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. Step 1: Recognize that the given parametric equations x = 2 \cos t and y = 2 \sin t represent a circle in the parametric form. The parameter t varies from 0 to 2\pi, which is a full rotation around the circle.
2. Step 2: To graph the plane curve, plot points for various values of t within the interval [0, 2\pi]. For example, calculate (x, y) for t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, and 2\pi. These points will help you sketch the circle.
3. Step 3: Notice that the parametric equations are similar to the standard form of a circle x = r \cos t and y = r \sin t, where r is the radius. Here, r = 2, indicating a circle with radius 2 centered at the origin.
4. Step 4: To find the rectangular equation, use the Pythagorean identity \cos^2 t + \sin^2 t = 1. Substitute x = 2 \cos t and y = 2 \sin t into this identity to eliminate the parameter t.
5. Step 5: Solve for the rectangular equation by substituting: \left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1, which simplifies to x^2 + y^2 = 4. This is the equation of a circle with radius 2 centered at the origin.

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 the parameter t, allowing for the representation of curves that may not be easily described by a single equation. Understanding how to manipulate and interpret these equations is crucial for graphing the curve.

Graphing Parametric Curves

Graphing parametric curves involves plotting points defined by the parametric equations over a specified range of the parameter. For the given equations x = 2 cos t and y = 2 sin t, as t varies from 0 to 2π, the points trace out a circle. Familiarity with the unit circle and the behavior of sine and cosine functions aids in visualizing the resulting graph.

Rectangular Equation

A rectangular equation eliminates the parameter by expressing the relationship between x and y directly. For the given parametric equations, using the identities cos²(t) + sin²(t) = 1 allows us to derive the rectangular equation x² + y² = 4. This transformation is essential for understanding the geometric properties of the curve in Cartesian coordinates.