Question

Given two complex number $z_1=5+\left(5\sqrt{3}\right)i$ and $z_2=\frac{2}{\sqrt{3}}+2i$, the argument of $\frac{z_1}{z_2}$ in degree is

• A. $0$
• B. $30$
• C. $60$
• D. $90$

Answer 1

The correct option is A $0$

$\frac{z_1}{z_2}=\frac{5+\left(5\sqrt{3}\right)i}{\frac{2}{\sqrt{3}}+2i}$
Rationalize $=\frac{(5+5\sqrt{3}i)(\frac{2}{\sqrt{3}}-2i)}{(\frac{2}{\sqrt{3}}+2i)(\frac{2}{\sqrt{3}}-2i)}$
$=\frac{\frac{10}{\sqrt{3}}-10i+10i-10\sqrt{3}{i}^2}{\frac{4}{3}-4i^2}$
$=\frac{\frac{10}{\sqrt{3}}+10\sqrt{3}}{\frac{4}{3}+4}=\frac{40}{\sqrt{3}}\times \frac{3}{16}=\frac{5\sqrt{3}}{2}$
$\frac{z_1}{z_2}$ is a real number
$\Rightarrow arg\left(\frac{z_1}{z_2}\right)=0$

Alternative solution:
$z_1=5+5\sqrt{3}i$
$are{z}_1=ta{n}^{-1}\left(\frac{5\sqrt{3}}{5}\right)={60}^o$
$z_2=\frac{2}{\sqrt{3}}+2i$
$arg{z}_2=ta{n}^{-1}\left(\frac{2}{2/\sqrt{3}}\right)=ta{n}^{-1}\left(\sqrt{3}\right)={60}^o$
$arg\left(\frac{z_1}{z_2}\right)=arg\left(z_1\right)-arg\left(z_2\right)={60}^o-{60}^o=0$