Prove: Product of Sin Values = $\frac{\sqrt{n}}{2^{n-1}}$

Thread starter Dragonfall
Start date Nov 7, 2007
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Identity

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The discussion focuses on proving the identity that the product of sine values, specifically $\prod_{k=1}^{n-1}\sin\frac{k\pi}{2n}$, equals $\frac{\sqrt{n}}{2^{n-1}}$. Participants suggest starting the proof by utilizing the primitive (2n)th root of unity, denoted as $\zeta = \exp(\pi i / n)$. A proposed approach involves simplifying the expression $|1 - \zeta^k|$ for positive integers $k$. Additionally, manipulating the polynomial $x^{2n} - 1$ is recommended as part of the solution strategy. The discussion emphasizes the need for a clear mathematical approach to establish the identity.

Nov 7, 2007

Dragonfall

Homework Statement

Show that \prod_{k=1}^{n-1}\sin\frac{k\pi}{2n}=\frac{\sqrt{n}}{2^{n-1}}

The Attempt at a Solution

I have no idea where to start.

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Nov 7, 2007

morphism

Science Advisor

Homework Helper

Let \zeta be the primitive (2n)th root of unity, i.e. \zeta = \exp(\pi i / n). Try to simplify |1 - \zeta^k| (where k is some positive integer). Then try playing around with x^(2n) - 1.

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