# How does one compute $\cos((\pi/4)(k-1 ))$?

How to compute $\cos \left( \frac{\pi}{4}(k-1) \right)$ ?

-

Think about it; $\pi/4$ is $45°$. With each different value of $k$ you're summing up $45°$. Since $\pi/4$ divides $2\pi$, you'll eventually get a cycle of values for $\cos(k(\pi/4))$, and then you can use this to derive an answer.
It's probably worthwhile pointing out that we're assuming that $k$ is an integer. This wasn't stated in the original post, but it isn't a very meaningful question otherwise. – cch May 27 '11 at 20:04
As a more difficult extension of fmartin's answer, we might consider by using the angle difference formula for cos: $\cos(\alpha \pm \beta) = \cos (\alpha ) \cos (\beta ) \mp \sin (\alpha ) \sin (\beta )$, which can be derived using complex numbers or matrices very quickly.
Then we consider $\cos (k \pi /4 - \pi /4) = \cos (k \pi/4) /\sqrt2 + \sin (k \pi /4)/\sqrt2$. Of course, we would then proceed by fmartin's answer, noting the cyclic nature for different integer values of k.