Question #117739
. If z1 = 1 + i

3 and z2 = 2i, find |z1| and arg z1, |z2| and arg z2. Using an Argand
diagram, deduce that arg(z1 + z2) = 5π/12. Hence show that tan(5π/12) = 2 + √
3
1
Expert's answer
2020-05-26T18:17:27-0400

z1=1+i3z_1=1+i\sqrt{3}

z1=1+3=2|z_1|=\sqrt{1+3}=2

cosθ1=1/2,sinθ1=3/2cos\theta_1=1/2, sin\theta_1=\sqrt{3}/2

θ1=π/3\theta_1=\pi/3


z2=2iz_2=2i

z2=2|z_2|=2

cosθ2=0,sinθ2=1cos\theta_2=0, sin\theta_2=1

θ2=π/2\theta_2=\pi/2


From Argand diagram:

arg(z1+z2)=θ1+θ2θ12=π3+π/2π/32=arg(z_1+z_2)=\theta_1+\frac {\theta_2-\theta_1}{2}=\frac {\pi}{3}+\frac {\pi/2-\pi/3}{2}=

=π3+π12=5π12=\frac {\pi}{3}+\frac {\pi}{12}=\frac {5\pi}{12}


tan(5π/12)=tan(arg(z1+z2))=3+21=3+2tan(5\pi/12)=tan(arg(z_1+z_2))=\frac {\sqrt{3}+2}{1}=\sqrt{3}+2


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