Question

The principal argument of $-2\sqrt{3}-2i$ is

• A. $-\frac{5\mathrm{\pi }}{6}$
• B. $\frac{5\mathrm{\pi }}{6}$
• C. $-\frac{2\mathrm{\pi }}{3}$
• D. $\frac{2\mathrm{\pi }}{3}$

Answer 1

The correct option is A

$-\frac{5\mathrm{\pi }}{6}$

Explanation for the correct option

The general form of complex number is written as: $z=r(\cos \theta +i\sin \theta )$ where $r$ is modulus and $\theta$ is its argument.

Given complex number $=-2\sqrt{3}-2i$

Let $r\cos \theta =-2\sqrt{3}...\left(i\right)$

$r\sin \theta =-2...\left(ii\right)$

Squaring and adding equation (i) and (ii)

$r^2({\cos }^2\theta +{\sin }^2\theta )={(-2\sqrt{3})}^2+2^2$

$\Rightarrow$ $r^2=16\left(\because {\cos }^2\theta +{\sin }^2\theta =1\right)$

$\Rightarrow$ $r=4$

Put $r$ in $r\cos \theta =-2\sqrt{3}$

$$\begin{gathered} \Rightarrow \cos \theta =\frac{-\sqrt{3}}{2} \\ and\sin \theta =\frac{-1}{2} \end{gathered}$$

$\theta$ lies in $3^{rd}$ quadrant

$$\begin{gathered} \theta =-\left(\mathrm{\pi }-\frac{\mathrm{\pi }}{6}\right) \\ =-\frac{5\mathrm{\pi }}{6} \end{gathered}$$

Hence the correct option is option(A)