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. Identify the given parametric equations: \(x = 2 + \sin t\) and \(y = 1 + \cos t\), where \(t\) ranges from \$0$ to \(2\pi\).
2. Recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). This identity will help us eliminate the parameter \(t\) to find a rectangular equation.
3. Express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\): from \(x = 2 + \sin t\), we get \(\sin t = x - 2\); from \(y = 1 + \cos t\), we get \(\cos t = y - 1\).
4. Substitute these expressions into the Pythagorean identity: \((x - 2)^2 + (y - 1)^2 = 1\).
5. Interpret this rectangular equation as a circle centered at \((2, 1)\) with radius \$1\(, which corresponds to the curve traced by the parametric equations for \)t$ in \([0, 2\pi]\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.

Rectangular Equation Conversion

Converting parametric equations to a rectangular equation involves eliminating the parameter t to find a direct relationship between x and y. This often uses trigonometric identities or algebraic manipulation to rewrite the curve in the standard Cartesian form.

Trigonometric Identities

Trigonometric identities, such as sin²t + cos²t = 1, are essential tools for eliminating parameters in parametric equations. They help relate sine and cosine terms to each other, enabling the derivation of a rectangular equation from parametric forms.