Sum of Sines of Arithmetic Sequence of Angles/Formulation 2

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Theorem

Let $\alpha \in \R$ be a real number such that $\alpha \ne 2 \pi k$ for $k \in \Z$.

Then:

\(\ds \sum_{k \mathop = 1}^n \map \sin {\theta + k \alpha}\) \(=\) \(\ds \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb\)
\(\ds \) \(=\) \(\ds \map \sin {\theta + \frac {n + 1} 2 \alpha}\frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }\)


Proof

From Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 2:

$\ds \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$

It is noted that, from Sine of Multiple of Pi, when $\alpha = 2 \pi k$ for $k \in \Z$, $\map \sin {\alpha / 2} = 0$ and the right hand side is not defined.


From Euler's Formula, this can be expressed as:

$\ds \sum_{k \mathop = 1}^n \paren {\map \cos {\theta + k \alpha} + i \map \sin {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$


Equating imaginary parts:

$\ds \sum_{k \mathop = 1}^n \map \sin {\theta + k \alpha} = \map \sin {\theta + \frac {n + 1} 2 \alpha} \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$

$\blacksquare$


Sources