Problem 18 - Precalculus with Calculus Previews

Question

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=-1+\cos t, y=2+\sin t, 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 + y^2 + 2x - 4y + 4 = 0 \).

Step 1: Understand the Parametric Relationships

We are given the parametric equations \( x = -1 + \cos t \) and \( y = 2 + \sin t \). These equations express \( x \) and \( y \) in terms of the parameter \( t \). Our task is to eliminate \( t \) to get an equation in terms of \( x \) and \( y \) only.

Step 2: Express trigonometric functions in terms of x and y

Rearrange the given parametric equations to express \( \cos t \) and \( \sin t \): \( \cos t = x + 1 \) and \( \sin t = y - 2 \).

Step 3: Use Pythagorean Identity

Recall the identity: \( \sin^2 t + \cos^2 t = 1 \). Substitute \( \cos t = x + 1 \) and \( \sin t = y - 2 \) into this identity: \[ (y - 2)^2 + (x + 1)^2 = 1 \]

Step 4: Simplify the Equation

Expand and simplify the equation: \[ (y - 2)^2 = y^2 - 4y + 4 \]\[ (x + 1)^2 = x^2 + 2x + 1 \]Adding these gives: \[ y^2 - 4y + 4 + x^2 + 2x + 1 = 1 \]

Step 5: Get the Rectangular Equation

Combine and simplify terms: \[ x^2 + y^2 + 2x - 4y + 4 = 0 \].This is the rectangular equation representing the same curve as the original parametric equations \( 0 \leq t \leq 2\pi \).

Key Concepts

Eliminating Parameter

When we have parametric equations like \( x = f(t) \) and \( y = g(t) \), our goal in eliminating the parameter is to express \( x \) and \( y \) without involving the parameter \( t \). This process converts the parametric equations into a rectangular equation, which involves only \( x \) and \( y \).

In this exercise, we are given the parametric equations:

• \( x = -1 + \cos t \)
• \( y = 2 + \sin t \)

The goal is to eliminate \( t \) from these expressions. By isolating the trigonometric expressions, we have:

• \( \cos t = x + 1 \)
• \( \sin t = y - 2 \)

This setup allows us to progress to the next step, which involves using mathematical identities to eliminate the parameter \( t \). The result will be a clear relationship between \( x \) and \( y \) alone.

Rectangular Equation

After eliminating the parameter \( t \), our aim is to find a rectangular equation that expresses the relationship between \( x \) and \( y \). This type of equation is called 'rectangular' because it does not involve the parameter \( t \), only the variables \( x \) and \( y \).

Let's continue from the versions we isolated,\( \cos t = x + 1 \) and \( \sin t = y - 2 \).We use these expressions in conjunction with mathematical identities, like the Pythagorean identity, to construct the rectangular equation. After substitution and algebraic manipulation, the rectangular equation we derive is:\[ x^2 + 2x + y^2 - 4y + 4 = 0 \].

This equation is now free from the parameter \( t \) and depicts the same geometric curve that the original parametric equations represented.

Pythagorean Identity

In trigonometry, the Pythagorean identity is a fundamental concept used to simplify expressions and solve equations. It states that for any angle \( t \), the identity \( \sin^2 t + \cos^2 t = 1 \) holds true.

This identity is especially useful when working to eliminate the parameter in trigonometric equations. In our case, substituting \( \cos t = x + 1 \) and \( \sin t = y - 2 \) into this identity helps to form the basis for simplifying into a rectangular equation:

\[ (x + 1)^2 + (y - 2)^2 = 1 \].This expression uses the identity to combine \( \sin t \) and \( \cos t \) into one relationship, giving us a pathway to our final rectangular equation. Understanding and employing the Pythagorean identity efficiently is key to transitioning from parametric to rectangular forms.