Yahoo Answers: Answers and Comments for Can someone find theta when r is maximized? [Mathematics]
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From Anonymous
enUS
Mon, 21 May 2012 16:00:09 +0000
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Yahoo Answers: Answers and Comments for Can someone find theta when r is maximized? [Mathematics]
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https://answers.yahoo.com/question/index?qid=20120521160009AAVzTB8
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From Jánošík: 1)
r = 4∙sin(θ)
r = 0 for
4∙sin(θ) = 0
sin(θ)...
https://answers.yahoo.com/question/index?qid=20120521160009AAVzTB8
https://answers.yahoo.com/question/index?qid=20120521160009AAVzTB8
Tue, 22 May 2012 09:56:54 +0000
1)
r = 4∙sin(θ)
r = 0 for
4∙sin(θ) = 0
sin(θ) = 0
θ = 0 or θ = π
r' = 4∙cos(θ)
r' = 0 for
4∙cos(θ) = 0
cos(θ) = 0
θ = π/2 or θ = 3π/2
4∙sin(π/2) = 4∙1 = 4
4∙sin(3π/2) = 4∙(1) = 4
so rmax at θ = π/2
Graph:
http://i48.tinypic.com/2ccmbrd.jpg
2)
r = 3 + 2∙cos(θ)
r = 0 for
3 + 2∙cos(θ) = 0
2∙cos(θ) = 3
cos(θ) = 3/2 < not possible (cos can not be less than 1), so r=0 never happens
r' = 2∙sin(θ)
r' = 0 for
2∙sin(θ) = 0
sin(θ) = 0
θ = 0 or θ = π
3 + 2∙cos(0) = 5
3 + 2∙cos(π) = 5
so rmax at θ = 0
Graph:
http://i48.tinypic.com/doauci.jpg
3)
r = 10∙(1  sin(θ))
r = 10  10∙sin(θ)
r = 0 for
10  10∙sin(θ) = 0
10∙sin(θ) = 10
sin(θ) = 1
θ = π/2
r' = 10∙cos(θ)
r' = 0 for
10∙cos(θ) = 0
cos(θ) = 0
θ = π/2 or θ = 3π/2
r = 10  10sin(π/2) = 0
r = 10  10sin(3π/2) = 20
so rmax at θ = 3π/2
Graph:
http://i46.tinypic.com/2qt9zdl.jpg
4)
r = 6 + 12∙sin(θ)
r = 0 for
6 + 12∙sin(θ) = 0
12∙sin(θ) = 6
sin(θ) = 1/2
θ = 5π/4 or θ = 7π/4
r' = 12∙cos(θ)
r' = 0 for
12∙cos(θ) = 0
cos(θ) = 0
θ = π/2 or θ = 3π/2
r = 6 + 12∙sin(π/2) = 18
r = 6 + 12∙sin(3π/2) = 6
so rmax at θ = π/2
Graph:
http://i49.tinypic.com/2aep1eo.jpg
5)
r = 4cos(3θ)
r = 0 for
4∙cos(3θ) = 0
cos(3θ) = 0
3θ = π/2 or 3θ = 3π/2
θ = π/6 or θ = π/2 or θ = 5π/6 or θ = 7π/6 or θ = 3π/2 or θ = 11π/6
r' = 12∙sin(3θ)
r' = 0 for
12∙sin(3θ) = 0
sin(3θ) = 0
3θ = 0 or 3θ = π
θ = 0 or θ = π/3 or θ = 2π/3 or θ = π or θ = 4π/3 or θ = 5π/3
r = 4cos(3∙0) = 4
r = 4cos(3∙π/3) = 4
r = 4cos(3∙2π/3) = 4
r = 4cos(3∙π) = 4
r = 4cos(3∙4π/3) = 4
r = 4cos(3∙5π/3) = 4
so rmax at θ = 0 or θ = 2π/3 or θ = 4π/3
Graph:
http://i48.tinypic.com/2my116g.jpg
6)
r = 5∙sin(2θ)
r = 0 for
5∙sin(2θ) = 0
sin(2θ) = 0
2θ = 0 or 2θ = π
θ = 0 or θ = π/2 or θ = π or θ = 3π/2
r' = 10∙cos(2θ)
r' = 0 for
10∙cos(2θ) = 0
cos(2θ) = 0
2θ = π/2 or 2θ = 3π/2
θ = π/4 or θ = 3π/4 or θ = 5π/4 or θ = 7π/4 or
r = 5∙sin(2π/4) = 5
r = 5∙sin(2∙3π/4) = 5
r = 5∙sin(2∙5π/4) = 5
r = 5∙sin(2∙7π/4) = 5
so rmax at θ = π/4 or θ = 5π/4
Graph:
http://i50.tinypic.com/30w5mh4.jpg