Question

Let $z$ be a complex number with modulus $2$ and argument $\frac{2\pi }{3}$ then $z$ is equal to:

• A. $-1+i\sqrt{3}$
• B. $\frac{-1+i\sqrt{3}}{2}$
• C. $-1-i\sqrt{3}$
• D. $\frac{-1-i\sqrt{3}}{2}$

Answer 1

The correct option is A

$-1+i\sqrt{3}$

Explanation for the correct option:

Finding the complex number $z$:

Given, modulus$=2$ and Argument$=\frac{2\mathrm{\pi }}{3}$

Therefore, $\theta$ lies in the second quadrant.

Now,

$$\begin{gathered} x=r\cos \theta ,y=r\sin \theta \\ \Rightarrow x=2\cos \frac{2\mathrm{\pi }}{3},y=2\sin \frac{2\mathrm{\pi }}{3} \\ \Rightarrow x=-1,y=\sqrt{3}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{cos}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{3}}\mathbf{=}\frac{\mathbf{-}\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathbf{sin}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{3}}\mathbf{=}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbf{]} \end{gathered}$$

$\begin{array}{rcl}& \Rightarrow & z=x+iy\\ & =& -1+i\sqrt{3}\end{array}$

Hence, the correct answer is option (A).