$e^{j\omega}$ on unit circle
Warning: $|e^{j\omega}|$ is equal to $1$ if and only if $\omega$ is a real number. More generally, for $z$ complex,  $|e^{z}|=e^{\Re{z}}$.

This really depends in your prior knowledge, because functions like exponentials, sines or cosines, can be developed in different ways. Here, I'll assume you know about sines  and cosines, as you refer to angular frequency. Then, an historical way to define the exponential is via Euler formula, for instance: 

$$e^{j\omega} = \cos \omega+j\sin \omega\,.$$

Then, through Argand's complex plane interpretation, you can interpret a complex number as a point with  2D coordinates $(x,y) \leftrightarrow  x+jy$: the real part refers to the $X$-axis, the imaginary one to the $Y$-axis. Thus, obviously, $e^{j\omega}$ has the coordinates of a point on the $(0,0)$-center circle with radius $1$, hence the magnitude  $|e^{j\omega}|$ is equal to the radius, or  $(\cos \omega)^2+(\sin \omega)^2=1^2$.

In more modern versions, $e^z$ is defined by a series (see Proof of complex conjugate symmetry property of DFT), and $\cos$ and $\sin$ are derived for it, so they don't come first.