How to evaluate $\cos(\frac{5\pi}{8})$?

Question

I'm sorry I don't know the way to input pie (3.14) don't have symbol on my pc

Answer 1

Note that $\frac58\pi=\frac12\pi+\frac18\pi$. You know $\sin\frac\pi2$ and $\cos\frac\pi2$, so once you know $\sin\frac\pi8$ and $\cos\frac\pi 8$ you can use the addition formulas. You should also have a formula to compute $\sin \frac\alpha2$ and $\cos\frac\alpha2$ if you know $\sin\alpha$ and $\cos\alpha$.

Answer 2

Recall the identity $$\cos 2\theta=2\cos^2 \theta-1.\tag{1}$$ Let $\theta=\frac{5\pi}{8}$. Then $\cos 2\theta=-\frac{1}{\sqrt{2}}$. To finish, note that $\cos(5\pi/8)$ is negative.

Answer 3

Just work out the details of the previous answer: $$\cos(\frac{5\pi}{8})=-2^{-\frac{3}{4}}\sqrt{\sqrt{2}-1}=-0.382683...$$