Answer to Question #349118 in Complex Analysis for Busi

Question #349118

(2<45°)/((3<120°)(4<180°)) leave your answer in rectangular form


1
Expert's answer
2022-06-08T18:04:23-0400
z1=2(22+22i)=2+2iz_1=2(\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{2}}{2}i)=\sqrt{2}+\sqrt{2}i




z2=3(12+32i)=32+332iz_2=3(-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)=-\dfrac{3}{2}+\dfrac{3\sqrt{3}}{2}i




z3=4iz_3=-4i


z2z3=(32+332i)(4i)=63+6iz_2z_3=(-\dfrac{3}{2}+\dfrac{3\sqrt{3}}{2}i)(-4i)=6\sqrt{3}+6i




z1z2z3=2(1+i)6(3+i)\dfrac{z_1}{z_2z_3}=\dfrac{\sqrt{2}(1+i)}{6(\sqrt{3}+i)}


=2(1+i)(3i)24=\dfrac{\sqrt{2}(1+i)(\sqrt{3}-i)}{24}


=62i+6i+224=\dfrac{\sqrt{6}-\sqrt{2}i+\sqrt{6}i+\sqrt{2}}{24}




=6+224+6224i=\dfrac{\sqrt{6}+\sqrt{2}}{24}+\dfrac{\sqrt{6}-\sqrt{2}}{24}i

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