Graph each plane curve defined by the parametric equations for t in [0, 2π] Then find a rectangular equation for the plane curve. See Example 3.


x = 2 + sin t , y = 1 + cos t

Verified step by step guidance

1. Step 1: Understand the parametric equations given: \(x = 2 + \sin t\) and \(y = 1 + \cos t\). These equations describe a curve in the plane as the parameter \(t\) varies from 0 to \(2\pi\).
2. Step 2: Recognize that \(\sin t\) and \(\cos t\) are trigonometric functions that describe a circle when combined. The standard form of a circle in parametric equations is \(x = a + r\cos t\) and \(y = b + r\sin t\).
3. Step 3: To find the rectangular equation, express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\). From the given equations, \(\sin t = x - 2\) and \(\cos t = y - 1\).
4. Step 4: Use the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\) to eliminate the parameter \(t\). Substitute \(\sin t = x - 2\) and \(\cos t = y - 1\) into this identity.
5. Step 5: Simplify the equation \((x - 2)^2 + (y - 1)^2 = 1\) to find the rectangular equation of the curve. This represents a circle centered at \((2, 1)\) with a radius of 1.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically denoted as 't'. In this case, x and y are defined in terms of 't', allowing for the representation of curves that may not be easily described by a single equation. Understanding how to manipulate and graph these equations is essential for visualizing the curve they represent.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate system based on the values derived from the parametric equations. For the given equations, one would calculate x and y for various values of 't' within the specified range [0, 2π] to create a visual representation of the curve. Familiarity with graphing tools and software can enhance this process, making it easier to visualize complex curves.

Rectangular Equation

A rectangular equation eliminates the parameter 't' to express the relationship between x and y directly. This is often achieved by solving one of the parametric equations for 't' and substituting it into the other. The resulting equation provides a more traditional representation of the curve, which can be useful for further analysis and understanding of its properties.