4unit is just too complex (1 Viewer)

LOL™ · Mar 23, 2010

Sorry if this question has been posted cause i couldnt one.

how do i do this question 4 of Ex 4J from patel

Show that the roots of (z-1)^6 + (z+1)^6 = 0 are +/-icot(<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDavid%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]π/12) and +/-icot(5[/FONT]<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDavid%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]π/12), +/-i ?

Expanding those binomials and adding would give z^6 + 15z^4 + 15z^2 +1
Then dividing by z^3 would give
(z^3 + 1/z^3) + 15(z + 1/z) = 0
Then using z^n + 1/z^n = 2cosn[/FONT]<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDavid%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>θ

So then the eq becomes
2cos3<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDavid%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>θ + 30cos<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDavid%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>θ = 0

(4cos^3<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDavid%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>θ - 3cos<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CDavid%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>θ) + 15cosθ = 0
cosθ(cos^2θ + 3)

And now im lost cause i definitely did something wrong.... And theres a test tommorow....

[FONT=&quot][/FONT]

Affinity · Mar 23, 2010

you can just substitute and verfy for this q.
if you want to solve it.. there are other ways:

alternate method:

(z+1)^6 = (-1)*(z-1)^6

taking 6ths roots we get,

z+1 = w*(z-1)
where w is a 6th root of -1
so z = (1+w)/(1-w)
let w' be the conjugate of w and w = cos(a) + isin(a)

z = [(1+w)(1-w')]/[(1-w)(1-w')]
= [1 + w - w' - 1] / [1 + 1 - w - w']
= [ 2isin(a)]/[2(1-cos(a)]
= i*[sin(a)/(1-cos(a)]
= i*[2sin(a/2)cos(a/2)] / [1-[(cos(a/2))^2 - (sin(a/2))^2] ]
= i *[2sin(a/2)cos(a/2)] / [2(cos(a/2))^2]
= i*cot(a/2)
then just substitute the values of a in there (pi/6 +n*pi/3) n = 0,1,2,3,4,5

LOL™ · Mar 23, 2010

thanks for that. Never thought of it that way before.

Affinity · Mar 23, 2010

your method is also ok. but "Then using z^n + 1/z^n = 2cosnθ" is wrong, it only works for z on the unit circle.

(z^3 + 1/z^3) + 15(z + 1/z) = 0
(z + 1/z)^3 + 12(z + 1/z) = 0
(z + 1/z)((z + 1/z)^2 + 12) = 0
(z + 1/z)((z + 1/z) - 2isqrt(3))((z + 1/z) + 2isqrt(3)) = 0
(z^2 + 1)(z^2 - 2isqrt(3)z + 1)(z^2 + 2isqrt(3)z + 1)

solving the quadratics:
z = i, -i, i(2 - sqrt(3)), i(-2 - sqrt(3)), i(2 + sqrt(3)), i(-2 + sqrt(3)),

note:
cot(A - B)
= cos(A-B)/sin(A-B)
= cos(A)cos(B)+sin(A)sin(B) / sin(A)cos(B) - cos(A)sin(B)
= cot(A)cot(B) + 1 / [cot(B) - cot(A)]
so cot(pi/12)
= cot(pi/3 - pi/4)
= [1 + 1/sqrt(3)]/[1- 1/sqrt(3)]
= (sqrt(3) + 1) / (sqrt(3) - 1)
= 2 + sqrt(3)
and cot(5pi/12) = tan(pi/12) = 1/cot(pi/12) = 2-sqrt(3)
etc.

4unit is just too complex (1 Viewer)

LOL™

I invented "lol"

Affinity

Active Member

LOL™

I invented "lol"

Affinity

Active Member

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