Problem 58 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=4 \cos 5 \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation, \(r = 4 \cos 5 \theta\), will look like a flower with 5 petals. The term '5θ' in the cosine function determines the number of petals in the graph. Each new petal starts being traced out when θ is an even multiple of π/5.

Step 1: Understanding Polar Equation

The first thing to understand is the given polar equation \(r = 4 \cos 5 \theta\). This equation represents a curve in a plane where every point on the curve is a certain distance 'r' from the origin (O), and the angle 'θ' that the line from the origin to the point makes with the positive x-axis.

Step 2: Translate Polar to Cartesian

Although we can directly graph polar equation in some graphing utilities, let's break it down to know how to translate it into Cartesian equivalance. We know that, in polar coordinates, \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Substituting \(r = 4 \cos 5 \theta\) in these equations, we get \(x = 4 \cos(5 \theta) \cos(\theta)\) and \(y = 4 \cos(5 \theta) \sin(\theta)\).

Step 3: Graphing the Equation

Now, we will use a graphing utility to graph the polar equation. In the graphing utility, select the mode for polar graphs, enter the equation and specify an appropriate range for 'θ'. The graph of polar equation, \(r = 4 \cos 5 \theta\), will look like a flower with 5 petals since the term '5θ' in the cosine function determines the number of petals in the graph and the number of direction changes in the form. The graph will give not only a visual understanding of the equation but also reveal some fundamental truths, for example, the graph starts tracing out each new petal when θ is an even multiple of π/5.