I was just wondering if there is any way to get an exact expression (with radicals) for $\sin \frac{3\pi}{8}$ and $\cos \frac{3\pi}{8}$. In case it's relevant, I want to express $z = \sqrt[4]{8} e^{\frac{5\pi}{8}i}$ in binomial form, and I know that
$ \begin{aligned} \cos \frac{5\pi}{8} &= - \cos \frac{3\pi}{8} \\ \sin \frac{5\pi}{8} &= \sin \frac{3\pi}{8} \end{aligned} $