Question

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

Accepted Answer

Identify the given parametric equations: $$x = 2 \sin t$$ and $$y = 2 \cos t$$, with the parameter $$t in $$the interval $$0 \leq t \u003c 2\pi. $$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 $$y. $$Express $$\sin t$$ and $$\cos t$$ from the parametric equations: $$\sin t = \frac{x}{2}$$ and $$\cos t = \frac{y}{2}. $$Substitute these expressions into the Pythagorean identity to get the rectangular equation: $$\left(\frac{x}{2}\$$right)^2$$ + \left(\frac{y}{2}\$$right)^2$$ = 1. $$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 $$y = 2 \cos t.$$

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Concetti chiave

Parametric Equations

Eliminating the Parameter

Orientation and Sketching of Parametric Curves