1. Let $a, b, c$ be real numbers, then for any real number $x$, the inequality
$$
a \sin x + b \cos x + c > 0
$$

always holds if and only if
(A) $a, b$ are both 0, and $c > 0$;
(B) $\sqrt{a^{2}+b^{2}} = c$;
(C) $\sqrt{a^{2}+b^{2}} < c$.
1. (C)

When $a, b$ are not both zero,
$$
\begin{aligned}
& a \sin x + b \cos x + c > 0 \\
\Leftrightarrow & \sqrt{a^{2} + b^{2}} \sin (x + \varphi) + c > 0 \\
\Leftrightarrow & \sin (x + \varphi) > -\frac{c}{\sqrt{a^{2} + b^{2}}}
\end{aligned}
$$

And the necessary and sufficient condition for (2) to hold is $-\frac{c}{\sqrt{a^{2} + b^{2}}}$ $\sqrt{a^{2} + b^{2}}$, which is included in (C). Therefore, the answer is (C).