Problem 54 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r=3 \cos (4 \theta) $$

Step-by-Step Solution

Answer

The polar equation describes a rose curve with 8 petals, each petal having a length of 3.

Step 1 - Understand the Polar Equation

Identify the type of polar equation given, which is of the form \( r = a \, \cos(k \, \theta) \). Here, \( a = 3 \) and \( k = 4 \). This is a rose curve.

Step 2 - Analyze the Rose Curve Parameters

Determine the number of petals of the rose curve. Since \( k = 4 \) is an even number, the rose curve will have \( 2k = 8 \) petals.

Step 3 - Determine the Length of Petals

The maximum value of \( r \) occurs when \( \cos(4 \theta) = 1 \). Since \( a = 3 \), the length of each petal is 3.

Step 4 - Plot Key Points

To plot the rose curve, calculate \( r \) for various values of \( \theta \). Example values: \( \theta = 0, \frac{\pi}{8}, \frac{\pi}{4}, ... \). For \( \theta = 0, \ r = 3 \), for \( \theta = \frac{\pi}{8}, r = 0 \), etc.

Step 5 - Sketch the Graph

Draw the polar graph by plotting the points calculated and sketching eight petals evenly distributed around the origin, each extending out to a radius of 3.

Step 6 - Review and Finalize Graph

Ensure that all petals are evenly spaced and symmetric around the origin. The complete graph should display eight petals.

Key Concepts

Rose Curves

Rose curves are a fascinating type of graph in polar coordinates. They often create beautiful, symmetrical patterns that resemble the petals of a flower. The general form for a rose curve is given by the polar equation: \( r = a \, \text{cos}(k \, \theta) \) or \( r = a \, \text{sin}(k \, \theta) \). Here, \(a\) determines the length of the petals and \(k\) influences the number of petals.

For a given \(k\):

• If \(k\) is even, the rose curve will have \(2k\) petals.
• If \(k\) is odd, the rose curve will have \(k\) petals.

In our example, since \(k = 4\) is even, the rose curve equation \( r=3 \, \text{cos}(4 \, \theta) \) results in 8 petals. Each petal extends out to a maximum length of 3 units. Understanding these properties helps you predict and sketch the graph effectively.

Polar Coordinates

Polar coordinates are a different way of representing points in a plane compared to the traditional Cartesian coordinates. Instead of using \( (x, y) \) pairs, polar coordinates use a radius \( r \) and an angle \( \theta \).

Here's how to understand polar coordinates:

• \(r\) is the distance from the origin to the point.
• \(\theta\) is the angle from the positive x-axis to the point, measured in radians.

For example, the polar coordinate (3, \(\frac{\pi}{4}\)) means you move 3 units away from the origin along the direction that makes an angle of \(\frac{\pi}{4}\) radians with the positive x-axis. This coordinate system is particularly useful for plotting curves like circles, spirals, and especially rose curves!

Graphing in Polar Form

Graphing in polar form can be a bit different from graphing in Cartesian form, but it's a valuable skill for visualizing certain types of functions such as those seen in rose curves.

Follow these steps for graphing in polar form:

• Identify the type of polar equation you're dealing with.
• Understand the parameters of the equation to know the shape and behavior of the graph.
• Choose a set of values for \(\theta\) with corresponding values for \(r\).

For example, consider the equation \( r=3 \, \text{cos}(4 \, \theta) \). By substituting various values of \(\theta\) (like 0, \(\frac{\pi}{8}\), \(\frac{\pi}{4}\), etc.) you can calculate the values of \(r\). Plot each point \((r, \theta)\) in polar coordinates and join these points to sketch out the curve.

Finally, ensure the graph makes sense with the predicted even spacing, symmetry, and length of petals to verify accuracy. This process transforms your understanding of abstract equations into tangible, visual forms.