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

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2 Answers 2

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.

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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.

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