Problem 21 - Algebra and Trigonometry

Question

\(3-24=A\) 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

Rectangular equation is \( xy = 1 \), in the first quadrant.

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = \tan t \) and \( y = \cot t \) with the parameter \( t \) in the range \( 0 < t < \frac{\pi}{2} \). Our task is to sketch the curve and find a rectangular equation by eliminating \( t \).

Step 2: Analyze the Range of the Functions

For \( x = \tan t \), as \( t \) approaches 0, \( x \) approaches 0. As \( t \) approaches \( \frac{\pi}{2} \), \( x \) approaches infinity. For \( y = \cot t \), as \( t \) approaches 0, \( y \) approaches infinity, and as \( t \) approaches \( \frac{\pi}{2} \), \( y \) approaches 0.

Step 3: Sketch the Curve

To sketch the curve, plot \( x = \tan t \) on the x-axis and \( y = \cot t \) on the y-axis for \( 0 < t < \frac{\pi}{2} \). Notice that as \( t \) increases from 0 to \( \frac{\pi}{2} \), \( x \) changes from 0 to infinity, while \( y \) changes from infinity to 0. The curve is a hyperbola-like curve in the first quadrant.

Step 4: Eliminate the Parameter \( t \)

To find the rectangular equation, eliminate \( t \) from the parametric equations. We know that \( x = \tan t \) and \( y = \cot t = \frac{1}{\tan t} \). Thus, \( y = \frac{1}{x} \). The rectangular coordinate equation is \( xy = 1 \).

Step 5: State the Domain

The domain of the rectangular coordinate equation \( xy = 1 \) is determined by the range of \( t \). As \( t \to 0^+ \), \( x \to 0^+ \) and \( y \to \infty \); as \( t \to (\frac{\pi}{2})^- \), \( x \to \infty \) and \( y \to 0^+ \). Therefore, \( x > 0 \) and \( y > 0 \).

Step 6: Conclusion

The curve represented by the parametric equations is a branch of the hyperbola given by the rectangular coordinate equation \( xy = 1 \), restricted to the first quadrant \( x > 0, y > 0 \).

Key Concepts

Rectangular Coordinate Equation

The concept of a rectangular coordinate equation comes into play when we translate parametric equations into standard Cartesian form. This is often necessary to understand the relationship between the x and y coordinates without referencing the parameter. In the provided exercise, we start with the parametric equations:

• \( x = \tan t \)
• \( y = \cot t \)

To eliminate the parameter \( t \) and find a rectangular equation, we utilize the mathematical identities of tangent and cotangent. Since \( y = \cot t \) is the reciprocal of \( x = \tan t \), we can rewrite it as \( y = \frac{1}{x} \). This transforms our parametric system into the rectangular coordinate equation \( xy = 1 \). This equation describes the relation between x and y in the form that most of us are familiar with, excluding the parameter \( t \).

Eliminating the Parameter

Eliminating the parameter is a crucial step in translating parametric equations into one concise rectangular equation. This is often a central step in understanding the overall behavior of a curve derived from parametric equations.

To eliminate the parameter \( t \) in the given exercise, we take the parametric equations \( x = \tan t \) and \( y = \cot t \). Our objective is to express x and y solely in terms of each other. With the identity \( y = \frac{1}{x} \), we can derive the rectangular coordinate equation \( xy = 1 \).

This demonstrates that by manipulating trigonometric identities and relationships, we can simplify a set of parametric equations into a single, more familiar form. This method is especially vital for sketching graphs and analyzing their properties without directly working with parameters.

Hyperbola

A hyperbola is a type of conic section that appears like two mirrored open curves, and it is often recognized by equations of the form \( xy = c \). In the scenario of the original exercise, we brainstormed a curve described by the rectangular equation \( xy = 1 \). This equation suggests a hyperbolic shape.

The characteristic presence of a rectangular hyperbola arises because the multiplication of x and y results in a constant. Here, the curve displayed behaves like a branch of a hyperbola situated entirely in the first quadrant of the Cartesian plane. Hyperbolas have their distinctive shape due to this constant relationship between x and y, with their arms stretching infinitely.Understanding the hyperbola aids in grasping more complex systems and recognizing these distinctive shapes in various mathematical contexts.

Domain of a Function

The domain of a function describes the set of possible input values (x-values) that make the function work correctly without leading to any undefined mathematical operations such as division by zero.

• For this exercise, the original domain of the parametric variable \( t \) was \( 0 < t < \frac{\pi}{2} \).
• In transforming to the rectangular equation \( xy = 1 \), this implies the function operates where both x and y are positive.

As \( x = \tan t \) and \( y = \cot t \), they're undefined when x or y equals zero, and thus the domain reflects only positive values for both variables in the first quadrant.

This careful determination of the domain is essential for understanding and mapping the range of the curve's valid and "real-world" application areas in a Cartesian plane.