Why is it possible to represent sin(x) = x + O(x^3)
thanks for any help
$\displaystyle \sin x = x + O(x^3) \Longleftrightarrow |\sin x - x| \leq M|x^3|\,\,\,as\,\, x \rightarrow 0\,\,\mbox{ and for some constant M}$
Using now the MacClaurin series of $\displaystyle \sin x$ that HallsofIvy wrote we get $\displaystyle \sin x -x=-\frac{x^3}{6}+\frac{x^5}{120}-...$ , and
this is less than some constant times $\displaystyle x^3$ when x is close enough to zero (can you see why?)
Tonio