Show that sin(x) = O(x) for all x in R.

From the defn, we must show

$\displaystyle \exists A$ s.t. $\displaystyle |\sin(x)| \le A|x|$

We know that |sin(x)| <= 1.

But now I'm stuck.

Similarly, how would I go about showing:

$\displaystyle \exists A>0,c\in \mathbb{R}$ s.t.

$\displaystyle |e^{-x}|\le A|x^{-m}|, m\in\mathbb{Z}^+,~c<x<\infty$

What's the general method for these things? (I know $\displaystyle e^x$ is always positive, so the mod signs are redundant)

(The above is derived from $\displaystyle e^{-x}=O(x^{-m})$, m positive integer.)