Question

lf $z=6e^{i\frac{\pi }{3}}$ then $|e^{iz}|=$

• A. $\frac{1}{e^{\sqrt{3}}}$
• B. $\frac{1}{e^{2\sqrt{3}}}$
• C. $\frac{1}{e^{3\sqrt{3}}}$
• D. $\frac{1}{e^{4\sqrt{3}}}$

Answer 1

Answer: (C)

$\frac{1}{e^{3\sqrt{3}}}$

$z=6e^{i\frac{\pi }{3}}$
$=6\left(cos\right(\frac{\pi }{3})+isin\frac{\pi }{3}))$
$=6\left(\frac{1+\sqrt{3}i}{2}\right)$
$=3(1+\sqrt{3}i)$
Hence
$iz=3(-\sqrt{3}+i)$
$e^{iz}$$=e^{3(-\sqrt{3}+i)}$
$=e^{-3\sqrt{3}}.e^{3i}$
Hence
$|e^{iz}|=e^{-3\sqrt{3}}$
$=\frac{1}{e^{3\sqrt{3}}}$