$\begin{aligned} & \text {If } z=r e^{i \theta}=r(\cos \theta+i \sin \theta) \\ & \Rightarrow i z=i r(\cos \theta+i \sin \theta)=-r \sin \theta+i r \cos \theta \\ & \text {or } e^{i z}=e^{(-r \sin \theta+i r \cos \theta)}=e^{-r \sin \theta} e^{r i \cos \theta} \\ & \text {or }\left|e^{i z}\right|=e^{-r \sin \theta} \| e^{r i \cos \theta}\left|=e^{-r \sin \theta}\right| e^{i r \cos \theta} \mid \\ & =e^{-r \sin \theta}\left[\left\{\cos ^2(r \cos \theta)+\sin ^2(r \cos \theta)\right\}\right]^{1 / 2}=e^{-r \sin \theta}\end{aligned}$