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Find parametric equations for the parth of a particle that moves along the circle $ x^2 + (y - 1)^2 = 4 $ in the manner described.(a) Once around clockwise, starting at $ (2, 1) $.(b) Three times around counterclockwise, starting at $ (2, 1) $.(c) Halfway around counterclockwise, starting at $ (0, 3) $.
(a) To get a clockwise orientation, we could change the equations to $x=2 \cos t, y=1-2 \sin t, 0 \leq t \leq 2 \pi$(b) To get three times around in the counterclockwise direction, we use the original equations $x=2 \cos t, y=1+2 \sin t$ withthe domain expanded to $0 \leq t \leq 6 \pi$(c) To start at (0,3) using the original equations, we must have $x_{1}=0 ;$ that is, $2 \cos t=0 .$ Hence, $t=\frac{\pi}{2} .$ So we use$x=2 \cos t, y=1+2 \sin t, \frac{\pi}{2} \leq t \leq \frac{3 \pi}{2}$Alternatively, if we want $t$ to start at $0,$ we could change the equations of the curve. For example, we could use$x=-2 \sin t, y=1+2 \cos t, 0 \leq t \leq \pi$
Calculus 2 / BC
Chapter 10
Parametric Equations and Polar Coordinates
Section 1
Curves Defined by Parametric Equations
Parametric Equations
Polar Coordinates
David C.
April 9, 2019
Hi, I was wondering about your thought process in knowing that x=cos(t) and y=2sin(t)+1?
Missouri State University
Harvey Mudd College
Idaho State University
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