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.