# How do you evaluate this cos((2pi)/7)*cos((3pi)/7)*cos((6pi)/7) ?

## Please, explain me EVERYTHING, I am pretty awful at trigonometry

Mar 22, 2017

$\cos \left(\frac{2 \pi}{7}\right) \cdot \cos \left(\frac{3 \pi}{7}\right) \cdot \cos \left(\frac{6 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{2 \pi}{7}\right)} \cdot 4 \cdot 2 \sin \left(\frac{2 \pi}{7}\right) \cos \left(\frac{2 \pi}{7}\right) \cdot \cos \left(\frac{3 \pi}{7}\right) \cdot \cos \left(\frac{6 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{2 \pi}{7}\right)} \cdot 2 \cdot 2 \sin \left(\frac{4 \pi}{7}\right) \cdot \cos \left(\frac{3 \pi}{7}\right) \cdot \cos \left(\frac{6 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{2 \pi}{7}\right)} \cdot 2 \cdot 2 \sin \left(\pi - \frac{3 \pi}{7}\right) \cdot \cos \left(\frac{3 \pi}{7}\right) \cdot \cos \left(\frac{6 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{2 \pi}{7}\right)} \cdot 2 \cdot 2 \sin \left(\frac{3 \pi}{7}\right) \cdot \cos \left(\frac{3 \pi}{7}\right) \cdot \cos \left(\frac{6 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{2 \pi}{7}\right)} \cdot 2 \sin \left(\frac{6 \pi}{7}\right) \cdot \cos \left(\frac{6 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{2 \pi}{7}\right)} \sin \left(\frac{12 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{2 \pi}{7}\right)} \sin \left(2 \pi - \frac{2 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{2 \pi}{7}\right)} \left(- \sin \left(\frac{2 \pi}{7}\right)\right)$

$= - \frac{1}{8}$