In Exercises 71–76, eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve. x = 3 + 2 cos t, y = 1+2 sin t; 0 ≤ t < 2π

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Identify the parametric equations given: \(x = 3 + 2 \cos t\) and \(y = 1 + 2 \sin t\), where \(0 \leq t < 2\pi\).

Recall the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\). Our goal is to eliminate the parameter \(t\) by expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\).

From the equations, isolate \(\cos t\) and \(\sin t\): \(\cos t = \frac{x - 3}{2}\) and \(\sin t = \frac{y - 1}{2}\).

Substitute these expressions into the Pythagorean identity: \(\left(\frac{x - 3}{2}\right)^2 + \left(\frac{y - 1}{2}\right)^2 = 1\).

Simplify the equation to get the Cartesian form of the curve: \(\frac{(x - 3)^2}{4} + \frac{(y - 1)^2}{4} = 1\). This represents a circle centered at \((3,1)\) with radius 2. To graph, plot this circle and use the parameter range to indicate orientation (counterclockwise as \(t\) increases from \(0\) to \(2\pi\)).

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주요 개념

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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 representation of more complex curves like circles and 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 the parameter and substituting into the other, or by using trigonometric identities to relate x and y directly.

Graphing and Orientation of Plane Curves

Graphing parametric curves requires plotting points for various parameter values and showing the direction of increasing parameter with arrows. Orientation indicates the path traced by the curve as the parameter increases, which is important for understanding motion or directionality.