If -1+-3= re ^ i , then is equal to - Vidyadip

MCQ

If $-1+\sqrt{-3}= re ^{ i \theta}$, then $\theta$ is equal to

A
$\frac{\pi}{3}$
B
$-\frac{\pi}{3}$
✓
$\frac{2 \pi}{3}$
D
$-\frac{2 \pi}{3}$

✓

Answer

Correct option: C.

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

(C)
Here, $-1+\sqrt{-3}= re ^{ i \theta}$
$\Rightarrow-1+ i \sqrt{3}= re ^{ i \theta}$
$=r \cos \theta+i r \sin \theta$
Equating real and imaginary parts, we get
$r \cos \theta=-1$ and $r \sin \theta=\sqrt{3}$
Hence, $\tan \theta=-\sqrt{3}$
$\Rightarrow \tan \theta=\tan \frac{2 \pi}{3}$
Hence, $\theta=\frac{2 \pi}{3}$

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