Intro
I am studying the mean from a random process $x(t)$ where $\theta$ is a random variable with pdf = $1/2\pi$
$x(t) = \cos(\omega t + \theta);$
and I go as this:
$E\{x(t)\} = E\{\cos(\omega t + \theta)\}$ which is $\int_{-A}^{A} \cos(\omega t + \theta) * 1/2\pi\, d\theta.$
So far, so good.
The problem is after doing the integration I end up with this:
$\sin(\omega t + \theta)$ where $\theta$ [-A, A] where A must be [A < $\pi$]
and I can not get to the correct result of:
$2 \cos[t \omega] \sin[A].$
(The result is from doing the integration, the $1/2\pi$ is outside the integration waiting)
My attempt was to use this Euler equation:
$\sin(a + b) = \sin(a)\cos(b) + \sin(a)\cos(b)$
The Question
Could you please tell me how to get to that result or what I am missing or need to know?
Thanks