Problem 43 - Algebra and Trigonometry

Question

A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=2^{\theta / 12}, \quad 0 \leq \theta \leq 4 \pi $$

Step-by-Step Solution

Answer

Convert to parametric: \( x = 2^{\theta/12} \cos \theta \), \( y = 2^{\theta/12} \sin \theta \); Graph this for \( 0 \leq \theta \leq 4\pi \).

Step 1: Convert to Parametric Equations - X

To express the given polar equation in parametric form, we first need to find the x-coordinate in terms of a parameter \( \theta \). Using the formula for conversion from polar to Cartesian coordinates, we have:\[x = r \cos(\theta)\]Substituting the expression for \( r \), the equation becomes:\[x(\theta) = \left(2^{\theta/12}\right) \cos(\theta)\]

Step 2: Convert to Parametric Equations - Y

Similarly, find the y-coordinate using the polar-to-Cartesian conversion formula:\[y = r \sin(\theta)\]Substitute the expression for \( r \) to obtain:\[y(\theta) = \left(2^{\theta/12}\right) \sin(\theta)\]

Step 3: Define the Parameter Range

With the parameter \( \theta \) now associated with the parametric equations:- \( x(\theta) = \left(2^{\theta/12}\right) \cos(\theta) \)- \( y(\theta) = \left(2^{\theta/12}\right) \sin(\theta) \)We set the range for \( \theta \) as given by the problem:\[0 \leq \theta \leq 4\pi\]

Step 4: Graph the Parametric Functions

Use a graphing device (or software like Desmos, GeoGebra) to plot the parametric equations.1. Enter the parametric equations: - \( x(\theta) = \left(2^{\theta/12}\right) \cos(\theta) \) - \( y(\theta) = \left(2^{\theta/12}\right) \sin(\theta) \)2. Set the parameter \( \theta \) to vary from 0 to \( 4\pi \).3. Observe the graph to see how the spiral behaves.

Key Concepts

Parametric Equations

Parametric equations are a fundamental way to represent a curve in the plane. Instead of describing a curve via a single equation relating the two coordinate variables directly, these equations express each coordinate as a separate function of an independent parameter, often denoted as \( \theta \). This method allows for more flexibility in describing complex curves.
For the given polar equation \( r=2^{\theta / 12} \), the parametric form involves finding expressions for \( x \) and \( y \). Here, we use the conversions:

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

Substituting \( r = 2^{\theta / 12} \) into these formulas gives:

• \( x(\theta) = \left(2^{\theta/12}\right) \cos(\theta) \)
• \( y(\theta) = \left(2^{\theta/12}\right) \sin(\theta) \)

These parametric equations define how both \( x \) and \( y \) change as \( \theta \) changes, effectively describing the full trajectory of the point \( (x, y) \) on the plane.

Polar to Cartesian Conversion

Converting between coordinate systems, such as from polar to Cartesian coordinates, is a critical skill in mathematics. This conversion is particularly useful when analyzing curves and plotting them with standard x-y axes. In polar coordinates, the position of a point is determined by a distance from the origin \( r \) and an angle \( \theta \), whereas in Cartesian coordinates, a point is defined by \( x \) and \( y \).
The conversion formulas:

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

allow us to translate these forms and better understand the geometry of the shapes involved. In the given polar equation \( r = 2^{\theta / 12} \), this approach lets us see how the spiral behaves in the more familiar Cartesian plane by plugging \( r \)'s expression into these equations. Thus, the curved path described in polar terms becomes a functional xy-plot through the parametric forms.

Graphing Techniques

Once parametric equations are established, graphing them effectively helps visualize the mathematical relationships they represent. This process involves using the parameter's values to plot points, revealing the curve's shape. Graphing devices and software are invaluable tools for this purpose.
For the parametric equations:

• \( x(\theta) = \left(2^{\theta/12}\right) \cos(\theta) \)
• \( y(\theta) = \left(2^{\theta/12}\right) \sin(\theta) \)

we systematically vary \( \theta \) over its range [0, \( 4\pi \)] to plot the corresponding \( (x, y) \) points. The resulting curve should reflect the pattern and growth of the given polar spiral.Using graphing tools like Desmos or GeoGebra can make this task easier, allowing one to visualize the intricate details of the spiral structure as \( \theta \) progresses. With these tools, students can manipulate and observe how changes in \( \theta \) affect the curve in real-time, providing deeper insights into its mathematical nature.