How to show if $x(t)=e^{\sin(t)}$ it is periodic or not?

Question

$$x(t)=e^{\sin(t)}$$

I have an idea about the way to show this is periodic, that is show $x(t+T) = x(t)$ and $T$ here is the period, but it is hard me to show $x(t+T) = e^{\sin(t)}$

Answer 1

Notice that $\sin(t)$ is periodic, and that is the only term dependent on $t$. So what happens when $T$ is equal to the period of $\sin(t)$?

Answer 2

For all $t \in \mathcal{R}$,

$$ x(t+2\pi) = e^{\sin(t+2\pi)} = e^{\sin(t)} = x(t) $$

Therefore, this function is periodic with period $T = 2\pi$.