Problem 61 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=4 \sin 6 \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r=4 \sin 6 \theta\) is a rose curve with six petals because the coefficient of \(\theta\) in the sine function determines the number of petals.

Step 1: Understanding the Polar Equation

The first step is to understand the polar equation \(r=4 \sin 6 \theta\). Here \(r\) is the distance from the origin and \(\theta\) is the angle measured counter-clockwise from the positive x-axis. The polar equation represents a graph that consists of plotting all points (r, \(\theta\)) such that the equation is satisfied.

Step 2: Converting into Cartesian Coordinates

The polar equation can be converted into Cartesian coordinates for easier visualization which is given by \(x = r\cos(\theta)\), \(y = r\sin(\theta)\). Hence the Cartesian equation becomes \(x= 4\sin(6\theta)\cos(\theta)\) and \(y= 4\sin(6\theta)\sin(\theta)\). This conversion can help to visualize and recognize the graph in a familiar format.

Step 3: Using Graphing Utility

Now, open a graphing utility and input the polar equation \(r=4 \sin 6 \theta\). Adjust the values of \(\theta\) to cover all possible values from 0 to 2\(\pi\). The utility will graph the polar equation, and you can see how the graph changes as \(\theta\) changes.

Step 4: Examining the Graph

The last step is to examine the characteristics of the graph. Since it's a sine graph with a given period, the graph will show multiple 'loops' or 'petals' formulated by the changes in \(\theta\) and \(r\). Notice also how the coefficient of \(\theta\) in the sine function, which is 6 in this case, will determine the number of 'loops' or 'petals' in the polar graph.