Problem 114 - Calculus Early Transcendentals

Question

Consider the polar curve \(r=\cos (n \theta / m)\) where \(n\) and \(m\) are integers. a. Graph the complete curve when \(n=2\) and \(m=3\) b. Graph the complete curve when \(n=3\) and \(m=7\) c. Find a general rule in terms of \(m\) and \(n\) (where \(m\) and \(n\) have no common factors) for determining the least positive number \(P\) such that the complete curve is generated over the interval \([0, P]\).

Step-by-Step Solution

Answer

Answer: The general rule to determine the least positive number P for a complete cycle of the polar curve r = cos(nθ/m) is P = 2πk, where k is the smallest integer such that mn divides km, and n and m have no common factors. This ensures that we get whole cycles of the cosine function, as well as integer multiples of angular cycles.

Step 1: Define the polar curve equation for this case

For this case, we have \(n = 2\) and \(m = 3\). The polar curve equation will be \(r = \cos(2\theta/3)\).

Step 2: Graph the polar curve

Using a graphing tool, plot the polar curve \(r = \cos(2\theta/3)\) for the interval \([0, 2\pi]\). Observe the obtained graph. #b. Graph the curve when n=3 and m=7#

Step 1: Define the polar curve equation for this case

For this case, we have \(n = 3\) and \(m =7\). The polar curve equation will be \(r = \cos(3\theta/7)\).

Step 2: Graph the polar curve

Using a graphing tool, plot the polar curve \(r = \cos(3\theta/7)\) for the interval \([0, 2\pi]\). Observe the obtained graph. #c. Find a general rule for determining the least positive number P#

Step 1: Observe the pattern in the polar curve graphs

Notice that for a complete graph of the polar curve, we need at least one full cycle of the cosine function in the radial direction and a whole number of cycles in the angular direction. From the examples, we see that we get a complete graph in the interval \([0, 2\pi]\) when the fractions \((2/3)\) and \((3/7)\) have integers in the numerator (\(n\)).

Step 2: Formulate the general rule

Since we know that the complete curve is generated in the interval where we have whole cycles of cosine and integer multiples of angular cycles, we can conclude that \(P=2\pi k\), where \(k\) is the smallest integer such that \(mn\) divides \(km\), and \(n\) and \(m\) have no common factors. This ensures that we get whole cycles of the cosine function, as well as integer multiples of angular cycles.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points in a plane. Unlike Cartesian coordinates, which use x and y to determine a location, polar coordinates use a radius and an angle. This system is especially useful in scenarios where circular, spiral, or radial patterns are involved.

Think of polar coordinates as a way of pinpointing something's location by telling you how far away it is from a central point and in which direction you need to head from that point. The radius (\(r\)) tells you the distance from the origin (the center), while the angle (\(\theta\)) indicates the direction.

• A point in polar coordinates is denoted as (\(r, \theta\)).
• \(r\) represents the radius or distance from the origin.
• \(\theta\) represents the angle in relation to the positive x-axis.

This system of coordinates is perfect for describing curves that loop around a central point, like the petals of a flower in our polar equations.

Cosine Function

The cosine function is pivotal in polar curves due to its oscillatory nature. This mathematical function appears frequently because it describes patterns of waves and cycles which are common in polar graphing.

The cosine function is defined mathematically as \(\cos(\theta)\). It produces values that oscillate between -1 and 1, based on the angle \(\theta\):

• When \(\theta = 0\), \(\cos(\theta) = 1\).
• When \(\theta = \pi/2\), \(\cos(\theta) = 0\).
• When \(\theta = \pi\), \(\cos(\theta) = -1\).

In polar equations like \(r = \cos(n\theta/m)\), the cosine function modulates how \(r\) changes as \(\theta\) varies. The parameters \(n\) and \(m\) adjust the frequency of the cycles, dictating how many peaks and troughs occur as one completes a full rotation around the origin.

Graphing Techniques

Graphing polar equations can be both fascinating and challenging. One must consider the oscillations and loops dictated by the equation, typically involving trigonometric functions like cosine.

To graph a polar curve such as \(r = \cos(n\theta/m)\), follow these steps:

• Identify the range of \(\theta\) where you want to graph the curve. Generally, \([0, 2\pi]\) is a good start for a full picture.
• Consider the effect of \(n\) and \(m\) on the frequency. For instance, a fraction of \(n/m\) like \(2/3\) may hint at cycles completing sooner or later than \(2\pi\).
• Use graphing tools to plot the curve over the chosen range. Observe repeated patterns or terminations.

Recognize that each cycle of the cosine component corresponds to dynamic changes in the radius \(r\), forming intricate designs like a star or spiral. Practice will increase proficiency in understanding these visual patterns.

Integer Factors

Understanding integer factors becomes essential when dealing with polar graphs containing integer ratios like \(n/m\). These factors help predict how the graph will look and when it will repeat itself.

In polar curves like \(r = \cos(n\theta/m)\), it's crucial to ensure that \(n\) and \(m\) have no common factors for simplicity. When these numbers are coprime, the curve's peculiarities emerge prominently:

• If \(n\) and \(m\) share any common integer factor other than 1, the period of repeating cycles shortens.
• Finding the least positive \(P\) for a complete graph involves understanding the smallest number of cycles needed.
• When \(m\) and \(n\) are coprime, the complete cycle often occurs at \(P = 2\pi m\).

This intuitive grasp of integer factors helps one predict the number of times a design will loop before fully completing, a key step in mastering polar curves.