TITLE: Absolute value of complex exponential
QUESTION [26 upvotes]: Can somebody explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook says.) 
For example:
$$|e^{-2i}|=1, i=\sqrt {-1}$$

REPLY [2 votes]: Definition of absolute value:
$$\left|a+i\ b\right|=\sqrt{a^2+b^2}$$
Euler's Formula:
$$e^{i\theta}=cos\ \theta + i\sin\ \theta$$
Trig Identity:
$$cos^{2} \theta + sin^{2} \theta = 1$$
Steps:
$$ \left|e^{-i2}\right| $$
$$\theta=-2$$
$$\left|e^{i\theta}\right|=\sqrt{\cos^2\ \theta + \sin^2\ \theta}$$
$$\left|e^{i\theta}\right|=\sqrt1$$
$$\left|e^{i\theta}\right|=1$$
$$ \left|e^{-i2}\right| = 1 $$