1. The maximum value of the real number $k$ for which the inequality $\frac{1+\sin x}{2+\cos x} \geqslant k$ has solutions for $x$ is ( ).
(A) $-\frac{4}{3}$
(B) $-\frac{3}{4}$
(C) $\frac{3}{4}$
(D) $\frac{4}{3}$
-.1.D.
This problem essentially seeks the upper limit of the range of $f(x)=\frac{1+\sin x}{2+\cos x}$.

View $f(x)$ as the slope of the line determined by point $A(\cos x, \sin x)$ and point $B(-2, -1)$. Point $A$ moves along the unit circle, and the slope takes its extreme values when $BA$ is tangent to the circle.
. From this, it is easy to find that $f_{\max }=\frac{4}{3}$.