Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 10 - Parametric Equations and Polar Coordinates - 10.1 Exercises - Page 665: 11

Answer

a) $y= cos(sin^{-1}x)$ or $x^{2}+y^{2}=1$, $y \geq 0$ b) (Graph) As θ increases, x increases and y increases to 1, then decreases to 0.
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Work Step by Step

a) $x= sin(\frac{1}{2}θ)$ $y= cos(\frac{1}{2}θ)$ $-π \leq θ \leq π$ $x= sin(\frac{1}{2}θ)$ $2sin^{-1}x=θ$ $y= cos(\frac{1}{2}θ)$ $y= cos(\frac{1}{2}2sin^{-1}x)$ $y= cos(sin^{-1}x)$ which is the same as, $x^{2}+y^{2}=1$, $y \geq 0$ b) When, θ=-π x=-1 y=0 θ=-π/2 $x=-(\sqrt 2)/2$ $y=(\sqrt 2)/2$ θ=0 x=0 y=1 θ=π/2 $x=(\sqrt 2)/2$ $y=(\sqrt 2)/2$ θ=π x=1 y=0 From this we can infer that as θ increases, x increases and y increases to 1, then decreases to 0.
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