs

Symmetry is a common feature in polar graphs, often introduced or reinforced by certain mathematical functions. This occurs when parts of the graph mirror each other around the origin or some axis, giving it a harmonious structure.
The term \sin^5 \frac{\theta}{12}\ in this function contributes to symmetry by offering repeated patterns, attributed to the periodic nature of the \(sin\) function. The slowing of changes, due to the \(\theta/12\) scaling, creates consistency in the appearance of the graph, ensuring that similar shapes manifest across various sections of the plot.
Symmetrical patterns in polar graphs aren't just visually appealing; they also provide insights into the underlying mathematical properties of the equation being studied. Observing these patterns helps understand the effects of different terms and how they interact to influence the overall graph.