Decoding the LaTeX that didn't render, we seek sum of the angles of the seventh roots of
[tex] - \frac 1 {\sqrt 2} - \frac i {\sqrt{2}} [/tex]
That's on the unit circle, 45 degrees into the third quadrant, aka 225 degrees.
The seventh roots will all be separated by 360/7, around 51 degrees. The first seventh root has
[tex] \theta_1 = 225^\circ / 7[/tex]
That's around 32 degrees.
The next angle is
[tex] \theta_2 = \frac{1}{7}( 225^\circ + 360^\circ)[/tex]
The next one is
[tex] \theta_3 = \frac{1}{7}( 225^\circ + 720^\circ)[/tex]
and in general
[tex] \theta_n = \frac{1}{7}( 225^\circ + (n-1)360^\circ) = \frac 1 7(-135^\circ + 360^\circ n)[/tex]
[tex]S = \displaystyle \sum_{n=1}^7 \theta_n = \frac{1}{7} \sum_{n=1}^7 -135^\circ + \frac{360}{7} \sum_{n=1}^7 n[/tex]
The first sum is just -135° since it's one seventh of the sum of seven -135s.
We have 1+2+3+4+5+6+7 = (1+7)+(2+6)+(3+5) + 4 = 28 so
[tex] S = -135^\circ + 360^\circ (\frac{28} 7) = 4(360)-135 = 1305^\circ[/tex]
If I didn't screw it up, that means the answer is
Answer: 1305°
