The amplitude of frac{1 + isqrt{3}}{sqrt{3} + | vedclass

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Question

(A) Let $z = \frac{1 + i\sqrt{3}}{\sqrt{3} + 1}$.Since the real part is $\frac{1}{\sqrt{3} + 1}$ and the imaginary part is $\frac{\sqrt{3}}{\sqrt{3} +

Vedclass · Accepted Answer

$\frac{\pi}{3}$

Vedclass · Answer

(A) Let $z = \frac{1 + i\sqrt{3}}{\sqrt{3} + 1}$.Since the real part is $\frac{1}{\sqrt{3} + 1}$ and the imaginary part is $\frac{\sqrt{3}}{\sqrt{3} + 1}$,both are positive.Thus,$z$ lies in the first quadrant.The argument (amplitude) is given by $	heta = 	an^{-1}\left(\frac{	ext{Im}(z)}{	ext{Re}(z)}ight)$.$	heta = 	an^{-1}\left(\frac{\sqrt{3}/(\sqrt{3} + 1)}{1/(\sqrt{3} + 1)}ight) = 	an^{-1}(\sqrt{3})$.Therefore,$	heta = \frac{\pi}{3}$.

The amplitude of $\frac{1 + i\sqrt{3}}{\sqrt{3} + 1}$ is

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