Answer
$[0,\infty)$
Work Step by Step
$\textbf{F}(t)+\textbf{G}(t)$ is defined when both $\textbf{F}(t)$ and $\textbf{G}(t)$ are defined. Thus, the domain of $\textbf{F}(t)+\textbf{G}(t)$ is the intersection of the domains of $\textbf{F}(t)$ and $\textbf{G}(t)$. For the domain of $\textbf{F}(t)$, trigonometric functions have domain $(-\infty,\infty)$ and $\sqrt{x}$ has domain $[0,\infty)$, so the domain of $\textbf{F}(t)$ is the intersection, or $[0,\infty)$. For the domain of $\textbf{G}(t)$, trigonometric functions have domain $(-\infty,\infty)$, so the domain of $\textbf{G}(t)$ is the intersection, or $(-\infty,\infty)$. Thus, the domain of $\textbf{F}(t)+\textbf{G}(t)$ is the intersection of $(-\infty,\infty)$ and $[0,\infty)$ which is equal to $[0,\infty)$.
