Problem 49 - Algebra and Trigonometry

Question

\(49-52\) . 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

Express as parametric: \( x = 2^{\theta/12} \cos \theta, \; y = 2^{\theta/12} \sin \theta \). Graph shows a spiral.

Step 1: Understand the Polar Equation

The given polar equation is \( r = 2^{\frac{\theta}{12}} \), where \( 0 \leq \theta \leq 4\pi \). In polar coordinates, \( r \) is the radial distance from the origin and \( \theta \) is the angle from the positive x-axis.

Step 2: Convert to Parametric Form

We need to express this polar equation in terms of \( x \) and \( y \) as parametric equations. The relationship between polar and Cartesian coordinates is given by \( x = r \cos \theta \) and \( y = r \sin \theta \). Substituting \( r = 2^{\theta/12} \) into these equations, we get:\[ x(\theta) = 2^{\frac{\theta}{12}} \cos \theta \]\[ y(\theta) = 2^{\frac{\theta}{12}} \sin \theta \]

Step 3: Specify the Range for Parametric Equations

The given range for \( \theta \) is \( 0 \leq \theta \leq 4\pi \). This range should be used for the parametric equations \( x(\theta) \) and \( y(\theta) \).

Step 4: Graph the Parametric Equations

To graph the parametric equations, use a graphing calculator or software. Input \( x(\theta) = 2^{\theta/12} \cos \theta \) and \( y(\theta) = 2^{\theta/12} \sin \theta \) with \( \theta \) ranging from \( 0 \) to \( 4\pi \) to visualize the curve. The graph will show a spiral stretching outward.

Key Concepts

Parametric Equations

Parametric equations are a way to express the coordinates of points on a curve through functions of a parameter, usually denoted as \( t \). In the context of converting polar equations to parametric equations, this parameter is often the angle \( \theta \). The parametric form splits the description of the curve into two separate functions: one for the \( x \)-coordinate and another for the \( y \)-coordinate. This helps in visualizing and analyzing curves that might be complex in polar form.
For the given polar equation \( r = 2^{\theta / 12} \), we convert it by using the equations for converting polar coordinates to Cartesian coordinates:

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

These equations specify the \( x \) and \( y \) positions as \( \theta \) varies, providing a full description of the curve in the Cartesian plane.

Graphing Polar Functions

Graphing polar functions often results in beautifully intricate patterns. In our example, the function \( r = 2^{\theta / 12} \) describes a spiral, which is commonly seen in nature such as in seashells and galaxies. Polar graphs involve plotting points with distances \( r \) from the origin at varying angles \( \theta \).
When we translate to parametric form, as with \( x(\theta) = 2^{\theta/12} \cos \theta \) and \( y(\theta) = 2^{\theta/12} \sin \theta \), we can use graphing technology to visualize these images more effectively. Simply input the parametric equations into a graphing device, ensuring that \( \theta \) varies from \( 0 \) to \( 4\pi \) to capture the full rotation of the spiral. This range allows us to see how the spiral unwinds in the plane, creating an expanding pattern.

Converting Polar to Cartesian Coordinates

Converting between polar and Cartesian coordinates is a fundamental skill in trigonometry and calculus. Polar coordinates describe points in terms of a distance from a central point and an angle from a reference direction. In contrast, Cartesian coordinates use perpendicular axes, typically named \( x \) and \( y \), to specify a point's exact location on the plane.
For the conversion, the key formulas are:

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

These equations help transform a point given in polar coordinates to its equivalent in the Cartesian system. This is especially useful in graphing and analyzing patterns that are more intuitive to describe in one coordinate system over another. Understanding how to navigate between these systems ensures flexibility and depth in mathematical analysis.