Of sum of cosines and the $7$th roots of unity In my solution here, it was shown that 
$$\omega+\omega^2+\omega^4=-\frac 12\pm\frac{\sqrt7}2\qquad\qquad (\omega=e^{i2\pi/7})$$
from which we know that 
$$\sin \frac{2\pi}7+\sin \frac{4\pi}7-\sin \frac{6\pi}7=\Im (\omega+\omega^2+\omega^4)=\frac{\sqrt7}2$$
This can also be verified easily by computation. 
By the same token it would appear that 
$$\cos \frac{2\pi}7+\cos \frac{4\pi}7-\cos \frac{6\pi}7=\Re (\omega+\omega^2+\omega^4)=\frac 12$$
However a quick computational check shows that the result is $1.3019...$ and not $\frac 12$.
Why is this so?

In that answer there is the following equality
$$\begin{align}
\sin\frac{2\pi}7+\sin\frac{4\pi}7-\sin\frac{6\pi}7
&=\Im(\omega^1+\omega^2-\omega^3)\\
&=\Im(\omega^1+\omega^2+\omega^4)\\
&=\Im(S)\\
&=\dfrac{\sqrt{7}}{2}\\
\end{align}.$$
For the real part you have
$$\begin{align}
-\dfrac{1}{2}&= 
\Re(S)\\
&=\Re(\omega^1+\omega^2+\omega^4)\\
&=\Re(\omega^1+\omega^2+\omega^3)\\
&=\cos\frac{2\pi}7+\cos\frac{4\pi}7+\cos\frac{6\pi}7
\end{align}$$
