Question

The butterfly curve of Example 8 may be enhanced by adding a term: $$r=e^{\sin \theta}-2 \cos 4 \theta+\sin ^{5}(\theta / 12), \quad \text { for } 0 \leq \theta \leq 24 \pi$$ a. Graph the curve. b. Explain why the new term produces the observed effect.

Short answer

Question: Explain the effect of the added term \(\sin^5(\frac{\theta}{12})\) on the butterfly curve polar equation \(r(\theta) = e^{\sin\theta} - 2\cos(4\theta) + \sin^5(\frac{\theta}{12})\). Answer: The added term \(\sin^5(\frac{\theta}{12})\) introduces a modulating effect on the butterfly curve, resulting in gentle variations and local changes in the original shape. This is due to its small amplitude and lower frequency, which creates subtle effects compared to the other terms in the equation.

Step-by-step solution

Graphing the Polar Equation

To graph the polar equation, we need to plot \(r(\theta)\) for \(\theta\) in the given interval \([0, 24\pi]\). You can use graphing software or graphing calculator to generate the polar plot of the equation: $$r(\theta) = e^{\sin\theta} - 2\cos(4\theta) + \sin^5(\frac{\theta}{12})$$ for \(0 ≤ \theta ≤ 24\pi\). You will observe the butterfly curve along with the effect of the added term.

Analyzing the New Term

To explain why the new term produces the observed effect, let's look at the components of the enhanced butterfly equation: $$r(\theta) = e^{\sin\theta} - 2\cos(4\theta) + \sin^5(\frac{\theta}{12})$$ The original butterfly curve equation is: $$r(\theta) = e^{\sin\theta} - 2\cos(4\theta)$$ The added term is: $$\sin^5(\frac{\theta}{12})$$ The \(\sin^5(\frac{\theta}{12})\) term has smaller oscillations compared to the \(2\cos(4\theta)\). When adding it to the original equation, it has a modulating effect, creating gentle variations along the curve which are observed as local changes in the original shape of the butterfly. Because the added term has a small amplitude and a lower frequency (due to the division by 12), its impact on the curve produces more subtle effects than the other terms. In conclusion, the addition of the term \(\sin^5(\frac{\theta}{12})\) introduces a modulating effect on the butterfly curve, resulting in gentle variations that create an enhanced curve.

Key concepts

Graphing Polar Equations

Graphing polar equations involves plotting points on a plane using coordinates that are determined by a distance from the origin and an angle from a reference direction. Unlike Cartesian coordinates, which use \(x\) and \(y\) axes, polar coordinates use \(r\) for radius and \(\theta\) for angle. Each equation in polar form \(r(\theta)\) defines a set of points where the radius changes as the angle changes.
When graphing the polar equation for the enhanced butterfly curve, \(r(\theta) = e^{\sin\theta} - 2\cos(4\theta) + \sin^5(\frac{\theta}{12})\), plot these values of \(r\) for angles from 0 to \(24\pi\). This creates a rich, intricate design that could be difficult to visualize without the use of technology, like graphing software, as manual plotting can be time-consuming and less precise.

• Always note the symmetry or patterns, as they can help in rapidly sketching the curve.
• Checking both the positive and negative values of \(r\) determines how the curve will loop or spiral.

Enhanced Butterfly Curve

The enhanced butterfly curve is a delightful mathematical creation which, in its conventional form, is expressed as \(r(\theta) = e^{\sin\theta} - 2\cos(4\theta)\). However, the addition of \(\sin^5(\frac{\theta}{12})\) introduces an even more captivating structure.
This new term creates subtle deviations from the original butterfly shape. The term's smaller amplitude allows it to affect the curve without overpowering the existing oscillations. The result is a visualization that mimics the gentle fluttering of butterfly wings in a breeze. While the foundational terms determine the primary oscillation, the enhanced term adds a layer of intricate movement and details.

• The \(e^{\sin\theta}\) component primarily handles the broad, wave-like spread of the wings.
• The \(-2\cos(4\theta)\) introduces symmetrical, sharp transitions that emulate the main shape of the wings.
• The added sine-power term results in minute adjustments that further beautify the pattern.

Polar Coordinates

Polar coordinates are a fantastic method for representing curves, especially when the radius and angle play a crucial role in the geometry of the curve. In a polar system, every point on the plane is defined by its distance from the origin (radius \(r\)) and an angle \(\theta\) from a fixed direction (usually the positive x-axis).
Using this system, curves such as circles, spirals, and even the enhanced butterfly curve can be more straightforwardly represented than in Cartesian coordinates. Distinctively, polar equations like \(r(\theta) = e^{\sin\theta} - 2\cos(4\theta) + \sin^5(\frac{\theta}{12})\) leverage trigonometric functions to describe complex curves through relatively simple expressions.

• Polar coordinates are inherently circular, making them ideal for equations involving angles and circles.
• They provide unique insights, particularly for spiral or radially symmetric shapes.
• Dependency on the angle makes measuring rotational symmetry or periodicity more intuitive.

Analysis of Polar Functions

Analyzing polar functions entails examining how the change in angle \(\theta\) affects the radius \(r\), and subsequently the shape of the curve. Each term in a polar function contributes uniquely, impacting the overall geometry.
For the enhanced butterfly curve, understanding each individual component is crucial:

• The term \(e^{\sin\theta}\) describes an exponential growth-modulation that affects the curve's spread.
• The \(-2\cos(4\theta)\) amplifies the symmetry and sharper transitions within the shape.
• The added \(\sin^5(\frac{\theta}{12})\) induces a more delicate, periodic modification.

Through such analysis, predicting and interpreting the behavior across various \(\theta\) values is possible. This approach helps deepen the understanding of how specific functions modify and influence complex graphical representations, leading to fascinating curves.