Modulus of Exponential of i z where z is on Circle/Examples/6 e^pi i over 3

Examples of Use of Modulus of Exponential of i z where z is on Circle
Let $z = 6 e^{\pi i / 3}$

Then:
 * $\cmod {e^{i z} } = e^{-3 \sqrt 3}$

Proof
From Modulus of Exponential of i z where z is on Circle:


 * $(1): \quad \cmod {e^{i z} } = e^{-R \sin \theta}$

where:
 * $z = R e^{i \theta}$

Here we have:
 * $R = 6$
 * $\theta = \dfrac \pi 3$

Hence:
 * $\sin \theta = \dfrac {\sqrt 3} 2$

and the result follows by substituting for $R$ and $\sin \theta$ in $(1)$.