Are there exact expressions for $\sin \frac{3\pi}{8}$ and $\cos \frac{3\pi}{8}$? 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
$$
\begin{aligned}
\cos \frac{5\pi}{8} &= - \cos \frac{3\pi}{8} \\
\sin \frac{5\pi}{8} &= \sin \frac{3\pi}{8}
\end{aligned}
$$

Yes, we can use the double angle formula
$$ 2\cos^2 x - 1 = \cos 2x$$
Pick $x = \frac{3\pi}{8}$ and you can solve it.
Once you have the cos value, you can easily get the sin value.
