Problem 50 - Precalculus with Calculus Previews

Question

Use \((4)\) to parameterize the curve whose polar equation is given. Use a graphing utility to obtain the graph of the resulting set of parametric equations. $$ r=2 \cos \frac{3 \theta}{2}, \quad 0 \leq \theta \leq 4 \pi $$

Step-by-Step Solution

Answer

Parameterize using \( x(t)=2 \cos \left( \frac{3t}{2} \right) \cos(t) \) and \( y(t)=2 \cos \left( \frac{3t}{2} \right) \sin(t) \).

Step 1: Understand Polar to Parametric

To convert a polar equation to parametric form, we use the relationships between Cartesian and polar coordinates: \[ x = r \cos(\theta), \quad y = r \sin(\theta) \] together with the given polar equation. Here, the polar equation is \[ r = 2 \cos\left( \frac{3\theta}{2} \right) \] which gives us that \( r \) changes with \( \theta \).

Step 2: Express x and y in terms of t

Let \( \theta = t \), then the parametric equations will be: \[ x(t) = r \cos(t) = 2 \cos\left( \frac{3t}{2} \right) \cos(t) \] \[ y(t) = r \sin(t) = 2 \cos\left( \frac{3t}{2} \right) \sin(t) \] This expresses \( x \) and \( y \) fully as functions of \( t \).

Step 3: Analyze the Range of t

Since \( \theta \) ranges from \( 0 \) to \( 4\pi \), \( t \) will also range between \( 0 \) and \( 4\pi \). This will ensure we cover the same interval as the original polar equation.

Step 4: Graph the Parametric Equations

Use a graphing utility to plot the parametric equations: \[ x(t) = 2 \cos \left( \frac{3t}{2} \right) \cos(t) \] \[ y(t) = 2 \cos \left( \frac{3t}{2} \right) \sin(t) \] for \( t \) ranging from \( 0 \) to \( 4\pi \). This will provide a visual representation of the parameterized curve.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way of representing a point on a plane through a distance and an angle. Instead of using the traditional Cartesian coordinates, which are pairs

• defined as
  • an x-coordinate (horizontal position) and
  • a y-coordinate (vertical position),

polar coordinates use

• a radius, \(r\) and
• an angle, \(\theta\).

These describe the point's distance from the origin and angle from the positive x-axis. This system is particularly useful for dealing with curves and equations that exhibit radial symmetry or involve complex rotations. When converting to parametric equations, it is essential to understand how the radius and angle change according to the given polar equation.
In our example, the curve is defined by \(r = 2 \cos\left( \frac{3\theta}{2} \right)\), showing that the radius varies with the angle \(\theta\), giving an insight into the shape of the curve.

Parametric Equations

Parametric equations express the coordinates of a curve as functions of a parameter, usually time, denoted as \(t\). In this system, both the

• x-coordinate
• y-coordinate

are described in terms of \(t\). Parametric equations are a powerful tool to describe complex curves that cannot be easily represented with a simple

• \(y = f(x)\) or
• \(x = g(y)\)

relationship.
In our exercise, we define \(\theta = t\). This means that our original polar equations are transformed into parametric equations,

• \(x(t) = r \cos(t) = 2 \cos\left( \frac{3t}{2} \right) \cos(t)\) and
• \(y(t) = r \sin(t) = 2 \cos\left( \frac{3t}{2} \right) \sin(t)\).

These functions translate the polar form into a two-dimensional plot where \(t\) runs from \(0\) to \(4\pi\). This allows us to graph the curve over its full range, showing how the point moves along the curve as \(t\) varies.

Graphing Utility

Graphing utilities are digital tools or graphing calculators that simplify the plotting of complex equations, particularly parametric ones. These tools allow students to visualize functions that could be challenging to graph by hand.
Using a graphing utility in our task helps in seeing the shape of the curve defined by the parametric equations:

• \(x(t) = 2 \cos\left( \frac{3t}{2} \right) \cos(t)\)
• \(y(t) = 2 \cos\left( \frac{3t}{2} \right) \sin(t)\).

With correctly set parameters, such as the range for \(t\) from \(0\) to \(4\pi\), the utility can display a complete and accurate graph of the curve. Graphing utilities can calibrate axes, check specific points, and even animate the progression of points across the graph, aiding in better comprehension of the curve's behavior over the parameter's interval.

Trigonometric Functions

Trigonometric functions are fundamental in connecting polar coordinates to parametric equations. These functions describe the relationships between the angles and lengths in right triangles. Understanding them is key to working effectively with curves in polar and parametric forms.
In our example, we used the cosine function in the polar equation \(r = 2 \cos\left( \frac{3\theta}{2} \right)\). The cosine function changes with \(\theta\), affecting the radius \(r\) and, consequently, the x and y positions in the parametric form.

• \(x(t) = 2 \cos\left( \frac{3t}{2} \right) \cos(t)\)
• \(y(t) = 2 \cos\left( \frac{3t}{2} \right) \sin(t)\).

These equations exploit trigonometric identities to express the parametric equations fully, while the sine and cosine functions dictate the curve's cyclical nature as \(t\) progresses. Understanding these functions helps decode how the curve traverses the plane, influenced by the oscillating values of trigonometric expressions.