prove-Cos-2pi-7-Cos-4pi-7-Cos-8pi-7-1-2-

Question Number 67464 by lalitchand last updated on 27/Aug/19

prove Cos (\frac{2 π}{7}) + Cos (\frac{4 π}{7}) + Cos (\frac{8 π}{7}) = - \frac{1}{2}

Answered by mind is power last updated on 27/Aug/19

Z^{7} - 1 = 0

\Rightarrow (z - 1) (1 + z + z^{2} + z 3 + z^{4} + z^{5} + z^{6}) = 0

z_{k} = e^{i \frac{2 k π}{7}} 0 ⩽ k ⩽ 6

1 + z + z^{2} + z^{3} + z^{4} + z^{5} + z^{6} = 0 \Leftrightarrow 1 ⩽ k ⩽ 6

\sum_{k ⩽ 6} e^{i \frac{2 k π}{7}} = - 1

\Rightarrow c o s (\frac{2 π}{7}) + c o s (\frac{4 π}{7}) + c o s (\frac{6 π}{7}) + c o s (\frac{8 π}{7}) + c o s (\frac{10 π}{7}) + c o s (\frac{12 π}{7}) = - 1

c o s (\frac{8 π}{7}) = c o s (\frac{14 π - 6 π}{7}) = c o s (2 π - \frac{6 π}{7}) = c o s (- \frac{6 π}{7}) = c o s (\frac{6 π}{7})

s a m e i d e a g i v e u s

c o s (\frac{2 π}{7}) = c o s (\frac{12 π}{7})

c o s (\frac{4 π}{7}) = c o s (\frac{10 π}{7})

\Rightarrow c o s (\frac{2 π}{7}) + c o s (\frac{4 π}{7}) + c o s (\frac{6 π}{7}) + c o s (\frac{8 π}{7}) + c o s (\frac{10 π}{7}) + c o s (\frac{12 π}{7}) =

2 c o s (\frac{2 π}{7}) + 2 c o s (\frac{4 π}{7}) + 2 c o s (\frac{8 π}{7}) = - 1

c o s (\frac{2 π}{7}) + c o s (\frac{4 π}{7}) + c o s (\frac{8 π}{7}) = - \frac{1}{2}

Commented by Kunal12588 last updated on 27/Aug/19

j u s t A M A Z I N G \dots

Commented by Kunal12588 last updated on 27/Aug/19

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