Answer to Question #117739 in Complex Analysis for Jason

Question

. 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

Answer 1

$z_1=1+i\sqrt{3}$

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

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

$\theta_1=\pi/3$

$z_2=2i$

$|z_2|=2$

$cos\theta_2=0, sin\theta_2=1$

$\theta_2=\pi/2$

From Argand diagram:

$arg(z_1+z_2)=\theta_1+\frac {\theta_2-\theta_1}{2}=\frac {\pi}{3}+\frac {\pi/2-\pi/3}{2}=$

$=\frac {\pi}{3}+\frac {\pi}{12}=\frac {5\pi}{12}$

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