Problem 20 - 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. $$x=\cot t, \quad y=\csc t, \quad 0 < t < \pi$$

Step-by-Step Solution

Answer

The curve is a hyperbola given by the equation \( y^2 = 1 + x^2 \).

Step 1: Understanding the Parametric Equations

The given parametric equations are \( x = \cot t \) and \( y = \csc t \). These functions are based on trigonometric identities. We know \( \cot t = \frac{\cos t}{\sin t} \) and \( \csc t = \frac{1}{\sin t} \). The parameter \( t \) is restricted to the interval \( 0 < t < \pi \).

Step 2: Sketching the Curve

For sketching, recall that \( x = \cot t \) describes a hyperbolic curve covering \( (-\infty, \infty) \) across \( t \) from 0 to \( \pi \), and \( y = \csc t \) gives the vertical position except \( t = \frac{\pi}{2} \) (as it approaches zero but \( \csc t \) tends to infinity). Sketch the graph by determining key points where \( t = \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{6} \) and noting the behavior as \( t \) approaches 0 and \( \pi \).

Step 3: Using the Trigonometric Identity

We use the identity \( \csc^2 t = 1 + \cot^2 t \) to eliminate the parameter \( t \). Substituting the given expressions, replace \( \cot t \) by \( x \): \( y^2 = \csc^2 t = 1 + \cot^2 t = 1 + x^2 \).

Step 4: Deriving the Rectangular Equation

We now have \( y^2 = 1 + x^2 \), which describes a hyperbola: rearrange to confirm the hyperbolic form. It is already in a form similar to \( x^2 - y^2 = -1 \), reflecting typical hyperbolic behavior.

Step 5: Graph Verification

Check points on your sketch for consistency: For instance, when \( t = \frac{\pi}{4} \), \( x = 1 \) and \( y = \sqrt{2} \), matching with the derived Cartesian equation \( y^2 = 1 + 1^2 = 2 \). This step validates the correct transformation from parametric to Cartesian coordinates.

Key Concepts

Rectangular Coordinates

The term "rectangular coordinates" refers to the standard coordinate system used in mathematics that uses two axes, often called the x-axis and the y-axis. This system helps in plotting and identifying the position of points on a plane. Here, the position is determined by an ordered pair of numbers \( (x, y) \), which are represented as distances along the horizontal and vertical axes respectively.

In the given parametric equations, we start with expressions for \( x = \cot t \) and \( y = \csc t \). Our task is to convert from this parametric form to the more familiar rectangular (or Cartesian) coordinates, which don't rely on the parameter \( t \). This process is called "eliminating the parameter."

To eliminate the parameter in our example, we utilized a trigonometric identity that connects the trigonometric functions of cotangent and cosecant without the presence of \( t \). Specifically, we use the identity \( \csc^2 t = 1 + \cot^2 t \), allowing us to express \( y \) in terms of \( x \): \( y^2 = 1 + x^2 \).

This conversion to the equation \( y^2 = 1 + x^2 \) represents the curve in rectangular coordinates, simplifying analysis and graphing as it provides a direct relationship between \( x \) and \( y \). This ability to switch forms showcases the flexibility and power of using both parametric and rectangular coordinate systems in mathematical analysis.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions (such as sine, cosine, tangent, cotangent, secant, and cosecant). They play a crucial role in simplifying expressions and solving equations that involve trigonometric terms.

In the context of our exercise, the parametric equations involved two trigonometric functions: \( \cot t = \frac{\cos t}{\sin t} \) and \( \csc t = \frac{1}{\sin t} \). By leveraging these identities, we could manipulate and eventually transform the given equations to a form which did not depend on the parameter \( t \).

A key identity used here is \( \csc^2 t = 1 + \cot^2 t \). This is a Pythagorean identity that directly connects cosecant and cotangent, allowing us to find a relationship between them without relying on \( t \). Using this identity, the parametric equations were seamlessly transitioned into the rectangular coordinate form \( y^2 = 1 + x^2 \).

Overall, trigonometric identities are invaluable tools for converting between different forms of equations and simplifying complex expressions. Their profound utility comes from their foundational relationships among trigonometric functions, deeply embedded within the framework of trigonometry and calculus.

Graph Sketching

Graph sketching is the process of creating a visual representation of mathematical functions or equations. It helps in visualizing the behavior of functions, checking for errors, and understanding relationships between variables.

In our exercise, we needed to sketch the curve defined by the parametric equations \( x = \cot t \) and \( y = \csc t \) within the interval \( 0 < t < \pi \). Understanding the behavior of these trigonometric functions over this interval is key to successfully graphing the curve.

• For \( x = \cot t \), the function describes a hyperbolic behavior, spanning both negative and positive infinity as \( t \) moves from near 0 to \( \pi \).
• Meanwhile, \( y = \csc t \) varies except at points where the sine function becomes zero (\