Problem 41 - College Trigonometry

Question

Every polar curve \(r=f(\theta)\) can be translated to a system of parametric equations with parameter \(\theta\) by \(\\{x=r \cos (\theta)=f(\theta) \cos (\theta), y=r \sin (\theta)=f(\theta) \sin (\theta) .\) Convert \(r=6 \cos (2 \theta)\) to a system of parametric equations. Check your answer by graphing \(r=6 \cos (2 \theta)\) by hand using the techniques presented in Section \(11.5\) and then graphing the parametric equations you found using a graphing utility.

Step-by-Step Solution

Answer

The parametric equations are \( x = 6 \cos(2\theta) \cos(\theta) \) and \( y = 6 \cos(2\theta) \sin(\theta) \).

Step 1: Identify the Polar Equation

The given polar equation is \( r = 6 \cos(2\theta) \). This will be used to create the parametric equations.

Step 2: Determine x in Parametric Form

Using the conversion formula \( x = r \cos(\theta) \), substitute \( r = 6 \cos(2\theta) \) to get \( x = 6 \cos(2\theta) \cos(\theta) \).

Step 3: Determine y in Parametric Form

Using the conversion formula \( y = r \sin(\theta) \), substitute \( r = 6 \cos(2\theta) \) to get \( y = 6 \cos(2\theta) \sin(\theta) \).

Step 4: Final Parametric Equations

The parametric equations are: \[ x = 6 \cos(2\theta) \cos(\theta) \] \[ y = 6 \cos(2\theta) \sin(\theta) \]

Step 5: Graph Polar Equation by Hand

Using polar graphing techniques, plot \( r = 6 \cos(2\theta) \). This curve appears as a four-leaved rose, with petals at every \( \frac{\pi}{2} \) interval.

Step 6: Graph Parametric Equations with a Graphing Utility

Use a graphing tool to plot the parametric equations: \ \( x = 6 \cos(2\theta) \cos(\theta) \) \ \( y = 6 \cos(2\theta) \sin(\theta) \). The graph should match the polar graph of the rose curve with four petals.

Key Concepts

Parametric Equations

In mathematics, parametric equations are an essential tool for describing curves and surfaces. These types of equations express the coordinates of points on a curve or surface as functions of a parameter. When dealing with polar coordinates, the parameter often involved is \(\theta\), the angle in radians.
In our exercise, we're asked to translate a polar equation, \( r = f(\theta) \), into parametric form.
This is achieved through the formulas:

• \( x = r \cos(\theta) = f(\theta) \cos(\theta) \)
• \( y = r \sin(\theta) = f(\theta) \sin(\theta) \)

By substituting \( r = 6 \cos(2\theta) \), we obtain the parametric equations:

• \( x = 6 \cos(2\theta) \cos(\theta) \)
• \( y = 6 \cos(2\theta) \sin(\theta) \)

These equations help us describe how the curve behaves in the Cartesian plane, providing a clear visual representation of the curve's path.

Graphing Polar Curves

Graphing polar curves may seem daunting at first, but with some practice, it becomes a useful skill, especially for visualizing complex patterns. Polar curves are often represented as \( r = f(\theta) \), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.
For the polar equation \( r = 6 \cos(2\theta) \), we identify its graph as a four-leaved rose. This is a specific type of polar curve with petal formations. Graphing it requires an understanding of how \( \theta \) affects the radius \( r \). As \( \theta \) varies from \( 0 \) to \( 2\pi \), \( r \) changes, creating a pattern with symmetry and repeating sections.
It's helpful to note:

• When \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), the curve's path reaches a maximum extending from the origin, marking the 'petals' of the rose.
• Negative values of \( r \) flip the point through the origin, reflecting the symmetry in polar curves.

Understanding these behaviors helps in manually plotting the curve using polar graphing techniques.

Conversion Between Coordinate Systems

Converting between polar and Cartesian coordinate systems is a fundamental skill in mathematics, bridging the gap between two distinct ways of describing a location in a plane. The conversion involves using trigonometric relationships that connect the polar coordinates \( (r, \theta) \) to Cartesian coordinates \( (x, y) \).
The core conversion formulas are:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(\frac{y}{x}) \)

In our exercise, converting the polar equation \( r = 6 \cos(2\theta) \) to parametric equations exemplifies the use of this conversion process, transitioning between polar and Cartesian systems.
This ability to switch between systems makes complex graphs easier to handle and provides a versatile approach to problem-solving. It helps in situations requiring integration or differentiation when one form is more suitable than the other.