Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Graph the curve \(r=\cos (8 \theta / 5)\) using the parametric graphing facility of a graphing calculator or computer. Notice that it is necessary to determine the proper domain for \(\theta\). Assuming that you start at \(\theta=0\), you have to determine the value of \(\theta\) that makes the curve start to repeat itself. Explain why the correct domain is \(0 \leq \theta \leq 10 \pi\)

Step-by-Step Solution

Answer

The curve \( r = \cos\left(\frac{8\theta}{5}\right) \) repeats every \( 10\pi \), hence the domain \( 0 \leq \theta \leq 10\pi \) is correct.

Step 1: Understand the Polar Equation

The given polar equation is \( r = \cos\left(\frac{8\theta}{5}\right) \). This represents a polar curve, where \( r \) is the radius (distance from the origin) as a function of the angle \( \theta \). In polar coordinates, \( \theta \) typically ranges from \( 0 \) to \( 2\pi \) to capture one full rotation around the circle.

Step 2: Analyze the Frequency of the Function

The function in question is \( \cos\left(\frac{8\theta}{5}\right) \). The argument of the cosine function is affected by the fraction \( \frac{8}{5} \), which alters the frequency. The value inside the cosine, \( 8\theta/5 \), completes one full cycle (i.e., increases by \( 2\pi \)) when \( 8\theta/5 = 2\pi k \) for integer \( k \). Solving for \( \theta \) gives \( \theta = \frac{5\pi k}{4} \).

Step 3: Determine the Smallest Repeating Interval

To find when the curve starts repeating, we set \( k = 5 \), since this would match one full cycle of \( \theta \) with the simplest integral multiple that includes an integer multiple of \( \pi \). Therefore, when \( k = 5 \), we have \( \theta = \frac{5\pi \times 5}{4} = \frac{25\pi}{4} \), which simplifies to \( \theta = 10\pi \) as a complete cycle of all possible repetitions of the pattern.

Step 4: Argument for Domain

Given the cycle calculated in the previous step, the correct domain for \( \theta \) should ensure that all pattern formations occur at least once without repetition. Thus, starting at \( \theta = 0 \), the domain that captures one complete repetitive pattern of the curve is \( 0 \leq \theta \leq 10\pi \), as this is when the pattern starts to repeat every \( 10\pi \).

Key Concepts

Parametric Graphing

Graphing equations with parameters means representing the equations in terms of a third variable, often called a parameter. In our example, the parameter is \(\theta\), which determines both the radial distance \(r\) from the origin and the angle of rotation. Parametric graphing in a polar coordinate system involves step-by-step instructions to plot points based on these parameters.
Here's how it works:

• Start with a set value of \(\theta\) (our parameter), often beginning at \(0\).
• Calculate the resultant \(r\) using the given polar equation, \(r = \cos\left(\frac{8\theta}{5}\right)\).
• Plot the point derived from the angle \(\theta\) and radius \(r\) on a graph.
• Continue the process as \(\theta\) changes, plotting multiple points to create the curve.

The smooth connectivity of points creates the full graph of the equation. Parametric graphing allows visualization of curves that aren't simple polynomials, giving insights into unique mathematical behaviors.

Polar Equations

Polar equations are a way to describe curves using polar coordinates. In contrast to Cartesian coordinates that use \(x\) and \(y\) axes, polar coordinates use radius \(r\) and angle \(\theta\) to determine the position of a point.
The polar equation \(r = \cos\left(\frac{8\theta}{5}\right)\) showcases how these two elements describe a set of points:

• The angle \(\theta\) represents the direction from the origin.
• The radius \(r\) indicates how far the point is from the center.

This mode of graphing offers cleaner numerical relationships for certain curves, especially spirals and roses. Using polar equations helps in capturing behaviors that aren't easily represented with traditional straight-line slopes or parabolas. Such equations often result in intricate and beautiful patterns.

Frequency of Trigonometric Functions

The function \(\cos\left(\frac{8\theta}{5}\right)\) underscores the importance of frequency in trigonometric functions. Frequency describes how often the cycles or patterns occur within a defined interval and is crucial in defining curves like polar graphs.
For \(\cos\left(\frac{8\theta}{5}\right)\), the argument of the cosine function involves a modification by \(\frac{8}{5}\), altering the regular cycle:

• An increase in frequency means the function completes more cycles over a given interval.
• Instead of a single cycle within \(2\pi\), as is standard, the function completes more due to the factor of \(\frac{8}{5}\).
• The cycle can be determined using the equation \(8\theta/5 = 2\pi k\), allowing calculations of when the function repeats.

Understanding frequency is key to mastering trigonometric functions in mathematical applications, particularly in oscillations and waveforms.

Cycle of Repetition in Trigonometric Functions

Trigonometric functions, particularly sine and cosine, have intrinsic patterns known as cycles. These cycles repeat every \(2\pi\) in a standard sine or cosine function, forming the basis for calculating repetition in altered form functions.
In \(r = \cos\left(\frac{8\theta}{5}\right)\), finding the cycle of repetition is crucial:

• The repeated cycle helps us determine the precise intervals where patterns begin recurring.
• Setting \(8\theta/5 = 2\pi k\) and solving for \(\theta\) gives the smallest \(\theta\) that makes the pattern repeat with \( k = 5 \).
• This results in \(\theta = 10\pi\), marking the interval where the graph completes a full cycle and begins again.

Grasping cycle repetition aids in creating accurate plots, ensuring graphs capture every unique instance of the function's behavior within a given domain. Recognizing these patterns transforms comprehension of how trigonometric functions govern complex curves.