13. Given $\sin ^{3} \theta+\cos ^{3} \theta=\frac{11}{16}$, find the value of $\sin \theta+\cos \theta$.
13. Let $\sin \theta+\cos \theta=t$, then $\sin \theta \cos \theta=\frac{1}{2}\left(t^{2}-1\right)$, so $\sin ^{3} \theta+\cos ^{3} \theta=t\left(1-\frac{1}{2} t^{2}+\frac{1}{2}\right)=$ $\frac{11}{16}$, simplifying gives $8 t^{3}-24 t+11=0$, which is $(2 t-1)\left(4 t^{2}+2 t-11\right)=0$. Since $t \in[-\sqrt{2}, \sqrt{2}]$, then $4 t^{2}+$ $2 t-11 \neq 0$, hence $2 t-1=0, t=\frac{1}{2}$. Therefore, $\sin \theta+\cos \theta=\frac{1}{2}$.