First slide

De-moivre's theorem

Question

The value of $S = \sum_{k = 1}^{6} (\sin \frac{2 π k}{7} - i \cos \frac{2 π k}{7})$

Moderate

A

-1
B

0
C

$- i$
D

$i$

Solution

$\begin{array}{l} \sin (\frac{2 π k}{7}) - i \cos (\frac{2 π k}{7}) \\ = - i [\cos (\frac{2 π k}{7}) + i \sin (\frac{2 π k}{7})] \end{array}$
$= - i ω^{k} [De Moivre's Theorem]$
where $ω = \cos (\frac{2 π}{7}) + i \sin (\frac{2 π}{7})$
Note that $ω^{7} = 1$
$∴$ $\begin{aligned} S = - i \sum_{k = 1}^{6} ω^{k} = \frac{- i ω (1 - ω^{6})}{1 - ω} \\ = \frac{- i (ω - ω^{7})}{1 - ω} = \frac{- i (ω - 1)}{1 - ω} = i \end{aligned}$

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