In the book "Sea Loads on Ships and Offshore Structures" by Faltinsen, page 60 there is an identity:
$$ \sin(\omega t - kR \cos(\theta)) \cos(\theta) d\theta $$
They claim that under long wavelengths (much larger than the radius $R$, so $kR$ is small), that we can approximate the above by:
$$ \cos(\theta) d\theta \sin(\omega t) - kR \cos^2(\theta) d\theta \cos(\omega t) $$
I see how - if you set $kR$ zero - you get:
$$ \sin(\omega t)\cos(\theta) d\theta $$
And if you then set $\omega t$ zero you get (via the sine near zero approximation):
$$\lim_{x\rightarrow0}\sin(x) = x$$
$$ \Rightarrow \sin(-kR \cos(\theta)) \cos(\theta) d\theta = -kR \cos^2(\theta) d\theta $$
But I'm confused as to what makes it valid to take compose these approximations together: what is it that makes them linear? I can picture a unit circle with the sine function, but won't the linear approximation be invalid near angles of $n\pi$?
$$f(x+h)\approx f(x)+f'(x)h.$$
In your case $f(x)=\sin(x)$, $x=\omega t$ and $h=-kR\cos(\omega)$.