Problem 18 - Precalculus: Mathematical 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=\cot t, \quad y=\csc t, \quad 0

Step-by-Step Solution

Answer

The curve is the upper half of a hyperbola, given by \(y = \sqrt{x^2 + 1}\).

Step 1: Understanding the Parametric Equations

The given parametric equations are \(x = \cot t\) and \(y = \csc t\), where \(0 < t < \pi\). The parameter \(t\) typically represents an angle measured in radians.

Step 2: Sketching the Curve

To sketch the curve, it's useful to calculate a few specific points by choosing values of \(t\) within the given interval \((0, \pi)\). For example:- At \(t = \frac{\pi}{4}\), \(x = \cot\left(\frac{\pi}{4}\right) = 1\) and \(y = \csc\left(\frac{\pi}{4}\right) = \sqrt{2}\).- As \(t\) approaches 0, \(x\) approaches \(+\infty\) and \(y\) approaches \(+\infty\).- As \(t\) approaches \(\pi\), \(x\) approaches \(-\infty\) and \(y\) approaches \(+\infty\).The curve generally decreases from \(+\infty\) to \(-\infty\) along the x-axis as \(t\) goes from 0 to \(\pi\). The y-values are always positive.

Step 3: Finding the Rectangular-Coordinate Equation

To eliminate the parameter \(t\), we use trigonometric identities. Recall that \(\cot^2 t + 1 = \csc^2 t\). Substituting the parametric expressions into this identity gives us:\[cot^2 t + 1 = \csc^2 t \implies x^2 + 1 = y^2.\]Thus, the rectangular-coordinate equation of the curve is \(x^2 + 1 = y^2\).

Step 4: Final Curve Analysis

Rewriting the rectangular equation \(y^2 = x^2 + 1\) can be transformed to \(y = \pm\sqrt{x^2 + 1}\). Considering only the positive root (since \(y = \csc t\) is positive for \(0 < t < \pi\)), we have \(y = \sqrt{x^2 + 1}\). This represents the upper half of a hyperbola, the graph of which stretches vertically upwards as x-values spread from negative to positive.

Key Concepts

Rectangular-Coordinate Equation

To understand how to find a rectangular-coordinate equation from parametric equations, we begin by recognizing what each form represents. Parametric equations describe a set of related quantities that rely on a common variable, here known as the parameter \( t \). In our exercise, these parametric equations are \( x = \cot t \) and \( y = \csc t \), each dependent on \( t \), a variable representing angles in radians.

Eliminating the parameter involves expressing the relationship between \( x \) and \( y \) directly, excluding \( t \). This transformation is accomplished by using a known trigonometric identity: \( \cot^2 t + 1 = \csc^2 t \). By substituting \( x = \cot t \) and \( y = \csc t \) into this identity, we get \( x^2 + 1 = y^2 \), which is the rectangular equation form. The conversion captures all x and y relationships that correspond to the collection of \( t \) values in the specified range \( 0 < t < \pi \).

Overall, finding the rectangular form is essential for analyzing curves using standard Cartesian concepts. It also simplifies graphing using commonly understood coordinates.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable within their domain. They play a crucial role in converting parametric equations into rectangular-coordinate equations as demonstrated in this exercise.

In our example, we utilized the identity \( \cot^2 t + 1 = \csc^2 t \) to link the equations \( x = \cot t \) and \( y = \csc t \). Understanding that \( \cot t \) is the reciprocal of \( \tan t \) and \( \csc t \) is the reciprocal of \( \sin t \) helps visualize these relationships.

• \( \cot t = \frac{\cos t}{\sin t} \)
• \( \csc t = \frac{1}{\sin t} \)

To handle more complicated expressions or equations, trigonometric identities like these allow for the substitution or simplification of terms. They are fundamental tools in calculus and algebra for integrating and differentiating trigonometric functions.

Graphing Curves

Graphing curves from parametric equations provides a visual understanding of the behavior of mathematical relationships. In the exercise, the parametric equations \( x = \cot t \) and \( y = \csc t \) describe a curve as \( t \) changes from \( 0 \) to \( \pi \).

The sketching of this curve involves computing specific points by plugging different \( t \) values. For instance:

• At \( t = \frac{\pi}{4} \), we find \( x = 1 \) and \( y = \sqrt{2} \).
• As \( t \) approaches 0, both \( x \) and \( y \) approach \( +\infty \).
• As \( t \) nears \( \pi \), \( x \) approaches \( -\infty \) while \( y \) remains \( +\infty \).

The curve rapidly descends along the x-axis from positive to negative infinity, with \( y \) values consistently positive. This is typical for hyperbolas which are often represented as "open" curves.

Graphing this type of curve helps students understand not only the geometric figure but also the nature of the function and its limits. Each plotted point gives insight into the curve's overall direction and shape.