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Find all cubic roots of z=1+i

 Dec 22, 2015

Best Answer 

 #2
avatar+118696 
+10

Hi Gibsonj338   laugh

 

Find all cubic roots of

 

 z=1+i distance of z to the origin =12+12=2cubic toots so there are 3 of them so they are 2π3radiansapart.z=2(cis3π4)3z=62(cis(3π4÷3)),62(cis((3π4+2π)÷3)),62(cis((3π4+4π)÷3))3z=62(cis(3π4÷3)),62(cis(11π4÷3)),62(cis(19π4÷3))3z=62(cis(3π12)),62(cis(11π12)),62(cis(19π12))3z=62(cis(π4)),62(cis(11π12)),62(cis(19π12))or 

 

3z=1+i32,62(cos(11π12)+isin(11π12)),62(cos(19π12)+isin(19π12))3z=1+i32,62(cos(π12)+isin(π12)),62(cos(5π12)isin(5π12))

 

 Dec 23, 2015
edited by Melody  Dec 23, 2015
 #1
avatar+1904 
+5

z=(1+i)(1/3)

 

r=((1)2+12)

 

r=(1+12)

 

r=(1+1)

 

r=(2)

 

tan(Θ)=1/1

 

tan(Θ)=1

 

Θ=tan1(1)

 

Θ=π/4

 

(2)e(π/4i)(1/3)

 

(2)(1/3)e(π/12i)

 

(2)(1/3)(cos(π/12)+isin(π/12))

 

(2)(1/3)(0.965925826289+isin(π/12))

 

(2)(1/3)(0.965925826289+i0.258819045103)

 

 1.414213562373095)(1/3)(0.965925826289+i0.258819045103)

 

 1.122462048309373(0.965925826289+i0.258819045103)

 

 1.084215081491274550528506797+i0.290514555507789375384650419

 

 1.084215081491274550528506797+0.290514555507789375384650419i

 

Answer 1: 1.0842150814912745505285067970.290514555507789375384650419i

 

(2)e(7π/4i)(1/3)

 

(2)(1/3)e(7π/12i)

 

(2)(1/3)(cos(7π/12)+isin(7π/12))

 

(2)(1/3)(0.258819045103+isin(7π/12))

 

(2)(1/3)(0.258819045103+i0.965925826289)

 

 1.414213562373095)(1/3)(0.258819045103+i0.965925826289)

 

 1.122462048309373(0.258819045103+i0.965925826289)

 

 0.290514555507789375384650419+i1.084215081491274550528506797

 

Answer 2: 0.290514555507789375384650419+1.084215081491274550528506797i

 

(2)e(15π/4i)(1/3)

 

(2)(1/3)e(15π/12i)

 

(2)(1/3)(cos(15π/12)+isin(15π/12))

 

(2)(1/3)(0.707106781187+isin(15π/12))

 

(2)(1/3)(0.707106781187+i0.707106781187)

 

 1.414213562373095)(1/3)(0.707106781187+i0.707106781187)

 

 1.122462048309373(0.793700525984607637192165751+i0.793700525984607637192165751)

 

 0.793700525984607637192165751+i0.793700525984607637192165751

 

 0.793700525984607637192165751+0.793700525984607637192165751i

 

Answer 3: 0.7937005259846076371921657510.793700525984607637192165751i

 Dec 22, 2015
edited by gibsonj338  Dec 22, 2015
 #2
avatar+118696 
+10
Best Answer

Hi Gibsonj338   laugh

 

Find all cubic roots of

 

 z=1+i distance of z to the origin =12+12=2cubic toots so there are 3 of them so they are 2π3radiansapart.z=2(cis3π4)3z=62(cis(3π4÷3)),62(cis((3π4+2π)÷3)),62(cis((3π4+4π)÷3))3z=62(cis(3π4÷3)),62(cis(11π4÷3)),62(cis(19π4÷3))3z=62(cis(3π12)),62(cis(11π12)),62(cis(19π12))3z=62(cis(π4)),62(cis(11π12)),62(cis(19π12))or 

 

3z=1+i32,62(cos(11π12)+isin(11π12)),62(cos(19π12)+isin(19π12))3z=1+i32,62(cos(π12)+isin(π12)),62(cos(5π12)isin(5π12))

 

Melody Dec 23, 2015
edited by Melody  Dec 23, 2015

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