Summing Scattered Sines

Relevant wiki: Proving Trigonometric Identities - Advanced

By Euler's formula, we have that $\sin(k \alpha) = \operatorname{Im} e^{i k \alpha} = \operatorname{Im} \zeta^k$ , where $\zeta = e^{i \alpha}$ . It is evident that $\zeta$ is a root of $x^{11} - 1 = 0$ , so it is an eleventh root of unity. The value of the following Gauss sum is well known:

$g_{11} = \sum_{k=1}^{10} \left( \frac{k}{11} \right) \zeta^k = \pm i\sqrt{11}$

where the symbol in parantheses is the Legendre symbol. From Vieta's formula applied to the polynomial $x^{11} - 1$ , we also have that

$\sum_{k=1}^{10} \zeta^k = -1$

Since $1, 3, 4, 5, 9$ are the quadratic residues modulo $11$ , from this it follows that

$\zeta + \zeta^3 + \zeta^4 + \zeta^5 + \zeta^9 = -\frac{1 \pm i\sqrt{11}}{2}$

Taking imaginary parts on both sides yields that our sum is $\pm \sqrt{11}/2$ . It is easy to see by inspection that the sum is positive, therefore its value is $\sqrt{11} / 2 \approx 1.658$ .