Question

What would the small angle approximation of [sin(a/2)cos(x)-cos(a/2)sin(x)] be given a is not a small angle and x is. More specifically, the solution given to me is -cos(a/2)x which I don't understand how it could be.

Thanks What would the small angle approximation of [sin(a/2)cos(x)-cos(a/2)sin(x)] be given a is not a small angle and x is. More specifically, the solution given to me is -cos(a/2)x which I don't understand how it could be.

Thanks @Physics

Answer 1

For small x, cos x ~ 1 and sin x ~ x. More formally,
\[ \lim_{x \rightarrow 0} \cos x = 1 \ \ \ \ \hbox{and} \ \ \ \ \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1\]
Hence
sin(a/2)cos(x)-cos(a/2)sin(x) ~ sin a/2 . 1 - cos a/2 . x = sin a/2 - x.cos a/2

So like you perhaps, I'm surprised the answer you cite doesn't have a sin a/2 term.