28. Given that $\mathbf{m}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j}$ and $\mathbf{n}=(\sqrt{2}-\sin \theta) \mathbf{i}+(\cos \theta) \mathbf{j}$, where $\mathbf{i}$ and $\mathbf{j}$ are the usual unit vectors along the $x$-axis and the $y$-axis respectively, and $\theta \in(\pi 2 \pi)$. If the length or magnitude of the vector $\mathbf{m}+\mathbf{n}$ is given by $\mathbf{m}+\mathbf{n}=\frac{8 \sqrt{2}}{5}$, find the value of $5 \cos \left(\frac{\theta}{2}+\frac{\pi}{8}\right)+5$.
28. Answer: 1
$$
\begin{aligned}
\frac{8 \sqrt{2}}{5}=\mathbf{m}+\mathbf{n} & =\sqrt{4+2 \sqrt{2} \cos \theta-2 \sqrt{2} \sin \theta} \\
& =2 \sqrt{1+\cos \left(\theta+\frac{\pi}{4}\right)} \\
& =2 \sqrt{2}\left|\cos \left(\frac{\theta}{2}+\frac{\pi}{8}\right)\right|
\end{aligned}
$$

Since $\pi<\theta<2 \pi$, we have $\frac{5}{3} \pi<\frac{\theta}{2}+\frac{\pi}{8}<\frac{9}{8} \pi$. Thus, $\cos \left(\frac{\theta}{2}+\frac{\pi}{8}\right)<0$. This implies that $\cos \left(\frac{\theta}{2}+\frac{\pi}{8}\right)=-\frac{4}{5}$ and hence
$$
5 \cos \left(\frac{\theta}{2}+\frac{\pi}{8}\right)+5=1
$$