Question

If \(-1+\sqrt{-3} = re^{iθ}\), then the value of \(θ\) is

• A. \(-\frac{2π}{3}\)
• B. \(\frac{π}{3}\)
• C. \(-\frac{π}{3}\)
• D. \(\frac{2π}{3}\)

Hint:

Write $-1+i\sqrt3$ in polar form and find its argument.

Answer 1

The Correct Option is D

Step 1: Factor out 2
$z=2\left(-\frac12+\frac{\sqrt3}{2}i\right)=2\left(\cos120^{\circ}+i\sin120^{\circ}\right)$.

Step 2: Read off
So $re^{i\theta}$ with $r=2$ and $\theta=120^{\circ}=\frac{2\pi}{3}$. Option (D).

Final Answer:
$\theta=\frac{2\pi}{3}$, option (D). \[ \boxed{\text{(D)}} \]