Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=8 \cos 2 \theta$$

Step-by-Step Solution

Answer

The graph is a rose curve with 4 petals on a viewing window ranging from \(\theta_{min} = 0\), \(\theta_{max} = 2\pi\), \(r_{min} = -8\) and \(r_{max} = 8\).

Step 1: Convert Polar to Cartesian Coordinates

We start by converting polar coordinates to Cartesian. The relationship between Polar and Cartesian coordinates is given by \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Now replace \(r\) from the given polar equation to these, we get coordinate equations in \(x\) and \(y\) as: \(x = 8 \cos^2(2\theta)\) and \(y = 8 \cos(2\theta) \sin(2\theta)\).

Step 2: Set-up the Graphing Calculator

Go into the settings of your graphing calculator and make sure you're in polar graphing mode, which is typically labeled as 'POL'. Enter the given polar equation \(r = 8 \cos 2 \theta\) into the function input field.

Step 3: Select Viewing Window

For polar coordinates, a complete view of the graph usually requires \(\theta\) values ranging from \(0\) to \(2\pi\), and an \(r\) range that covers the extent of the r-values generated by the equation. In this case, choose \(\theta_{min} = 0\), \(\theta_{max} = 2\pi\), and adjust \(r_{min}\) and \(r_{max}\) to ensure the entire graph fits in the window. In general, for \(r = cos 2 \theta\) a good starting point is \(r_{min} = -8\) and \(r_{max} = 8\).

Step 4: Graph the Polar Equation

With the equation inputted and the viewing window set, proceed to graph the equation. You should see a rose-like graph with 4 petals.

Key Concepts

Polar to Cartesian Coordinates

When it comes to understanding polar equations, one of the fundamental skills is converting polar coordinates to Cartesian coordinates. Polar coordinates represent points on a plane using a distance from a reference point and an angle from a reference direction, whereas Cartesian coordinates use a grid of x and y values. To transform from polar to Cartesian, we employ the relationships
\(x = r \times \text{cos}(\theta)\) and \(y = r \times \text{sin}(\theta)\).
Using the example from our exercise, \(r = 8 \text{cos} 2\theta\), we substitute \(r\) into these equations to get the Cartesian form, \(x = 8 \text{cos}^2(2\theta)\) and \(y = 8 \text{cos}(2\theta) \text{sin}(2\theta)\). This conversion is vital for visualizing complex polar graphs in a more familiar Cartesian coordinate system or for performing calculations that require Cartesian form.

Graphing Calculator Setup

Properly setting up your graphing calculator is a key step in graphing polar equations. Different calculators have varying interfaces, but the general steps are similar. Firstly, access the mode or settings section of your calculator. You'll want to select the 'polar graphing mode,' often denoted as 'POL' or something similar. This mode will allow you to input polar equations directly and view them as they are meant to be seen in the polar coordinate system. Once you've switched to polar graphing mode, you can enter the provided equation \(r = 8 \text{cos} 2\theta\) into the function input. This prepares the calculator to graph the equation correctly once you've set your viewing window.

Polar Graphing Mode

Using the polar graphing mode on your calculator is essential for accurately rendering polar equations. Unlike rectangular coordinates, which graph according to x and y axes, polar graphing mode plots points based on their distance \(r\) from the origin and their angle \(\theta\) from the positive x-axis.
This mode interprets the inputted polar equation and plots the corresponding points accordingly. For the exercises' equation, \(r = 8 \text{cos} 2\theta\), the polar graphing mode takes into account the variable radius as the angle \(\theta\) changes from 0 to \(2\pi\), creating a visual representation distinct from Cartesian plots. Your calculator handles the intricate process of plotting each point based on these two variables to give you a complete and accurate graph.

Polar Coordinates Visualization

Visualizing polar coordinates involves understanding how points are plotted based on a radius and an angle. When graphing the given polar equation \(r = 8 \text{cos} 2\theta\), the resulting visualization is a rose curve, which is a common shape in polar coordinate graphing. This specific equation produces a rose with 4 petals because the coefficient of \(\theta\) in the cosine function determines the number of petals.
With a graphing calculator, once you input the equation and define the viewing window, the visualization aspect becomes much more intuitive. You'll be able to observe how the radius changes with each increment of the angle as the generous mechanism in the polar graphing mode translates these values into the beautiful patterns characteristic of polar graphs. With practice, interpreting and predicting these visual patterns becomes a fascinating and valuable skill in mathematics.