Question

Let $z_{1}and{z}_2$be two complex numbers with alpha and beta as their principal arguments such that $\alpha +\beta >\mathrm{\pi }$, then principal $arg\left(z_1{z}_2\right)$ is given by

• A. $\alpha +\beta +\pi$
• B. $\alpha +\beta -\pi$
• C. $\alpha +\beta -2\pi$
• D. $\alpha +\beta$

Answer 1

The correct option is C

$\alpha +\beta -2\pi$

Explanation for correct option:

The given information are as follows:

$\begin{array}{rcl}arg{z}_1& =& \alpha \\ arg{z}_2& =& \beta \\ \alpha +\beta & >& \mathrm{\pi }\end{array}$

Use the concept of the argument of a complex number:

$\begin{array}{rcl}-\mathrm{\pi }& <& \arg z\le \mathrm{\pi }\\ a\mathrm{rg}\left({\mathrm{z}}_1{\mathrm{z}}_2\right)& =& a\mathrm{rg}{z}_1+a\mathrm{rg}{z}_2\\ & =& \mathrm{\alpha }+\mathrm{\beta }-2\mathrm{\pi }\end{array}$

Hence, the principal argument is $\alpha +\beta -2\mathrm{\pi }$

Therefore option (C) is the correct answer.