Problem 48 - Calculus Early Transcendentals

Question

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r=2 \sin 5 \theta\)

Step-by-Step Solution

Answer

Question: Sketch the polar graph of the equation \(r=2\sin(5\theta)\). Answer: The polar graph of this equation consists of five loops, with alternating positive and negative values of \(r\). Key points include \((\frac{\pi}{10},2)\), \((\frac{3\pi}{10},-2)\), \((\frac{4\pi}{10},0)\), \((\frac{5\pi}{10},2)\), \((\frac{7\pi}{10},-2)\), \((\frac{9\pi}{10},0)\), \((\frac{11\pi}{10},2)\), and \((\frac{13\pi}{10},-2)\).

Step 1: Observe the equation

Observe the given polar equation, \(r=2\sin(5\theta)\). This is a sinusoidal function with a coefficient of 5 in the argument of the sine function. That means there are five periods in one complete circle (\(2\pi\)).

Step 2: Determine key points

To determine the key points on the graph, we need to consider the values of \(\theta\) that yield either a maximum or minimum value of \(r\). To do this, consider the points at which \(\sin(5\theta)=1\) and \(\sin(5\theta)=-1\). Set \(5\theta = k\pi + \frac{n\pi}{2}\), where \(k,n\in \mathbb{Z}\). Notice that if \(n\) is odd, then \(5\theta\) corresponds to the maximum or minimum of the sine function. Therefore, we have \(\theta = \frac{k\pi + \frac{n\pi}{2}}{5}\), where \(n\) is odd.

Step 3: Make a table of key points and plot them

Using the key points determined in Step 2, construct a table of values and plot them on a polar grid: \(\theta\) | \(r\) ------------|---- \(\frac{\pi}{10}\) | \(2\) \(\frac{3\pi}{10}\)| \(-2\) \(\frac{4\pi}{10}\) | \(0\) \(\frac{5\pi}{10}\)| \(2\) \(\frac{7\pi}{10}\)| \(-2\) \(\frac{9\pi}{10}\) | \(0\) \(\frac{11\pi}{10}\)| \(2\) \(\frac{13\pi}{10}\)| \(-2\)

Step 4: Sketch the graph

Using the key points from Step 3, sketch a curve in the polar grid. The graph consists of five loops, with alternating positive and negative values of \(r\).

Step 5: Check using a graphing utility

To check our work, input the equation \(r = 2\sin(5\theta)\) into a graphing utility. Compare your hand-drawn graph to the one generated by the utility. Ensure that the key points and overall shape match.

Step 6: Finalize the graph

Based on your comparison of the hand-drawn graph and the graph generated by the graphing utility, make any necessary adjustments to your graph. If your hand-drawn graph is accurate, you have successfully completed this exercise.

Key Concepts

Sinusoidal Function

In mathematics, a sinusoidal function is a type of periodic wave that includes sine and cosine functions. These functions are significant in studying periodic phenomena because of their wave-like characteristics.
The sinusoidal function can be represented as either sine: \(y = a \sin(bx + c) + d\) or cosine: \(y = a \cos(bx + c) + d\).

• In polar form, you often see sinusoidal functions written as \(r = a \sin(n\theta)\), like in the equation \(r = 2 \sin(5\theta)\). Here, \(a\) adjusts the amplitude (the maximum value), and \(n\) affects the frequency (how many cycles occur within \(2\pi\)).
• For \(r = 2 \sin(5\theta)\), the number 5 indicates there are five cycles of the sine wave as \(\theta\) completes a full circle of \(0\) to \(2\pi\).
• The amplitude \(2\) means the wave will reach a maximum value of \(2\) and a minimum value of \(-2\).

Sinusoidal functions naturally occur in signals, sound waves, and tides due to their repetitive nature. Understanding these waves is essential for fields like physics, engineering, and various technical subjects.

Polar Graphing

Polar graphing is an alternative way of plotting data using a different coordinate system. Instead of using the standard Cartesian coordinates (x, y), polar graphs use coordinates \((r, \theta)\).

In polar coordinates, \(r\) represents the distance from the origin (center point), while \(\theta\) is the angle in radians measured from the positive x-axis.

• A key feature of polar graphs is that they can easily represent curves that are challenging to show in Cartesian coordinates.
• For the equation \(r = 2 \sin(5\theta)\), a looped pattern emerges. As \(\theta\) varies, \(r\) fluctuates between positive and negative values.
• This specific function constructs multiple loops or petals around the pole (origin).

Polar graphing is particularly useful in scenarios where symmetry around a central point is key, such as in circular motion, radar plots, and certain signal processing tasks.

Graphing Utility

A graphing utility is a tool—often software or a calculator—that helps visualize mathematical equations by automatically plotting their graphs.
These tools enable users to input equations quickly, without manually calculating numerous points alone.

• For complex equations like \(r = 2 \sin(5\theta)\), using a graphing utility simplifies the plotting process and provides accurate visual representations of the graph's shape.
• By checking your hand-drawn work against a graphing utility, you can verify the correctness of your plot and ensure key features, like intercepts or curves, match expected outcomes.
• Popular graphing utilities include handheld graphing calculators, online tools like Desmos, or software such as GeoGebra.

Whether in class or while working on complex homework, graphing utilities play a crucial role in understanding mathematical functions and their behaviors, making the learning experience both easier and more interactive.