14. Given that the sum of the sines of any 5 interior angles and the cosines of the remaining 4 interior angles of a convex nonagon equals a constant value $\lambda$. If one of the nine interior angles is $120^{\circ}$, find the value of the constant $\lambda$.
Let $\lambda=\sin x_{1}+\sin x_{2}+\sin x_{3}+\sin x_{4}+\sin x_{5}+\cos y_{1}+\cos y_{2}+\cos y_{3}+\cos y_{4}$,
According to the problem, if we only swap $x_{1}, y_{1}$, we also have
$$
\lambda=\sin y_{1}+\sin x_{2}+\sin x_{3}+\sin x_{4}+\sin x_{5}+\cos x_{1}+\cos y_{2}+\cos y_{3}+\cos y_{4},
$$

Thus, $\sin x_{1}+\cos y_{1}=\sin y_{1}+\cos x_{1} \Rightarrow \sin x_{1}-\cos x_{1}=\sin y_{1}-\cos y_{1}$
$$
\Rightarrow \sin \left(x_{1}-45^{\circ}\right)=\sin \left(y_{1}-45^{\circ}\right) \Rightarrow x_{1}=y_{1} \text { or }\left(x_{1}-45^{\circ}\right)+\left(y_{1}-45^{\circ}\right)=180^{\circ} \Rightarrow x_{1}+y_{1}=270^{\circ} \text {, }
$$

This indicates that the angles that can be replaced with $120^{\circ}$ are only $120^{\circ}$ and $150^{\circ}$.
Let there be $x$ angles of $120^{\circ}$ and $y$ angles of $150^{\circ}$ in the convex nonagon, then we have
$$
\left\{\begin{array} { l } 
{ 1 2 0 x + 1 5 0 y = 1 2 6 0 , } \\
{ x + y = 9 }
\end{array} \Rightarrow \left\{\begin{array} { l } 
{ 4 x + 5 y = 4 2 , } \\
{ x + y = 9 }
\end{array} \Rightarrow \left\{\begin{array}{l}
x=3, \\
y=6 .
\end{array}\right.\right.\right.
$$

Therefore, $\lambda=5 \cdot \frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{3}{2}=1-\frac{\sqrt{3}}{2}$.