Why is $\sin(t)$ a stationary process?
$\sin(t)$ is no random process, because there's nothing random about it. You could add a random amplitude to get a random process:

$$x(t)=A\sin(t)\tag{1}$$

This is a random process because $A$ is a random variable. However, $x(t)$ is not stationary, but it is cyclostationary, i.e., its statistical properties vary periodically. You can make the process $x(t)$ stationary by adding a random phase:

$$\tilde{x}(t)=A\sin(t+\phi)\tag{2}$$

The phase $\phi\in [0,2\pi]$ is a uniformly distributed random variable that is independent of $A$. It can be shown that the statistical properties of $\tilde{x}(t)$ given by $(2)$ are independent of $t$, and hence, the process is stationary.