find acute angle x such that
$\displaystyle \cos x+\cos 2x+\cos3x=-0.5$
Alternatively, note that the roots of the complex equation $\displaystyle z^7-1=0$ are
$\displaystyle z\ =\ 1,\ \cos{\frac{2\pi}{7}}\pm\mathrm{i}\,\sin{\frac{2\pi }{7}},\ \cos{\frac{4\pi}{7}}\pm\mathrm{i}\,\sin{\frac{4\pi }{7}},\ \cos{\frac{6\pi}{7}}\pm\mathrm{i}\,\sin{\frac{6\pi }{7}}$
Sum the roots and equate to 0.