Question

If $-1+\sqrt{-3}=r{e}^{\iota \theta }$, then $\theta$ is equal

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

Answer 1

The correct option is C

$\frac{2\mathrm{\pi }}{3}$

To find the value of $\theta$:

Step 1: Calculate the value of $r$:

The given equation is $-1+\sqrt{-3}=r{e}^{\iota \theta }$

$\begin{array}{rlll}-1+\sqrt{-3}& =r{e}^{\iota \theta }& & \\ -1+\sqrt{-3}& =r(\cos \theta +\iota \sin \theta )& & \\ -1+\sqrt{3}& \iota =r(\cos \theta +\iota \sin \theta )& & \\ r\cos \theta & =-1& & \\ r\sin \theta & =\sqrt{3}& & \\ \text{Squaring and adding}& & & \\ r^2({\cos }^2\theta +{\sin }^2\theta )& =1+3& & \\ r^2& =4& & (\because {\cos }^2\theta +{\sin }^2\theta =1)\\ r& =2& & \end{array}$

Step 2: Calculate the value of $\theta$

$\begin{array}{rl}\cos \theta & =\frac{-1}{2}\\ \sin \theta & =\frac{\sqrt{3}}{2}\\ \theta liesin& secondquadrant\\ \theta & =\pi -\frac{\pi }{3}\\ & =\frac{2\pi }{3}\end{array}$

Hence option C: $\frac{2\mathrm{\pi }}{3}$ is the correct option.