Why is $| \exp [i c (\cos \theta + i \sin \theta)] | \leq \exp[-c \sin \theta] $?

Question

Why is $\left| \exp [i c (\cos \theta + i \sin \theta)] \right| \leq \exp[-c \sin \theta] $, where $i$ is the imaginary unit and $c$ is a real constant?

Answer 1

Because

$\vert \exp(ic(\cos \theta + i \sin \theta)) \vert = \vert \exp (ic\cos \theta - c\sin \theta) \vert$ $= \vert \exp(-c\sin \theta) \exp (ic \cos \theta) \vert = \vert \exp (-c\sin \theta) \vert \vert \exp (ic\cos \theta) \vert$ $= \exp(-c\sin \theta) \vert \exp(ic \cos \theta) \vert = \exp(-c\sin \theta), \tag 1$

where we have used

$\vert \exp(-c \sin \theta) \vert = \exp(-c\sin \theta), \tag 2$

since for real $c$ and $\theta$

$\exp(-c\sin \theta) > 0, \tag 3$

and

$\vert \exp(ic \cos \theta) \vert = 1, \tag 4$

since

$\exp(ic \cos \theta) = \cos(c \cos \theta) + i\sin (c\cos \theta), \tag 5$

and

$\cos^2 x + \sin^2 x = 1 \tag 6$

for any real $x$.

To re-iterate and clarify, I have made liberal use of the assumption that

$c, \theta \in \Bbb R. \tag 7$