Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation for each curve and graph the curve. $$x=2+\sin t, y=1+\cos t ; \text { for } t \text { in }[0,2 \pi]$$

Step-by-Step Solution

Answer

The rectangular equation is \( (x-2)^2 + (y-1)^2 = 1 \), which is a circle.

Step 1: Identify Parametric Equations

The given parametric equations are \( x = 2 + \sin t \) and \( y = 1 + \cos t \). They describe the curve in terms of the parameter \( t \).

Step 2: Isolate Sine and Cosine

To convert these equations into a rectangular form, isolate \( \sin t \) and \( \cos t \) in terms of \( x \) and \( y \): \( \sin t = x - 2 \) and \( \cos t = y - 1 \).

Step 3: Use Pythagorean Identity

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

Step 4: Simplify and Identify Curve

The equation \( (x - 2)^2 + (y - 1)^2 = 1 \) is a circle with center at (2, 1) and radius 1.

Step 5: Graph the Equation

Graph a circle centered at (2, 1) with a radius of 1 to represent the rectangular equation of the parametric curve.

Key Concepts

Rectangular Equation

In mathematics, a rectangular equation expresses a relationship between two variables in the Cartesian coordinate system. Parametric equations, on the other hand, define these variables in terms of one or more parameters. This approach is useful in tracing paths or curves, like circles or ellipses, which aren't easily described using standard Cartesian equations.

To convert from parametric to a rectangular form, the goal is to eliminate the parameter and create an equation solely involving the Cartesian coordinates, like \(x\) and \(y\). This often helps to identify the nature of the curve, such as straight lines or circles.

For example, given the parametric equations \(x = 2 + \sin t\) and \(y = 1 + \cos t\), we can see how the parameters relate directly to trigonometric functions. By cleverly manipulating these equations, using concepts like trigonometric identities, we can derive a single rectangular equation that encapsulates the relationship between \(x\) and \(y\).

Thus, these manipulations and conversions are crucial for understanding the shape and characteristics of complex curves, making it easier to graph and analyze them.

Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric property stating that \(\sin^2 t + \cos^2 t = 1\). This identity springs from the Pythagorean theorem applied in the context of a unit circle. It's incredibly handy when converting parametric equations to a rectangular form.

For instance, in our example, the parametric equations \(x = 2 + \sin t\) and \(y = 1 + \cos t\) need to be translated to a rectangular form. By isolating \(\sin t\) and \(\cos t\) from these equations, we have \(\sin t = x - 2\) and \(\cos t = y - 1\).

Substituting these into the Pythagorean Identity gives us \((x - 2)^2 + (y - 1)^2 = 1\). This showcases the power of the identity to merge parametric components into a recognizable geometric form—in this case, the equation of a circle.

Parametric to Cartesian Conversion

Converting a parametric equation to a Cartesian equation involves rephrasing the relationships typically linked by a parameter, into equations describing the y-axis versus the x-axis directly. This conversion helps in understanding the geometric nature of the curve.

Let's walk through the given problem where we started with \(x = 2 + \sin t\) and \(y = 1 + \cos t\). The goal is to express \(y\) directly in terms of \(x\) without involving \(t\).

By using relationships like \(\sin t\) and \(\cos t\) from our parametric to Cartesian strategy, we easily manipulated the parametric equations and substituted into a Pythagorean context. Therefore, \((x - 2)^2 + (y - 1)^2 = 1\) is the resulting Cartesian equation.

This captures the essence of the conversion: reducing complexity of trigonometric expressions into a clearer picture, a picture that, once graphed, clearly displays its circular nature centered at (2, 1) with radius 1. Understanding this conversion is key to analyzing the shape and dimensions of curves within the Cartesian plane.