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(a) Eliminate the parameter to find a Cartesian equation of the curve.

(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.

$ x = \sin\dfrac{1}{2}\theta $, $ \quad y = \cos\dfrac{1}{2}\theta $, $ \quad -\pi \leqslant \theta \leqslant \pi $

(a) (a) $x=\sin \frac{1}{2} \theta, y=\cos \frac{1}{2} \theta,-\pi \leq \theta \leq \pi$

$x^{2}+y^{2}=\sin ^{2} \frac{1}{2} \theta+\cos ^{2} \frac{1}{2} \theta=1 .$ For $-\pi \leq \theta \leq 0,$ we have

$-1 \leq x \leq 0$ and $0 \leq y \leq 1 .$ For $0<\theta \leq \pi,$ we have $0<x \leq 1$

and $1>y \geq 0 .$ The graph is a semicircle.

(b) Graph

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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