Question

If $x+270\left[cos\frac{2\pi }{7}+cos\frac{4\pi }{7}+cos\frac{6\pi }{7}\right]=3746$ then the value of $x$ is

Answer 1

Given $x+270[cos\frac{2\pi }{7}+cos\frac{4\pi }{7}+cos\frac{6\pi }{7}]=3746$ ...(1)
Consider, $\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}$
$=\frac{1}{2\sin \frac{2\pi }{7}}\times [\sin \frac{4\pi }{7}+\sin \frac{6\pi }{7}-\sin \frac{2\pi }{7}+\sin \frac{8\pi }{7}-\sin \frac{4\pi }{7}]$
$=\frac{1}{2\sin \frac{2\pi }{7}}[-\sin \frac{2\pi }{7}][\therefore \sin \frac{6\pi }{7}=-\sin \frac{8\pi }{7}]$
$=-\frac{1}{2}$
Substitute this value in (1)
$\Rightarrow x=3746+135=3881$.