Problem 49 - Precalculus Mathematics for Calculus

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). $$r=2^{a / 12}, \quad 0 \leq \theta \leq 4 \pi$$

Step-by-Step Solution

Answer

The parametric form is \(x(\theta) = 2^{a/12} \cos(\theta)\) and \(y(\theta) = 2^{a/12} \sin(\theta)\).

Step 1: Express Polar Equation in Parametric Form

To convert the polar equation \( r = 2^{a / 12} \) to parametric form, we use the relationships \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Define two new parameters based on \(r\) and \(\theta\): \[ x(\theta) = 2^{a/12} \cos(\theta) \] and \[ y(\theta) = 2^{a/12} \sin(\theta) \]. Here, \(\theta\) serves as the parameter, and the equations are valid for \(0 \leq \theta \leq 4\pi\).

Step 2: Prepare Graphing the Parametric Equations

The parametric equations to be graphed are \( x(\theta) = 2^{a/12} \cos(\theta) \) and \( y(\theta) = 2^{a/12} \sin(\theta) \). To graph these equations, use a graphing calculator or software. Set the parameter \(\theta\) to vary from \(0\) to \(4\pi\) for the full graph.

Step 3: Graph the Parametric Equations

Use a graphing tool to plot the equations \( x(\theta) = 2^{a/12} \cos(\theta) \) and \( y(\theta) = 2^{a/12} \sin(\theta) \). Adjust the viewing window to ensure that both vertical and horizontal axes are visible over the range specified by \(\theta\). The graph should display a spiral form as \( \theta \) increases due to the exponential growth of \( r \).

Key Concepts

Parametric Equations

Parametric equations are a powerful way to express curves using parameters. When dealing with polar equations, like the one given in the exercise, converting them to parametric form helps in plotting and analyzing the curve.
To convert the polar equation, we utilize two key expressions:

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

In these, \( r \) is the radius that describes the distance from the pole, and \( \theta \) is the angle. By substituting the given equation \( r = 2^{a/12} \) into these expressions, we've defined our parametric equations:

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

These equations describe the x-coordinate and y-coordinate of each point on the curve, with \( \theta \) as the controlling parameter. By expressing the polar equation in parametric form, we create a pair of equations that can be more readily plotted, allowing for visual representation and analysis.

Graphing

Graphing parametric equations is an effective way to visually understand mathematical concepts. With the parametric form derived, the next step is to plot the curve for the specified range of \( \theta \), which in this exercise is from 0 to \( 4\pi \).
To do this, we utilize a graphing calculator or software that can handle parametric inputs. Set the parameter \( \theta \) to increment within the specified interval, and the equations

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

are plotted.
The purpose of graphing these equations is to visually capture the nature of the curve, ensuring the full extent of its behavior is observable. Proper graphing also requires setting the axes limits broad enough to accommodate the possible range of x and y values produced by the function.

Trigonometric Functions

Trigonometric functions are at the heart of converting polar equations to parametric forms. In this exercise, we use the functions \( \cos(\theta) \) and \( \sin(\theta) \) to split the radial and angular components of the polar form into Cartesian coordinates.
These functions help us determine the location of each point relative to the center or pole, by breaking down the radius into a horizontal component \((x)\) and a vertical component \((y)\).

• \( \cos(\theta) \): Determines the projection of the radius onto the horizontal (x-axis).
• \( \sin(\theta) \): Determines the projection of the radius onto the vertical (y-axis).

Both functions repeat their values over every full rotation of \( 2\pi \), which aligns perfectly with the cyclical nature of rotations in polar coordinates. Understanding the behavior of these trigonometric functions is critical for proper graphing and analysis of any polar equation.

Spiral Graphs

When graphing the given equations, one can expect to see a spiral shape forming. This happens because the value of \( r \) grows exponentially as \( \theta \) increases due to the function \( r = 2^{a/12} \).
Spiral graphs are fascinating because they illustrate a continual outward movement as the curve rotates about the pole. Each loop formed by the spiral grows larger as the exponential factor in the equation increases the radius for each successful angle.
To fully understand the nature of spiral graphs:

• As the angle \( \theta \) increases, \( r \) increases, causing the curve to expand outward.
• Exponential growth guarantees that each loop will be larger than the previous, accentuating the spiral effect.

These properties underline the importance of using appropriate tools to observe the behavior of such curves, especially in dynamic, parametric graphing scenarios.