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 cos t , y = 2 sin t

Verified step by step guidance

1. Identify the given parametric equations: \(x = 2 \cos t\) and \(y = 2 \sin t\), where \(t\) ranges from \$0$ to \(2\pi\).
2. Recall the Pythagorean identity: \(\cos^2 t + \sin^2 t = 1\). This identity will help us eliminate the parameter \(t\) to find a rectangular equation.
3. Express \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\): from \(x = 2 \cos t\), we get \(\cos t = \frac{x}{2}\); from \(y = 2 \sin t\), we get \(\sin t = \frac{y}{2}\).
4. Substitute these expressions into the Pythagorean identity: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1\).
5. Simplify the equation to get the rectangular form: \(\frac{x^2}{4} + \frac{y^2}{4} = 1\), which represents the equation of the curve in the \(xy\)-plane.

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.

Graphing Parametric Curves

To graph parametric curves, plot points (x(t), y(t)) for values of t within the given interval. This approach helps visualize the path traced by the parameter, revealing shapes like circles, ellipses, or other plane curves.

Converting Parametric to Rectangular Equations

Converting parametric equations to a rectangular form involves eliminating the parameter t to find a direct relationship between x and y. For trigonometric parametrics like x = 2 cos t and y = 2 sin t, using identities such as cos²t + sin²t = 1 helps derive the rectangular equation.