This is more a formulation question. I need help making a sales pitch (lol). I am working on an practical engineering problem where I encounter functions of the form:
$\cos(\phi + d_\phi)$, $ \tan(\phi+d_{\phi}) $ etc.
where $\phi$ is a deterministic variable taking values to the open interval $(-\pi/2, \pi/2)$ and $d_{\phi}$ is a zero mean random variable, possibly Gaussian, but not necessarily.
Under what assumptions, or how should I phrase/justify the following:
$E[\cos(\phi + d_{\phi})] \approx E[\cos(\phi)]$
$E[\tan(\phi + d_{\phi})] \approx E[\tan(\phi)]$
$E[\sec(\phi + d_{\phi})] \approx E[\sec(\phi)]$
where $E$ represents the expectation operator. I did plot the above using Matlab and qualitatively the above approximations look OK. I am looking for some form analytical justification. Please feel free to make any assumptions in your response. For example, I started with the most obvious and conservative assumption on the noise magnitude:
$|d_{\phi}|<c$ where $c$ is some small positive constant.
Any other suggestion? Much obliged.
Use trig identities such as $\cos (A +B) = \cos A \cos B-\sin A\sin B$
Then, as they are independent: $\mathsf E(\cos (A +B)) = \mathsf E(\cos A)\mathsf E(\cos B)-\mathsf E(\sin A)\mathsf E(\sin B)$
Thus if $B$ is zero mean and symmetrically distributed, you have $\mathsf E(\sin B)= 0$ (as Sine is an odd function).
You must also show that for your particular distribution $\mathsf E(\cos B) \approx 1$ (which is not generally true).