Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

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 of \(r=2 \cos (3 \theta-2)\) creates a polar curve with three loops or petals. The viewing window was set from 0 to \(2\pi/3\) (for \(\theta\)) and from -2 to 2 (for \(r\)).

Step 1: Understand the Polar Equation

The given polar equation is \(r=2 \cos (3 \theta-2)\). This describes a polar curve, which is a shape traced by a point whose distance \(r\) from the origin and angle \(\theta\) from the positive x-axis satisfy the given equation. Here, the number 2 adjusts the size of the graph, while 3 multiplies the angle by 3, thus giving three loops or petals to the polar graph. The number 2, subtracted from 3\(\theta\), shifts the graph to the right.

Step 2: Graphing the Polar Equation

Now, use a graphing utility that can graph polar equations, such as Desmos or Grapher on a Mac. Enter the polar equation \(r=2 \cos (3 \theta-2)\) as is in the graphing tool. Choose the polar graphing mode if it's not selected by default.

Step 3: Setting the Viewing Window

The viewing window should be adjusted for a clear visualization of the polar curve. For the \(\theta\)-values, it's advisable to view at least one complete cycle of the polar curve. Given the periodic nature of the cosine function and the multiplier of 3 in the argument, \(\theta\)-values from 0 to \(2\pi/3\) will suffice. For \(r\)-values, considering the range of the cosine function (-1 to 1), multiplied by the constant 2 in the equation, a logical range would be from -2 to 2.

Step 4: Observing and Describing the Graph

Observe the graph, identify the shapes and features shown, and describe notable points. In this case, the graph will have three loops or petals due to the argument being multiplied by 3. The graph is also expanded and shifted due to the constant multipliers.