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When equal and equi-spaced forces are summed on y-axis what is vector sum? How do we derive the formula

$$ \sum_{k=1}^{n-1}\sin\frac{\pi k}{n} = \cot \frac{\pi}{2 n} $$

( Formula given by Marco Cantarini in comments below. )

By a similar token, can

$$ \prod_{k=1}^{n-1}\sin\frac{\pi k}{n}=\frac{2n}{2^n} $$

represent some physics force multiplication situation or any generalized law in which

this analogue is valid? (Formula mentioned by Jack D'Aurizio in a recent thread

Geometric proof of $\frac{\sin{60^\circ}}{\sin{40^\circ}...}$).

Narasimham
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  • Be careful with your first formula, $\sin x \leq 1$ implies that the first sum can be at most $n$ – Alberto Debernardi Mar 02 '15 at 15:13
  • It is not correct, is a total guess,so I placed a question mark there wanting to derive it now, shall remove the 2 ok?.. – Narasimham Mar 02 '15 at 15:16
  • The identities are $$\sum_{k=1}^{n-1}\sin\left(\frac{\pi k}{n}\right)=\cot\left(\frac{\pi}{2n}\right)$$ $$\prod_{k=1}^{n-1}\sin\left(\frac{\pi k}{n}\right)=\frac{2n}{2^{n}} $$ so there is something wrong. – Marco Cantarini Mar 02 '15 at 15:22
  • @MarcoCantarini: Thanks. I corrected the Sum formula as you have given. – Narasimham Mar 02 '15 at 16:02
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    @Narasimham, See http://math.stackexchange.com/questions/17966/how-can-we-sum-up-sin-and-cos-series-when-the-angles-are-in-arithmetic-pro and http://math.stackexchange.com/questions/8385/prove-that-prod-k-1n-1-sin-frack-pin-fracn2n-1 – lab bhattacharjee Mar 03 '15 at 01:26
  • @labbhattacharjee Thanks for bringing to notice..I know about sum formula of sine of angles in AP, it is is even listed in Loney's book.The statics principle of addition of force components along y-axis leads to this formula. However, is there existence no such physics principle about vector absolute value products? I am amazed that the derivation stemming from Euler's identity $ e^{i \theta } $ has no corresponding physics originating principle attached to it. – Narasimham Mar 03 '15 at 01:52

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With some preliminary manipulations, both the identities can be derived by regarding

$$\zeta_k = \sin\frac{\pi k}{n}$$ as roots of a suitable Chebyshev polynomial, then applying Vieta's formulas - relations between the roots and the coefficients of a monic polynomial.

Jack D'Aurizio
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  • Does the product formula have a statics/optics/communication origin or application? – Narasimham Mar 02 '15 at 17:43
  • @Narasimham: honestly I don't know, but I know for sure that the product formula is involved in the broken stick problem, i.e. on the expected value of the length of a stick broken in $n$ random points. – Jack D'Aurizio Mar 02 '15 at 18:19