Question

If $z_1 = \sqrt{3} - i$ and $z_2 = 1 + i \sqrt{3}$, then amp $(z_1 + z_2) = $

• A. $\frac{\pi}{12}$
• B. $\frac{\pi}{15}$
• C. $\frac{\pi}{6}$
• D. $\frac{\pi}{4}$

Answer 1

Answer: (A)

We have , $z_1 = \sqrt{3} - i$ and $z_2 = 1 + i \sqrt{3}$
So, $z_{1} +z_{2} =\left(\sqrt{3} +1\right)+ i\left(\sqrt{3} -1\right)$
Putting $ r \cos \theta =\sqrt{3} +1$ and $r \sin \theta =\sqrt{3} -1$
$ \tan \theta =\frac{\sqrt{3} - 1}{3+1} \times\frac{\sqrt{3} -1}{\sqrt{3} - 1}$
$= \frac{\left(\sqrt{3} -1\right)^{2}}{3-1} = \frac{3+1-2\sqrt{3}}{2}$
$ \tan \theta =\frac{4-2\sqrt{3}}{2}= 2-\sqrt{3}$
$ \tan \theta =\tan \frac{\pi}{12}$
amplitude $ \left(\theta\right) = \frac{\pi}{12}$