Problem 21 - Precalculus Mathematics for Calculus

Question

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\tan t, \quad y=\cot t, \quad 0 < t < \pi / 2$$

Step-by-Step Solution

Answer

The rectangular equation is \( y = \frac{1}{x} \). Sketch starts at \((0, \infty)\) and approaches \((\infty, 0)\).

Step 1: Understand the Parametric Equations

We are given two parametric equations: \( x = \tan t \) and \( y = \cot t \), where \( t \) is the parameter that ranges from \( 0 < t < \pi/2 \). This means as \( t \) changes within the given interval, \( x \) and \( y \) will change correspondingly.

Step 2: Sketch the Curve

To sketch the curve, we need to analyze how \( x \) and \( y \) behave as \( t \) varies from 0 to \( \pi/2 \). \( \tan t \) increases from 0 to \( \infty \), while \( \cot t \) decreases from \( \infty \) to 0. Hence, the curve will start from the point \((0, \infty)\) and will approach \((\infty, 0)\). The sketch would show the path approaching a horizontal asymptote at \( y = 0 \) and a vertical asymptote at \( x = 0 \).

Step 3: Eliminate the Parameter

To eliminate \( t \), use the identities of \( \tan t \) and \( \cot t \). Recall that \( y = \cot t = \frac{1}{\tan t} \). Thus, substitute the expression for \( \tan t \) in terms of \( x \): \( y = \frac{1}{x} \). This is the rectangular-coordinate equation.

Step 4: Conclude the Rectangular Equation

With the above substitution, the rectangular-coordinate equation of the curve is \( y = \frac{1}{x} \). This tells us that the curve is a hyperbola in rectangular coordinates.

Key Concepts

Eliminating the Parameter

Eliminating the parameter in parametric equations involves finding a way to relate the parametric variables directly without involving the parameter itself. In our original exercise, we have the equations \( x = \tan t \) and \( y = \cot t \), where \( t \) serves as the parameter.
First, note that \( \tan t \) and \( \cot t \) are reciprocal identities, meaning \( \cot t = \frac{1}{\tan t} \). By understanding this reciprocal relationship, we can express \( y \) directly in terms of \( x \) by substituting \( x \) for the \( \tan t \). This gives us:

• \( y = \cot t = \frac{1}{\tan t} = \frac{1}{x} \)

Through this substitution, we've successfully eliminated the parameter \( t \) and found a direct relationship between \( x \) and \( y \): the rectangular-coordinate equation \( y = \frac{1}{x} \). This equation is much simpler to analyze and visualize in a Cartesian coordinate system.

Sketching Curves

Sketching parametric curves is about visualizing how the point \((x, y)\) moves as the parameter changes. In this exercise, we are analyzing the curve generated by \( x = \tan t \) and \( y = \cot t \) as \( t \) varies from \( 0 \) to \( \pi/2 \).
When \( x = \tan t \), the values of \( x \) start from 0 and increase to infinity. Conversely, \( y = \cot t \) starts from infinity and decreases to 0. This specific behavior hints at the presence of asymptotic behavior:

• The curve starts at the point \((0, \infty)\).
• It approaches the horizontal asymptote as \( y = 0 \), since \( y \) gets closer to zero as \( t \) approaches \( \pi/2 \).
• There’s also a vertical asymptote at \( x = 0 \), as \( x \) tends towards zero when \( t \) is near the lower limit.

By sketching with these behaviors in mind, we understand the nature of the curve, exposing the hyperbolic shape described by the rectangular equation \( y = \frac{1}{x} \).

Rectangular-Coordinate Equation

A rectangular-coordinate equation expresses the relationship between two variables \( x \) and \( y \) in a Cartesian plane with no parameters involved. In our problem, starting with the parametric equations \( x = \tan t \) and \( y = \cot t \), we identified that a direct relationship can be derived.
Using the identity \( \cot t = \frac{1}{\tan t} \), we rearranged the equation to express \( y \) solely in terms of \( x \), resulting in \( y = \frac{1}{x} \). This conversion from a parametric form to a rectangular form simplifies understanding and handling of such equations.
Such a rectangular equation simplifies visualization. In particular, \( y = \frac{1}{x} \) is known as a hyperbola in typical Cartesian coordinates. Hyperbolas have unique properties that affect their graph:

• They display two asymptotes, which are lines that the curve approaches but never touches.
• The equation \( y = \frac{1}{x} \) specifically features a hyperbola with its asymptotes as the \( x \)- and \( y \)-axes.

This shift from parametric to rectangular form not only aids in graphing but also deepens our understanding of the underlying relationships between \( x \) and \( y \) without needing to revert back to a parameter.