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We use the fact that
cos(ϴ) = -cos(180°-ϴ) or cos(ϴ)+cos(180°-ϴ) = 0
cos(1°) + cos(179°) = 0
cos(2°) + cos(178°) = 0
cos(3°) + cos(177°) = 0
...
cos(87°) + cos(93°) = 0
cos(88°) + cos(92°) = 0
cos(89°) + cos(91°) = 0
cos(181°) + cos(359°)
and
We use the fact that
cos(180°+ϴ) = -cos(360°-ϴ) or cos(180°+ϴ)+cos(360°-ϴ) = 0
cos(181°) + cos(359°) = 0
cos(182°) + cos(358°) = 0
cos(183°) + cos(357°) = 0
...
cos(267°) + cos(273°) = 0
cos(268°) + cos(272°) = 0
cos(269°) + cos(271°) = 0
So all those cosines in the sum cancel out.
Thus the only cosines in that sum that we haven't
considered are
cos(90°) = 0, cos(180°) = -1, and cos(270°) = 0
And their sum is -1. So that's the answer, -1.
Edwin
