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: Write the number
\(\sqrt{-3}=i\sqrt3\), so the number is \(z=-1+i\sqrt3\). Its real part is \(-1\) and imaginary part is \(\sqrt3\).

Step 2: Find modulus
\(r=\sqrt{1+3}=2\).

Step 3: Find the argument
\(\cos\theta=\frac{-1}{2}\) and \(\sin\theta=\frac{\sqrt3}{2}\). Both conditions hold for the second quadrant angle \(\theta=\frac{2\pi}{3}\).
The reference angle is \(\tan^{-1}\sqrt3=\frac{\pi}{3}\), and in the second quadrant \(\theta=\pi-\frac{\pi}{3}=\frac{2\pi}{3}\).

Step 4: Why not the others
\(-\frac{2\pi}{3}\) is in the third quadrant, \(\frac{\pi}{3}\) is in the first quadrant and \(-\frac{\pi}{3}\) is in the fourth. None of those has a negative real part with a positive imaginary part, except \(\frac{2\pi}{3}\).

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