Is it possible to express $\sin(\frac{k\pi}{4})$ as something of the form $(-1)^{f(k)}$?

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For example $\cos(k\pi)=(-1)^k$ similarily, $\sin(\frac{k\pi}{2})=(-1)^\frac{k-1}{2}$ when k is odd, and 0 otherwise. Is there a similar representation for $\sin(\frac{k\pi}{4})$