Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify and sketch the graph of the polar equation. Identify any symmetry and zeros of \(r .\) Use a graphing utility to verify your results. $$r=\sin 5 \theta$$

Step-by-Step Solution

Answer

The zero of \(r\) are at multiples of \(\pi/5\). The graph of \(r=\sin 5\theta\) doesn't have symmetry with respect to the polar axis or the origin. A sketch of the graph is made by plotting points for various values of \(\theta\) from 0 to \(2\pi\) and connecting those points smoothly. The graph should be verified using a graphing utility.

Step 1: Identify the zeros of r

The zeros of r are values of \(\theta\) for which \(r= 0\). So, setting \(r=\sin 5\theta=0\), we know \(\sin\) function is 0 at \(0, \pi, 2\pi\) etc. So, the values of \(\theta\) that would make r zero are all multiples of \(\pi/5.\)

Step 2: Identify the symmetry

The graph of \(r= \sin 5\theta\) would be symmetric with respect to the polar axis if it is possible to replace \(\theta\) with \(-\theta\). However, since this replacement would yield \(r=\sin -5\theta= -\sin 5\theta\), so the graph doesn't have symmetry with respect to the polar axis; it is not symmetric with the origin either.

Step 3: Sketch the graph

To sketch the graph of \(r=\sin 5\theta\), plot the points for various values of \(\theta\) and connect those points smoothly. Since \(\theta\) varies from 0 to \(2\pi\), make sure you cover the whole range of \(\theta\).

Step 4: Verifying by using a graphing utility

Use a graphing utility with the polar mode to check your graph. Input \(r=\sin 5\theta\) and compare the output with your sketch.

Key Concepts

Graph Symmetry

When examining polar equations like \( r = \sin 5\theta \), it's essential to explore their symmetry properties. In this context, symmetry often refers to how the graph looks when flipped or rotated. For the equation \( r = \sin 5\theta \), checking for symmetry involves substituting \( \theta \) with \( -\theta \).
\[ r(\theta) = \sin(-5\theta) = -\sin(5\theta)\]
This tells us that the function is not symmetric with respect to the polar axis, because the sign changes. Similarly, it is not symmetric with the origin, since there’s no alteration transforming it back into its original form when either \( \theta \) or \( r \) is replaced by negative.
Sometimes, a graph might exhibit symmetry regarding the vertical line \( \theta = \frac{\pi}{2} \); however, this is not the case here. Understanding these properties helps us predict graph behavior which can streamline the plotting process.

Zeros of Functions

The zeros of a function are crucial as they indicate where the graph crosses the origin in polar coordinates. For the function \( r = \sin 5\theta \), the zeros occur where \( r = 0 \). In the sine function, this happens at the integer multiples of \( \pi \).
Substituting into our function, we find:

• \( r = 0 \) when \( \sin 5\theta = 0 \).

The result is that \( \theta = \frac{k\pi}{5} \), where \( k \) is an integer (e.g., 0, 1, 2,...). These points are valuable for sketching and understanding the graph's structure, as each zero represents an intersection through the pole, which is the origin (0,0).

Graphing Utilities

Graphing utilities have become pivotal tools in visualizing complex equations like polar graphs. For the equation \( r = \sin 5\theta \), they allow students to easily check the structure and symmetry of the graph they've sketched manually.
To use a graphing utility:

• Switch the calculator or software to polar mode.
• Input the equation \( r = \sin 5\theta \).
• Observe the graph's symmetry and behavior.

Employing graphing utilities ensures accuracy and can provide a deeper understanding of how the mathematical equation translates into a visual representation. It's also an excellent way to verify manual sketches, reinforcing learning.

Polar Graphs

Understanding polar graphs is fundamental in exploring advanced mathematical topics. Unlike Cartesian graphs, polar graphs use circles and angles to plot points.
The equation \( r = \sin 5\theta \) represents a polar graph that features petals, a common pattern when dealing with sine or cosine functions in polar coordinates. Here are key points to help you understand polar graphs:

• \( r \) is the radial distance from the pole (origin).
• \( \theta \) represents the angle from the polar axis (positive x-axis).
• The graph often repeats patterns every full rotation of \( 2\pi \).
• Equations like \( \sin n\theta \) produce \( n \) petals if \( n \) is odd, or \( 2n \) petals if \( n \) is even.

These points help students visualize and understand how polar graphs are constructed, making it easier to transition from Cartesian plotting to polar coordinates.