Problem 82 - Algebra and Trigonometry

Question

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$\begin{aligned}&r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15} \quad \text { (Use } \quad \theta \min =0 \quad \text { and }\\\&\theta \max =20 \pi .)\end{aligned}$$

Step-by-Step Solution

Answer

The butterfly curve can be graphed using a graphing utility and the polar coordinates equation \(r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7}\frac{\theta}{15}\). By setting appropriate range and step for \(\theta\), a quality butterfly graph can be plotted.

Step 1: Understand the equation

The butterfly curve is given by the equation \(r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15}\). Here, \(r\) and \(\theta\) are polar coordinates, where \(r\) is the radial coordinate (distance from the origin) and \(\theta\) is the angular coordinate, measured counter-clockwise from the x-axis. The range for \(\theta\) is specified to be from 0 to \(20\pi\).

Step 2: Input the equation into the graphing utility

Open the graphing utility and input the given equation. Make sure you select 'polar' as the coordinate system, since we are working with polar coordinates here.

Step 3: Set the range and step for theta

Set the values of \(\theta_{min}\) and \(\theta_{max}\) to 0 and \(20\pi\) respectively. The step for \(\theta\) represents the incremental value with which \(\theta\) increases as the graph is plotted. Experiment with small step values to create a smooth graph. A general recommendation is to start with a step of \(\pi/180\) or even smaller, depending on the graphing utility used.

Step 4: Plot the graph

Once you have input the equation and set the range and step for \(\theta\), plot the graph. A shape that resembles a butterfly should emerge.

Key Concepts

Graphing Utilities

Graphing utilities are tools that greatly simplify the task of visualizing complex mathematical equations. When dealing with polar coordinates, it is crucial to use a graphing utility that supports plotting in polar mode. In this mode, equations are expressed using a radius, \(r\), and an angle, \(\theta\). This contrasts with the more common Cartesian coordinate system that uses \(x\) and \(y\) coordinates.

Using a graphing utility, one can quickly input an equation and see its graphical representation. This is particularly helpful for checking work, identifying interesting patterns, and experimenting with different mathematical parameters. When graphing the butterfly curve, make sure to switch the utility to polar mode to correctly plot the curve.

Butterfly Curve

The butterfly curve is a fascinating example of a polar graph with intricate and aesthetically pleasing shapes. The equation provided to plot a butterfly curve is \(r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15}\). This curve is named so because its graph resembles the wings of a butterfly, with smooth, symmetrical patterns very distinct from typical Cartesian graphs.

In polar coordinates, complex shapes are often produced by mathematical operations involving trigonometric functions like sine and cosine, as well as exponents. When graphing this curve, be prepared for an intricate design that captures the beauty and complexity of mathematics. The outcome heavily depends on setting appropriate bounds for \(\theta\) and using a fine step size to reveal all the details of the curve.

Polar Equation

A polar equation is a type of mathematical representation where the position of a point is defined by its distance from a reference point and its angular displacement from a reference direction. In the equation for the butterfly curve, \(r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15}\), \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis.

Polar equations are useful for describing phenomena that have rotational symmetry, such as flowers, spirals, or in this case, the butterfly curve. The blend of sine and cosine in this equation introduces periodicity and symmetry, generating a comprehensive and striking visual design when graphed. Familiarizing yourself with how to manipulate polar equations can enhance your ability to explore and represent rotationally symmetrical patterns more effectively.

Theta Range Adjustment

Adjusting the \(\theta\) range is essential when plotting polar graphs like the butterfly curve. The range of \(\theta\) in this case is from 0 to \(20\pi\), which encompasses multiple rotations around the origin. By choosing this extensive range, the entire pattern of the butterfly shape can be revealed.

When plotting, it's important to not only set your \(\theta_{min}\) and \(\theta_{max}\) accurately but also to choose a suitable step size. A small step, such as \(\pi/180\), allows for a detailed and smooth curve. A larger step might render a less smooth curve, potentially obscuring the delicate features of the graph. Experimenting with the step size can enhance the quality of the final visual.