In which interval the function $f(x)=\sin x$ is uniformly continuous

Question

Suppose $f(x)=\sin x$ then how can I know in which interval it is uniformly continuous. I can solve easily by prove $|f(x_2)-f(x_1)|<ε$ for $|x_2-x_1|<\delta$ but I can't understand the same function is not uniformly continuous on other interval

Answer 1

Since $f'$ is a bounded function, $f$ is uniformly continuous on $\mathbb R$ and therefore on every interval of $\mathbb R$.

Answer 2

As $f'(x)= \cos(x)$ is bounded, $f$ is uniformly continuous on the whole of $\mathbb{R}$.