Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r^{2}=25 \cos 2 \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r^{2}=25 \cos 2 \theta\) is a rose curve with 4 petals each of length 5. The curve is traced once for the interval \(\theta = [0, \pi]\).

Step 1: Understand the Equation

Recognize that the given equation is a rose curve polar equation of the form \(r^{2}=a^2 cos(2\theta)\), where \(a = 5\). It means, the length of each petal is \(a = 5\) and there are 4 petals because the coefficient of \(\theta\) is an even integer.

Step 2: Find the interval for \(\theta\)

We need to find the interval of \(\theta\) in which the graph is traced only once. For such rose curves, the curve completes one complete cycle in the interval \(\theta = [0, \pi]\). Thus to ensure the graph is traced only once, we should take \(\theta\) in this interval.

Step 3: Graph the Equation

Graph the polar equation using a suitable graphing utility. Remember to set the range of \(\theta\) as obtained in Step 2. That is, \(\theta\) must be limited to the interval \([0, \pi]\) in the settings of the graphing utility. We will get four petals extending in the directions of \(\theta = 0, \(\frac{\pi}{2}, \(\pi\), \(\frac{3\pi}{2}\) and each of length 5.

Step 4: Interpret the graph

Each petal corresponds to the polar expression within an interval of length \(\pi/2\). The graph makes one complete cycle as \(\theta\) varies from 0 to \(\pi\). When \(\theta\) varies from \(\pi\) to \(2\pi\), the same graph is retraced.

Key Concepts

Rose Curve Polar Equation

The polar equation you're working with, \r^2 = 25 \[cos 2\theta\], belongs to a category known as rose curve polar equations. These equations have a unique, flower-like pattern consisting of petals radiating from the origin. The general form of a rose curve is r = a cos(k\theta) or r = a sin(k\theta), where a is the amplitude influencing the petal's length, and k determines the number of petals.

In this case, with a = 5 and k = 2, the rose curve will have 4 petals due to the cosine function with an even coefficient. Understanding the properties of rose curves is crucial, as it helps in visualizing the graph before using a graphing utility and confirming its accuracy after plotting.

Polar Coordinates

Polar coordinates inherently differ from the Cartesian coordinates you might be more familiar with, which use (x, y) points to denote location. Polar coordinates instead express locations based on the angle \theta and the distance r from a central point known as the pole, similar to how you might describe the location of a star in the night sky.

In our problem, the radius r depends on the angle \( \theta \), which means the distance from the origin will change as the angle increases, tracing out the captivating shape of the rose curve.

Graphing Utility Usage

Using a graphing utility is key in plotting complex polar equations like rose curves. These utilities allow for precise and quick rendering of mathematical patterns. When setting up your graphing tool, ensure that the \( \theta \) interval aligns with the necessary range to display the curve accurately, in this case, \theta = [0, \(\theta\)].

Remember to take advantage of the utility's features such as adjusting the scale, viewing the graph from various angles for better understanding, and using trace functions to follow the curve's path as \( \theta \) changes. This will aid in confirming that each petal is graphed correctly and that the overall shape aligns with the expected rose curve.

Polar Equation Intervals

Choosing the correct interval for \( \theta \) in polar equations is crucial to accurately graphing the curve without repetition. In the rose curve equation r^2 = 25 cos(2\theta), the interval \theta = [0, \pi] ensures that the graph is traced only once. This is because within this range the function goes through all the necessary angles to plot each unique petal of the rose. After \pi, the values of cosine begin to repeat, which would retrace the existing graph.

It is important not to extend the interval beyond \pi unless required by the problem, as this would lead to overlapping graphics and might confuse the true shape of the polar curve being studied.