Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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.

share|cite|improve this 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.

And if k is not an integer, then this question is just sort of asking how to evaluate cos(x)? With a Taylor series, I suppose.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.