Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. $$ r=\cos (13 \theta / 5) $$

Step-by-Step Solution

Answer

Graph the polar equation for \( \theta \) from 0 to 10 for a complete pattern.

Step 1: Understanding the Equation

The given equation is written in polar coordinates: \( r = \cos \left( \frac{13\theta}{5} \right) \). In polar equations, \( r \) represents the radius from the origin to the point on the curve, and \( \theta \) is the angle.

Step 2: Choosing the Parameter Interval

The function inside the cosine, \( \frac{13\theta}{5} \), suggests a periodic behavior. The basic period of \( \cos(x) \) is \( 2\pi \). Therefore, setting \( \frac{13\theta}{5} = 2\pi \), we solve for \( \theta \) to find the period of this curve in terms of \( \theta \): \[ \theta = \frac{10\pi}{13} \times 2 \approx 4.84 \]. To capture the entire pattern, plot \( \theta \) from 0 to \( 4.84 \). Extend beyond one period (0 to 10) for a fuller view.

Step 3: Graphing the Equation

Using a graphing calculation tool or software, input the equation \( r = \cos \left( \frac{13\theta}{5} \right) \). Set the range for \( \theta \) from 0 to 10 to capture more than a complete cycle of the curve. Ensure the tool is set to plot in polar coordinates.

Step 4: Analyzing the Graph

Observe the plotted graph for the given range of \( \theta \). You should see a star-like pattern with symmetry about the pole, reflecting the periodicity and complexity introduced by the coefficients \( 13/5 \) in the argument of the cosine function.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for visualizing mathematical functions, especially those involving complex equations and graphs. In the context of polar equations such as \( r = \cos \left( \frac{13\theta}{5} \right) \), a graphing calculator helps to precisely plot the curve and understand its shape. When working with graphing calculators, you must input the equation correctly, ensure the mode is set to polar, and select an appropriate parameter interval.

• Start by switching the calculator to polar mode to correctly interpret equations given in polar coordinates.
• Input the equation exactly as presented to avoid errors in plotting.
• Choose a range for \( \theta \) that includes at least one full period of the function to see the complete pattern.

By observing the resulting graph, you can gain insights into the geometric properties of the equation such as symmetries, cycles, and intersections with axes. This visualization makes graphing calculators invaluable for deeply understanding complex polar functions.

Periodic Function

Periodic functions are functions that exhibit a repeating pattern over regular intervals. For the equation \( r = \cos \left( \frac{13\theta}{5} \right) \), we recognize the periodic nature due to the cosine component.

• The basic period of the cosine function \( \cos(x) \) is \( 2\pi \).
• In this equation, the angle input is modified by the fraction \( \frac{13}{5} \), affecting the period length.

To determine the actual period in terms of \( \theta \), solve \( \frac{13\theta}{5} = 2\pi \) to find the length of one cycle in \( \theta \), which is around \( 4.84 \). By understanding that this function repeats every \( 4.84 \) units, you can accurately choose plotting intervals and predict the function's behavior beyond the first cycle. Observing several cycles can help identify symmetries and repetitive patterns that are characteristic of periodic functions.

Cosine Function

The cosine function, one of the fundamental trigonometric functions, is essential in various mathematical contexts, particularly in polar equations. In our specific equation \( r = \cos \left( \frac{13\theta}{5} \right) \), cosine affects how results are interpreted and graphed.

• Cosine oscillates between -1 and 1, creating a wave-like pattern.
• Its wave-like behavior contributes to the periodic nature of the function, evident in the repeating star-like pattern of the graph.
• Amplitude is not directly applicable in this context as \( r \) itself is controlled by cosine values instead.

The application of \( \cos \) in polar coordinates distinguishes these graphs from more common Cartesian plots, highlighting intricate curves and symmetrical patterns. In our example, the cosine function, modified by the fraction \( 13/5 \), creates unique intersections and looping patterns due to its inherent cyclic nature.

Polar Equation

Polar equations represent relationships in polar coordinates where each point on the plane is determined by a radius \( r \) and an angle \( \theta \). In the equation \( r = \cos \left( \frac{13\theta}{5} \right) \), the position of each point is derived through its polar form.

• In polar coordinates, \( r \) measures the distance from the origin, and \( \theta \) denotes the angle from the positive x-axis.
• Polar equations allow for the representation of curves that twist and overlap, which can be difficult to describe in Cartesian terms.

This particular polar equation showcases patterns that change with variations in \( \theta \). Using a graphing tool to visualize, you can see how these relationships configure the space, creating beautiful and complex motifs typical of polar graphs. Polar equations like this one highlight the versatility and elegance of describing forms in mathematical terms beyond simple linear graphs.