In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.


x = 2 sin t, y = 2 cos t; 0 ≤ t < 2π

Verified step by step guidance

1. Identify the given parametric equations: \(x = 2 \sin t\) and \(y = 2 \cos t\), with the parameter \(t\) in the interval \(0 \leq t < 2\pi\).
2. Recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). This identity will help us eliminate the parameter \(t\) by expressing \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\).
3. Express \(\sin t\) and \(\cos t\) from the parametric equations: \(\sin t = \frac{x}{2}\) and \(\cos t = \frac{y}{2}\).
4. Substitute these expressions into the Pythagorean identity to get the rectangular equation: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1\).
5. Simplify the equation to the standard form of a circle: \(\frac{x^2}{4} + \frac{y^2}{4} = 1\). This represents a circle centered at the origin with radius 2. To sketch the curve, draw this circle and use the parameter interval to determine the orientation, noting that as \(t\) increases from \(0\) to \(2\pi\), the point moves clockwise because \(x = 2 \sin t\) and \(y = 2 \cos t\).

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. Understanding how to manipulate these equations is key to analyzing the curve's shape and behavior.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to form a single equation in x and y, removing t. This is done by expressing t from one equation and substituting into the other or using trigonometric identities. This step converts the parametric form into a rectangular (Cartesian) equation, simplifying graphing and analysis.

Orientation and Sketching of Parametric Curves

Orientation refers to the direction in which the curve is traced as the parameter t increases. When sketching, arrows indicate this direction, helping to understand the curve's dynamic behavior. Recognizing the interval of t and how x and y change with t is essential for accurate graphing and interpretation.