Question

Use a graphing utility to graph the following equations. In each case, give the smallest interval \([0, P]\) that generates the entire curve. \(r=\cos \frac{3 \theta}{5}\)

Short answer

Based on the step by step solution, the smallest interval on which the entire curve of the polar equation \(r=\cos \frac{3\theta}{5}\) is generated is \([0, \frac{5\pi}{3}]\).

Step-by-step solution

Graph the equation

Graph the given polar equation \(r=\cos \frac{3\theta}{5}\) using a graphical calculator or online graphing tool. Observe the pattern of the curve to determine the shape and symmetries of the polar plot.

Identify the symmetries

As this is a cosine function, it has even symmetry about the polar axis (the real axis). Therefore, the shape of the curve repeats every 180 degrees or \(\pi\) radians.

Determine the smallest interval \([0, P]\) that generates the entire curve

To find the smallest interval that generates the entire curve, we will look for the lowest value of \(\theta\) such that the pattern repeats. Keep in mind that as an even function, the interval must include \(0\) and \(P\) radians. Since the argument of the cosine function is scaled by \(\frac{3}{5}\), the whole curve will repeat after an angular distance of \(\theta = \pi \cdot \frac{5}{3}\). Therefore, the smallest interval that generates the entire curve is \([0, \frac{5\pi}{3}]\).

Key concepts

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions, such as those in polar coordinates. Using a graphing calculator or an online graphing tool, you can easily plot equations to explicitly see their shapes and patterns. For polar equations like \[ r = \cos \frac{3\theta}{5} \] staying true to the function’s unique characteristics becomes crucial. These tools allow you to plot the functions by inputting them and observing their graph, which unveils details like symmetries and repetitions.

This visual aid can quickly show intricate details about the curve, such as areas where the curve self-intersects or spots that repeat regularly. Aligning with mathematical principles, graphing tools help you forecast the real behavior of these curves with precision. Students can rely on these utilities to cross-verify manual plotting steps, thereby embedding more confidence in their understanding.

Symmetry in Graphs

Symmetry is key in grasping the full extent of polar graphs effectively. The function \( r = \cos \frac{3\theta}{5} \) shows even symmetry, which means it is symmetric about the polar (or real) axis. Recognizing symmetry aids in reducing computational effort as it brings down the need to compute over a wider angle range. Instead, computations can be analyzed for a smaller region and then mirrored accordingly.

This is profound in polar graphs since symmetries allow specific insights into how a curve is organized. If a curve is symmetric around the polar axis or any other axis, it implies that the graph looks the same on both sides of that axis. This feature can simplify calculations and graphing efforts since once you've understood a portion of the curve, the repeating symmetry gives you a complete picture of the graph. Additionally, recognizing symmetry supports predicting where intersections and complete rotations take place, ultimately leading to better comprehension of the plotted curve.

Angle Intervals

Understanding angle intervals in polar coordinates is integral to fully visualizing the complete graph of a polar equation. For the equation \( r = \cos \frac{3\theta}{5} \), identifying the smallest interval \( [0, P] \) that completes the curve helps in seeing the totality of the graph without unnecessary repetitions.

Here, you take into account how the argument within the cosine function is scaled. Due to the factor \( \frac{3}{5} \), the function completes a full rotation around at \( \theta = \pi \cdot \frac{5}{3} \), which becomes the minimal angle range \( [0, \frac{5\pi}{3}] \). Working out this interval is a crucial step as it depicts the simplest angle range required to recreate the curve entirely.

By managing angle intervals sensibly, not only do you simplify the graphing process, but you also gain an intricate understanding of the periodic nature of trigonometric functions and their graphical representations. Recognizing the precise interval helps in constructing the graph accurately for both academic and practical efforts.