First slide

Trigonometric Identities

Question

The value of $\cos \frac{2 π}{7} + \cos \frac{4 π}{7} + \cos \frac{6 π}{7} + \cos \frac{7 π}{7}$ is

Moderate

A

1
B

-1
C

1/2
D

-3/2

Solution

$\cos \frac{2 π}{7} + \cos \frac{4 π}{7} + \cos \frac{6 π}{7} + \cos \frac{7 π}{7}$
$\begin{array}{l} = \frac{1}{2 \sin \frac{π}{7}} [2 \sin \frac{π}{7} \cos \frac{2 π}{7} + 2 \sin \frac{π}{7} \cos \frac{4 π}{7} + 2 \sin \frac{π}{7} \cos \frac{6 π}{7}] - 1 \\ = \frac{1}{2 \sin \frac{π}{7}} [\sin \frac{π}{7} \frac{3 π}{7} - \sin \frac{π}{7} + \sin \frac{5 π}{7} - \sin \frac{3 π}{7} + \sin \frac{7 π}{7} - \sin \frac{5 π}{7}] - 1 \end{array}$
$= \frac{1}{2 \sin \frac{π}{7}} (- \sin \frac{π}{7}) - 1 = - \frac{1}{2} - 1 = - \frac{3}{2}$ .

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