Problem 62 - Precalculus with Limits

Question

In Exercises 59-64, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. \(r=3\ \cos \left(\dfrac{5\theta}{2}\right)\)

Step-by-Step Solution

Answer

The interval for \(\theta\) for which the polar equation \(r=3\ \cos \left(\dfrac{5\theta}{2}\right)\) is traced only once is \(0 \leq \theta < \dfrac{4\pi}{5}\).

Step 1: Graph the Polar Equation

Use a graphing utility to graph the polar equation \(r=3\ \cos \left(\dfrac{5\theta}{2}\right)\). Inspect the graph to understand the behaviour of the polar equation.

Step 2: Identify the Pattern

Upon inspecting the graph, it can be noticed that the graph patterns seems to repeat itself every once a full cycle (360 degrees or \(2\pi\) radians) of \(\dfrac{5\theta}{2}\) has been made. Largely due to the periodic nature of cosine function. The period of \( \cos x \) is \(2\pi\), so the period of \( \cos \left(\dfrac{5x}{2}\right) \) would be \( \dfrac{4\pi}{5} \).

Step 3: Determine the Interval for \(\theta\)

Since the graph of the polar equation repeats itself every \( \dfrac{4\pi}{5} \) radians, we can find an interval for \(\theta\) by starting from \(\theta = 0\), which is usually the starting point for analysing polar graphs. Therefore, \(0 \leq \theta < \dfrac{4\pi}{5}\) is an interval for \(\theta\) for which the polar equation \(r=3\ \cos \left(\dfrac{5\theta}{2}\right)\) is traced only once.

Key Concepts

Graphing Utility

A graphing utility is a vital tool for visualizing equations, especially in polar coordinates. To use it effectively for polar equations, one must understand its capabilities in displaying data in the polar form. Here's a helpful guide on what you can do with a graphing utility:

• Input the equation directly, as provided. For example, you can enter the equation as \( r = 3\cos\left(\frac{5\theta}{2}\right) \).
• Ensure that the utility is set to polar mode, as opposed to Cartesian mode.
• Watch how the graph draws itself based on these inputs. This will give insights into its structure and shape.

Using the utility, you can observe the ray-like structures emanating from the pole, which are the graph of the polar equation. You can also adjust parameters like angle range and scale to better interpret the graph. Analyzing the resulting visual can help deduce repeating patterns and symmetries inherent in the polar function.

Interval for \(\theta\)

Determining the correct interval for \(\theta\) is essential for understanding where the polar graph traces out its path once completely. For the equation \( r = 3 \cos \left( \frac{5\theta}{2} \right) \), the interval for \( \theta \) can be found by considering the period of the function.A cosine function itself has a natural period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radian increment. However, due to the multiplication inside the argument, \(\frac{5\theta}{2}\), we need to adjust this.
To find the period of \( \cos \left( \frac{5\theta}{2} \right) \), solve the equation for one complete cycle:

• Set \( \frac{5\theta}{2} = 2\pi \)
• Solving for \( \theta \) yields \( \frac{4\pi}{5} \)

Thus, the interval for \( \theta \) is from 0 to \( \frac{4\pi}{5} \). Within this interval, the graph completely traces itself once, providing a full picture of the pattern before potentially repeating itself.

Periodic Function

The concept of a periodic function is crucial in polar equations. A periodic function is one that repeats after a fixed interval known as the period. For example, the basic cosine function, \( \, \cos \theta \, \), repeats every \( 2\pi \) radians.
In the case of \( r = 3\cos(\frac{5\theta}{2}) \), the periodic nature is slightly altered by the coefficient \(\frac{5}{2}\) within the cosine function.

• This coefficient affects the argument \(\theta\) by stretching or compressing the cosine wave.
• The new period becomes \( \frac{4\pi}{5} \), as deduced from setting the equation \( \frac{5\theta}{2} = 2\pi \).

This tells us the graph will repeat itself every \( \frac{4\pi}{5} \) radians. By understanding this periodic nature, you can predict the structure and form of the polar graph without needing to plot every possible angle visually.