Big-Oh $\sin$ Proof

I want to prove that $\sin(x)=O(1)$ as $x \rightarrow \infty$

I know there are different methods of solving. Are all these appropriate?

Method $1$:I want to prove that $|\sin(x)| \leq K|1|$ whenever $x>M$ where there exist constants $K,M>0$ satisfying this.

I do $|\sin(x)| \leq 1 = 1 \cdot |1| $, so there exist constants $K=1$ and $M$ can be any positive number satisfying this.

Method $2$:Observe that $\lim _{x \rightarrow \infty} |\frac{\sin{x}}{1}| = \lim _{x \rightarrow \infty} |\sin{x}| \leq 1$

As this limit is finite, then this means $\sin{x}$ is bounded by a multiple of the function $1$, so $\sin(x)=O(1)$ as $x \rightarrow \infty$.