Problem 52 - 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). $$r=2^{\text {sin } \theta}$$

Step-by-Step Solution

Answer

(a) Parametric equations: \(x = 2^{\sin t} \cdot \cos t\), \(y = 2^{\sin t} \cdot \sin t\). (b) Graph these for \(0 \leq t \leq 2\pi\).

Step 1: Understand Polar to Parametric Conversion

In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angular coordinate. To convert this into parametric form, we express both \(x\) and \(y\) in terms of \(t\), where \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\).

Step 2: Express Polar Equation in Terms of \(\theta\)

Given the polar equation \(r = 2^{\sin \theta}\), it's already given with \(\theta\) as the parameter. We will use \(\theta\) itself as our parameter \(t\).

Step 3: Find Parametric Equations

Using \(r = 2^{\sin \theta}\), we have the parametric equations: \[ x = 2^{\sin t} \cdot \cos t \] \[ y = 2^{\sin t} \cdot \sin t \] where \(t = \theta\).

Step 4: Graph the Parametric Equations

Using a graphing device or software, input the parametric equations: \(x(t) = 2^{\sin t} \cdot \cos t\) and \(y(t) = 2^{\sin t} \cdot \sin t\). Set the parameter \(t\) to vary over an adequate range, such as from 0 to \(2\pi\), to see the complete shape of the curve.

Key Concepts

Parametric Equations

Parametric equations are a way of expressing mathematical equations using parameters instead of variables. This approach allows the representation of complex curves and motions by introducing a third variable, often called "parameter" or "time," that bridges other variables.

In a two-dimensional plane, a parametric equation will have two separate equations: one for the x-component and one for the y-component. These equations typically use the same parameter. For example, a point \((x, y)\) on a curve can be defined in terms of a parameter \(t\) where \(x = f(t)\) and \(y = g(t)\).

• This method is especially useful if curves cannot be represented as a single function of \(x\) or \(y\).
• Parametric representation also helps in describing paths of moving points.

Overall, parametric equations provide flexibility, simplify complex expressions, and facilitate graphing when movement is expressed in terms of time or another changing quantity.

Graphing Parametric Equations

Graphing parametric equations involves plotting points calculated from parametric formulas, which express x and y coordinates as functions of a parameter, typically denoted as \(t\).

To graph parametric equations:

• Identify the parameter range. For continuous curves, \(t\) could range from 0 to \(2\pi\), \(-\infty\) to \(\infty\), or another finite interval.
• Compute several values for \(x(t)\) and \(y(t)\) using chosen \(t\) values.
• Plot these coordinates on a coordinate plane.
• Connect the dots to see the curve.
  
  It’s often useful to use a graphing calculator or software tool as it can handle large numbers of computations quickly and offers visual aids like tracing.

While manual calculations give insight into the graphing process, software tools provide efficiency and creativity for complex curves. This approach is vital for visualizing relationships among mathematical equations, especially in applications like physics and engineering.

Conversion Between Polar and Parametric Form

The conversion between polar coordinates and parametric equations is beneficial in providing alternate forms that can be easier to manipulate or interpret.

Polar coordinates describe a point using a radius \(r\) and angle \(\theta\), whereas parametric equations use a parameter \(t\) to describe the same point. The conversion process involves transforming the polar definitions to parametric components using trigonometric relationships. Specifically:

• Convert the polar angle \(\theta\) to a parameter \(t\).
• Utilize the relationships \(x = r \, \cos(\theta)\) and \(y = r \, \sin(\theta)\) to express coordinates in parametric form.

Take the polar equation \(r = 2^{\sin \theta}\). Here, using \(t = \theta\), you can find the parametric form as:

• \[x(t) = 2^{\sin t} \cdot \cos t\]
• \[y(t) = 2^{\sin t} \cdot \sin t\]

Converting from polar to parametric is a powerful tool, facilitating graphing and further computations by leveraging the strengths of both coordinate systems.