k=cos2π3−cos(π)+cos4π3−cos5π3+cos(2π)−cos7π3+⋯+cos40π3−cos41π3
=cos2π3+cos4π3+cos(2π)+⋯+cos40π3 −(cos(π)+cos5π3+cos7π3+⋯+cos41π3)
=19∑n=0cos(2π3+cos2nπ3)−19∑n=0cos(π+cos2nπ3)
=sin(80π3)cos(2π3+19π3)sin(π3)−sin(80π3)cos(π+19π3)sin(π3)
=sin(π3)cos(7π)sin(π3)−sin(π3)cos(2π3)sin(π3)
=−1−(−12)
=−12
∴|6k|=3