Question

Find the four angles that define the fourth roots of the number z1 = 1 + sqrt 3 * i

Answer 1

\[z1 =r (\cos \theta+\iota \sin \theta) \]
\[r \cos \theta=1\]
\[r \sin \theta=\sqrt {3}\]
square and add
\[r^2=1+3=4\]
r=2
divide
\[\tan \theta=\sqrt{3}=\tan 60=\tan (180n+60)\]
\[\theta=180n+60\]
\[z1=r(\cos \theta+\iota \sin \theta)=re^{\iota \theta}\]
\[z1=2e^{\iota (180n+60)}\]
\[z1^{\frac{ 1 }{ 4 }}=2^{\frac{ 1 }{ 4}}e ^{\frac{ 180n+60 }{ 4 }}\]
\[z1^{\frac{ 1 }{ 4 }}=2^{\frac{ 1 }{ 4 }}e ^{45n+15}\]
put n=0,1,2,3
fourth roots are
\[2^{\frac{ 1 }{ 4 }}e ^{15 \iota },2^{\frac{ 1 }{ 4 }}e ^{60 \iota} ,2^{\frac{ 1 }{ 4}}e ^{150 \iota},2^{\frac{ 1 }{ 4 }}e ^{195 \iota }\]

Answer 2

correction last line
fourth roots are
\[2^{\frac{ 1 }{ 4 }}e ^{15 \iota},2^{\frac{ 1 }{ 4 }}e ^{ 60 \iota},2^{\frac{ 1 }{4 }}e ^{105 \iota },2^{\frac{ 1 }{ 4 }}e ^{150 \iota }\]