Problem 53 - 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. $$r=2 \cos \frac{3 \theta}{2}$$

Step-by-Step Solution

Answer

The interval for \(\theta\) over which the graph of the polar equation \(r=2 \cos \left(\frac{3\Theta}{2}\right)\) is traced only once is \(\left[-\frac{2\pi}{3}, \frac{2\pi}{3}\right]\)

Step 1: Graphing the polar equation

Start by graphing the polar equation \(r=2 \cos \left(\frac{3\Theta}{2}\right)\). Use a graphing calculator or computer algebra system to plot the polar function. In polar coordinates, \(r\) represents the distance from the origin and \(\theta\) the angle formed with the positive x-axis. Note how the function behaves when \(\theta\) is in the interval [-2π, 2π].

Step 2: Identify the period of the function

The next step is to determine the interval over which the graph is traced only once by identifying the period of the cosine function. The period of a function \(\cos(ax)\) is \(\frac{2\pi}{|a|}\). Applying this formula here, we get the period of the function to be \(\frac{2\pi}{3/2} = \frac{4\pi}{3}\). Therefore, the function will start repeating itself after this interval.

Step 3: Define the interval for \(\theta\)

Now, we need to specify an interval for \(\theta\) over which the graph is traced only once. It can be any interval as long as its length is \(\frac{4\pi}{3}\). We can center this interval about zero, resulting in the interval \(\left[-\frac{2\pi}{3}, \frac{2\pi}{3}\right]\). So, in this interval, the function will not repeat its values and the graph will not overlap itself.

Key Concepts

Understanding Polar Coordinates

Polar coordinates offer a unique and efficient way to represent points in a plane using the distance from a central point (the pole) and an angle from a fixed direction. This system contrasts with Cartesian coordinates, which use two perpendicular axes to specify points based on x and y coordinates.

In polar coordinates, a point is described by a pair \( (r, \theta) \), where \( r \) is the radial distance from the pole, and \( \theta \) is the angular component, usually measured in radians from the positive x-axis. This approach is particularly useful when dealing with circular or spiral patterns, common in natural phenomena and engineering applications.

A key to mastering polar equations is visualizing how the angle and radial distance interact to form the shape of the graph. Students often find graphing polar coordinates to be a challenging transition from the more familiar Cartesian system. It's helpful to practice converting between these systems and plotting points to gain intuition for polar graphs.

Cosine Function Periodicity

The cosine function, a fundamental concept in trigonometry, is known for its periodic nature. The 'period' of a function refers to the interval over which the function cycles through a complete set of values before starting to repeat itself. For the standard cosine function \( \cos(x) \), the period is \( 2\pi \), meaning that every \( 2\pi \) radians along the x-axis, the function's values start repeating.

When the cosine function contains a coefficient within the argument, as in \( \cos(ax) \), the period changes to \( \frac{2\pi}{|a|} \). Therefore, for the function \( r=2 \cos\left(\frac{3\Theta}{2}\right) \), the period is \( \frac{4\pi}{3} \) radians. Recognizing the period is crucial when graphing trigonometric functions, as it helps identify the interval within which the graph is unique and doesn't overlap itself. This understanding also aids in solving trigonometric equations and in modeling cyclic phenomena.

Using Graphing Utilities

Graphing utilities, such as calculators and computer algebra systems, play a vital role in creating visual representations of mathematical equations, particularly when manual sketching is complex or impractical. These tools are indispensable when working with polar coordinates because they can efficiently handle the intricate curves and loops associated with polar graphs.

When entering a polar equation into a graphing utility, one must often set the mode to 'polar'. However, it's essential to understand the underlying functions to interpret the generated graph correctly. For the exercise \( r=2 \cos\left(\frac{3\Theta}{2}\right) \) , a graphing utility will help capture the subtle changes as \( \theta \) varies and illustrate the period of the cosine function clearly.

Moreover, students using these utilities should learn to specify appropriate intervals for \( \theta \) and scaling for \( r \) within the tool's settings to accurately display specific portions of the graph. Ultimately, while graphing utilities offer great assistance, they serve best when combined with a firm conceptual understanding of the material.