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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} $

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    @BenCrowell that's why we leave it *as a comment*, not as an *answer*. This is a collaborative website. **All relevant information are relevant,** and it will benefit future visitors of this post to see every possible relevant solutions and answers in one place, including W|A output.2012-03-13

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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.

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    @JavierBadia: You are welcome!2012-03-13
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The following is a beautiful old theorem of Gauss. Let $n$ be a positive integer. Then there is an expression for $\cos(2\pi/n)$ (and thus for $\sin(2\pi/n)$) in terms of the ordinary arithmetical operations augmented by $\sqrt{\hphantom{aa}}$ (of non-negative quantities) if and only if $n$ is of the shape $n=2^e p_1p_2\cdots p_k,$ where $e$ is a non-negative integer, and $\{p_1,p_2,\dots,p_k\}$ are distinct Fermat primes. (There might be no Fermat primes in the collection.)

The Fermat primes are the primes of the shape $2^{2^m}+1$. There are only five Fermat primes known at this time. They are $3$, $5$, $17$, $257$, and $65537$.