Problem 56 - Precalculus

Question

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=2 \cos (3 \theta-2)$$

Step-by-Step Solution

Answer

The graph is a series of loops around the origin, with peaks when the cosine function reaches its maxima and minima. This occurs with a periodicity depending on 3\(\theta\) - 2 due to the cosine multiplier. A suitable viewing window would be one where the x and y-axis ranges from -2.5 to 2.5.

Step 1: Convert Polar Equation To Cartesian Coordinates

If using a graphing utility that does not support polar coordinates, the equation must first be converted to Cartesian coordinates. For any point in polar coordinates \((r, \theta)\), the corresponding Cartesian coordinates \((x, y)\) are given by \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Thus, applying the given polar equation \(r = 2\cos(3\theta - 2)\), you can rewrite it in Cartesian coordinates as \(x = 2\cos(3\theta - 2)\cos(\theta)\), \(y = 2\cos(3\theta - 2)\sin(\theta)\).

Step 2: Graph the Equation

Using the graphing utility, plot the equation from step 1. This is usually done by entering the equation into the utility and setting the appropriate scales and limits. As the range of cosine is between -1 and 1, the radius r will oscillate between -2 and 2. Theta can be from 0 to \(2\pi\) as it represents a full circle in polar coordinates.

Step 3: Describe the Viewing Window

The viewing window should be set to accommodate the full range of x and y as determined in Step 2. As the radius r varies between -2 and 2 and theta is between 0 to \(2\pi\), the x and y values will fall within the range of -2 to 2. Therefore, a viewing window that shows x and y values from -2.5 to 2.5 would be a good choice in order to capture the full graph while leaving a small margin around the edges.

Key Concepts

Polar to Cartesian Coordinates

Understanding the relationship between polar and Cartesian coordinates is fundamental for graphing equations that are initially presented in a polar form, like the polar equation given: \(r=2\cos(3\theta-2)\).

In polar coordinates, a point is determined by two values: the radius \(r\), which measures how far away the point is from the origin, and the angle \(\theta\), which represents the direction from the origin to the point. To convert a point from polar to Cartesian coordinates, which consist of the \(x\) and \(y\) values we are more familiar with, you can use these formulas: \[x = r\cos(\theta)\] and \[y = r\sin(\theta)\].

When you convert the given polar equation into Cartesian coordinates, you'll compute the \(x\) and \(y\) values for several values of \(\theta\), typically between 0 and \(2\pi\) radians. For example, to convert the endpoint of the vector with a polar equation \(r=2\cos(3\theta-2)\) to Cartesian coordinates when \(\theta=0\), you'd calculate \(x = 2\cos(-2)\cos(0)\) and \(y = 2\cos(-2)\sin(0)\).

Plotting Polar Functions

Once you've understood the conversion of polar to Cartesian coordinates, plotting polar functions becomes more approachable. Essentially, to plot polar equations, you would typically use a graphical representation where the horizontal axis represents the cosine function and the vertical axis represents the sine function.

By substituting various values for \(\theta\) into the polar equation, you will obtain different values for \(r\), the distance from the origin. These values will then be plotted corresponding to the angle \(\theta\). For the equation in question, \(r=2\cos(3\theta-2)\), the graph will reflect the trigonometric property of the cosine function, causing a looping or petal-like structure due to the multiplicative factor of 3 in the cosine argument.

When plotting, each point in the polar graph represents the end of a vector originating at the pole (or origin) at an angle of \(\theta\) from the positive x-axis (polar axis), with a length of \(r\). The complete graph of the polar function will reveal a pattern that can be easily identified once all the necessary points have been plotted.

Graphing Utility Usage

Utilizing a graphing utility greatly simplifies the process of visualizing complex polar equations. Instead of plotting each individual point manually, a graphing utility can generate a precise graph of the polar function almost instantaneously.

To graph the polar equation \(r=2\cos(3\theta-2)\) using a utility, you can directly input the polar equation if the utility supports polar coordinates. If not, as in Step 1 of our exercise, you would first convert the polar equation to its Cartesian form. Next, you will set appropriate scales and limits to display the range of the function. Consider that since \(r\) can oscillate between -2 and 2, therefore, your x and y-axis should accommodate slightly more than this range to account for the curve's full breadth.

The choice of a viewing window is essential; you typically want a window that captures the entire graph of the equation. For the given exercise, setting the viewing window to show \(x\) and \(y\) values from -2.5 to 2.5 ensures that the entire graph is visible with a margin for clarity. By following these steps and adjusting the graphing utility settings, students can achieve a clear visualization of complex polar graphs.