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 = 1 + 3 cos t, y = 2 + 3 sin t; 0 ≤ t < 2π

Verified step by step guidance

1. Identify the parametric equations given: \(x = 1 + 3 \cos t\) and \(y = 2 + 3 \sin t\), with \(0 \leq t < 2\pi\).
2. Recall the Pythagorean identity: \(\cos^2 t + \sin^2 t = 1\). This will help eliminate the parameter \(t\) by expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\).
3. Rewrite \(\cos t\) and \(\sin t\) from the parametric equations: \(\cos t = \frac{x - 1}{3}\) and \(\sin t = \frac{y - 2}{3}\).
4. Substitute these expressions into the Pythagorean identity to get the rectangular equation: \(\left(\frac{x - 1}{3}\right)^2 + \left(\frac{y - 2}{3}\right)^2 = 1\).
5. Recognize that this equation represents a circle centered at \((1, 2)\) with radius \(3\). To sketch the curve, draw this circle and use the parameter interval \(0 \leq t < 2\pi\) to determine the orientation, which corresponds to increasing \(t\) moving counterclockwise starting from the point \((4, 2)\).

Key Concepts

Parametric Equations

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

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to form a single equation in x and y. This is done by solving one equation for t or using trigonometric identities, enabling the curve to be expressed in rectangular form for easier analysis and graphing.

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 motion along the curve and the behavior of the parameter over its interval.