Question

find the value of Cos pi/ 14 + cos 2 pi/14 + cos 3 pi /14 +........ upto 14 terms ​

Answer 1

we have to find the value of cos(π/14) + cos(2π/14) + cos(3π/14) + cos(4π/14) + ....... + cos(14π/14)

we know, cos(π - θ) = -cosθ

so, cos(8π/14) = cos(π - 6π/14) = - cos(6π/14)

cos(9π/14) = cos(π - 5π/14) = - cos(5π/14)

cos(10π/14) = cos(π - 4π/14) = - cos(10π/14)

cos(11π/14) = cos(π - 3π/14) = - cos(3π/14)

cos(12π/14) = cos(π - 2π/14) = - cos(2π/14)

cos(13π/14) = cos(π - π/14) = -cos(π/14)

now cos(π/14) + cos(2π/14) + cos(3π/14) + cos(4π/14) + ....... + cos(14π/14)

= cos(π/14) + cos(2π/14) + cos(3π/14) + cos(4π/14) + ....+ cos(7π/14) - cos(6π/14) - cos(5π/14) .....- cos(π/14) + cos(14π/14)

= [cos(π/14) - cos(π/14)] + [cos(2π/14) - cos(2π/14) ] + [cos(3π/14) - cos(3π/14) ] + [cos(4π/14) - cos(4π/14)] + [cos(5π/14) - cos(5π/14)] + [cos(6π/14) - cos(6π/14)] + cos(7π/14) + cos(14π/14)

= 0 + 0 + 0 + 0 + 0 + 0 + cos(π/2) + cos(π)

= -1

therefore cos(π/14) + cos(2π/14) + cos(3π/14) + cos(4π/14) + ....... + upto 14 terms = -1