Question

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r^{2}=16 \sin 2 \theta\)

Short answer

Question: Sketch the graph of the polar equation \(r^2 = 16\sin{2\theta}\) and describe the process you used. Answer: To sketch the graph of the polar equation \(r^2 = 16\sin{2\theta}\), I followed these steps: 1. I found the relationship between r and θ by rewriting the equation as \(r = \pm \sqrt{16 \sin 2\theta}\). 2. I created a table for different angles of θ, and found the corresponding radius values using the equation from Step 1. 3. I plotted the points from the table on the polar coordinate plane, considering both positive and negative values of r. 4. I connected the points to sketch the graph with smooth curves for both positive and negative values of r. 5. I verified my sketch using a graphing utility, such as Desmos or Wolfram Alpha. 6. Finally, I produced a neat and clean final graph with correct labels for the axes, and displayed both positive and negative values of r for the given angles of θ.

Step-by-step solution

Find the relationship between r and θ

First, determine the relationship between \(r\) and \(\theta\) by rewriting the given equation as \(r = \pm \sqrt{16 \sin 2\theta}\). So, we will have two sets of points: one for positive values of r and another for negative values of r.

Make a table for angles and corresponding radius values

Create a table of values for \(\theta\), and find the corresponding values of \(r\) using the equation \(r = \pm \sqrt{16 \sin 2\theta}\). It's better to start with some common angles such as 0, π/4, π/2, 3π/4, π, and so on.

Plot the points

Using the values from the table in Step 2, plot the points on the polar coordinate plane. Remember to plot both positive and negative values of r.

Connect the points to sketch the graph

Connect the plotted points to sketch the graph of the equation \(r^2 = 16\sin{2\theta}\). Make sure to create smooth curves for both the positive and negative values of \(r\).

Verify your graph using a graphing utility

Now that you have sketched the graph, it's essential to check your work using a graphing utility. There are many graphing utilities available online, such as Desmos or Wolfram Alpha, that can plot polar equations. Input the equation \(r^2 = 16\sin{2\theta}\) in the graphing utility and see if your sketch matches the graph produced by the utility.

Produce the final graph

If your sketch matches the graph generated by the graphing utility, you can be confident that you have graphed the equation correctly. Now, create a neat and clean final graph, either by hand or using the graphing utility's output. Remember to include correct labels for the axes, and display both positive and negative values of r for the given angles of θ.

Key concepts

Polar Equations

Polar equations are a way to describe curves using the polar coordinate system. Instead of using the traditional cartesian coordinates, which consist of x and y values, polar coordinates use the distance from the origin (denoted as r) and the angle from a reference direction (denoted as \( \theta \)).
This means any point in a plane can be represented as \( (r, \theta) \). A polar equation defines a relationship between r and \( \theta \), allowing us to explore the behavior and characteristics of curves in a circular manner.
For example, in the given equation \( r^2 = 16 \sin 2\theta \), there's a unique connection between the radius r and the angle \( \theta \). By understanding how to manipulate these expressions, we can plot a wide variety of curves. Unlike cartesian equations, where we deal with straight lines and parabolas, polar equations often depict circular or spiral graphs, providing a comprehensive way to study curved shapes using angles.

Graphing Polar Curves

Graphing polar curves involves a series of steps to accurately display an equation in the polar coordinate system. First, you identify a relationship between r and \( \theta \). In our exercise, by rewriting \( r^2 = 16 \sin 2\theta \) as \( r = \pm \sqrt{16 \sin 2\theta} \), we can determine the radius values for different angles.
Begin by creating a table with common angles such as \( 0, \pi/4, \pi/2, 3\pi/4, \pi \), and compute the corresponding value of r using the equation.

• Consider both the positive and negative square roots to ensure the complete graph is represented.
• Plot these points on a polar grid, accounting for both directions (since \( r \) can be positive or negative).

Next, connect these points smoothly to illustrate the curvature of the graph, often resulting in intricate shapes like lemniscates or roses. Finally, use a graphing utility to verify the accuracy of your hand-drawn graph and make any necessary adjustments.

Trigonometric Functions

Trigonometric functions play a significant role in plotting polar equations because they introduce cyclical or oscillating behavior that is pivotal in the graphing process. Common functions like sine (\( \sin \)) and cosine (\( \cos \)) determine the wave-like components of a polar graph, leading to aestheticallycompelling patterns.
In the exercise, the function \( \sin 2\theta \) dictates how values of \( r \) vary with changes in \( \theta \).As \( \theta \) alters, the sine function transitions between -1 and 1, affecting the radius and thus the overallshape of the polar curve.

• Sine functions result in symmetric lines or loops due to their periodic nature, making them common in many polar graphs.
• It is crucial to test angles where the sine function completes a full cycle, typically every \( 2\pi \).

By understanding how trigonometric functions modify polar equations, students can predict the resulting patterns and gain insights into graph behaviors, reinforcing the link between algebraic manipulation and geometric visualization.