Problem 19 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\tan t, \quad y=\cot t, \quad 0

Step-by-Step Solution

Answer

The curve is a hyperbola with the equation \(xy = 1\).

Step 1: Understand the Problem

We have two parametric equations: \(x = \tan t\) and \(y = \cot t\), while the parameter \(t\) is between 0 and \(\pi/2\). We need to sketch this curve and then find its rectangular coordinate equation.

Step 2: Analyze the Range of \(t\)

For \(0 < t < \pi/2\), as \(t\) approaches 0, \(\tan t\) approaches 0 and \(\cot t\) approaches infinity. As \(t\) approaches \(\pi/2\), \(\tan t\) approaches infinity, and \(\cot t\) approaches 0. This informs us about the behavior of the curve's endpoints.

Step 3: Sketch the Curve

Using the information from step 2, sketch the curve. It starts at \((0, \infty)\) and progresses towards \((\infty, 0)\) as \(t\) increases from 0 to \(\pi/2\). The curve is in the first quadrant, since both \(x = \tan t\) and \(y = \cot t\) are positive in this range.

Step 4: Eliminate the Parameter

Use the identities \(x = \tan t\) and \(y = \cot t\). Recall \(\tan t = \frac{\sin t}{\cos t}\) and \(\cot t = \frac{\cos t}{\sin t}\). Thus, \(xy = \tan t \cdot \cot t = 1\), since \(\tan t \cdot \cot t = \frac{\sin t}{\cos t} \cdot \frac{\cos t}{\sin t} = 1\).

Step 5: Write the Rectangular Equation

The rectangular coordinate equation by eliminating the parameter \(t\) is \(xy = 1\). This is a hyperbola.

Key Concepts

Rectangular-coordinate Equation

In mathematics, parametric equations use parameters to define a set of related quantities. To gain a different perspective, we often translate these into rectangular-coordinate equations. For the given parametric equations,

• \( x = \tan t \)
• \( y = \cot t \)

the goal is to find a relationship that connects \( x \) and \( y \) without considering the parameter \( t \). This process involves using trigonometric identities or algebraic manipulation to eliminate \( t \). In this exercise, multiplying the two equations

• \( \tan t \times \cot t \)
<\/ul>produces the constant \( 1 \). So, the derived rectangular-coordinate equation is \( xy = 1 \). This equation signifies a hyperbola, reflected by its symmetry and opposite asymptotic behavior, crossing axes at points like \((1, 1)\) and \((-1, -1)\). Understanding these transformations provides a clearer geometric interpretation by changing perspectives from parametric curves.

Eliminating the Parameter

Eliminating the parameter is a critical step in transforming parametric equations into their rectangular-coordinate form. Essentially, we want to remove the variable \( t \) to discover a direct relationship between \( x \) and \( y \). For our current problem,

• \( x = \tan t \)
• \( y = \cot t \)

We apply trigonometric identities: \[\tan t = \frac{\sin t}{\cos t}\] and \[\cot t = \frac{\cos t}{\sin t}\]. By substituting these identities and then multiplying \( \tan t \) and \( \cot t \), we obtain the equation \( xy = \tan t \cdot \cot t = 1 \). This approach is useful because it shows how the functions relate directly to one another without involving the excess detail of the parameter \( t \). Such a method is not only a timesaver but also a revelation in simplifying complex sets of equations.

Sketching Curves

Sketching curves from parametric equations can provide a visual representation that aids our understanding of how the functions behave. When working with the given parametric equations \( x = \tan t \) and \( y = \cot t \), it's important to recognize the range limitations: \( 0 < t < \frac{\pi}{2} \). This informs us of the curve's path and its location within the first quadrant. As \( t \) approaches 0, \( x \) starts at 0 and moves towards infinity, whereas, \( y \) begins at infinity and approaches 0.

• The curve starts at point \((0, \infty)\)
• Progresses through the first quadrant
• Ends approaching \((\infty, 0)\)

Sketching involves plotting these routes visually, emphasizing the infinity boundaries. Given this range and the equation \( xy = 1 \), the graph reveals a hyperbola. Recognizing this characteristic through sketching can improve the intuitive understanding of the mathematical transformations and behaviors of parametric equations.