if cos (β−γ)+cos( γ−α)+cos (α−β)=−(3/2) so proof it, Σcos α=0,Σsin α=0


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Question Number 20646 by oyshi last updated on 30/Aug/17

$i f \cos (β - γ) + \cos (γ - α) + \cos (α - β) = - \frac{3}{2}$ $s o p r o o f i t,$ $Σ \cos α = 0, Σ \sin α = 0$
	Answered by $@ty@m last updated on 31/Aug/17
	$Given, cos (β−γ)+cos( γ−α)+cos (α−β)=−(3/2) ⇒2cos (β−γ)+2cos( γ−α)+2cos (α−β)=−3 ⇒Σ(2cosβcosγ+2sinβsinγ) +3=0 ⇒Σ(2cosβcosγ+2sinβsinγ) +Σ(sin^2 α+cos^2 α)=0 ⇒{Σcos^2 α+Σ2cosβcosγ}+{Σsin^2 α+Σ2sinβsinγ}=0 ⇒(Σcos α)^2 +(Σsin α)^2 =0 ⇒Σcos α=0,Σsin α=0$
	$G i v e n,$ $\cos (β - γ) + \cos (γ - α) + \cos (α - β) = - \frac{3}{2}$ $\Rightarrow 2 \cos (β - γ) + 2 \cos (γ - α) + 2 \cos (α - β) = - 3$ $\Rightarrow Σ (2 c o s β c o s γ + 2 s i n β s i n γ) + 3 = 0$ $\Rightarrow Σ (2 c o s β c o s γ + 2 s i n β s i n γ) + Σ ({s i n}^{2} α + {c o s}^{2} α) = 0$ $\Rightarrow {Σ {c o s}^{2} α + Σ 2 c o s β c o s γ} + {Σ {s i n}^{2} α + Σ 2 s i n β s i n γ} = 0$ $\Rightarrow {(Σ \cos α)}^{2} + {(Σ \sin α)}^{2} = 0$ $\Rightarrow Σ \cos α = 0, Σ \sin α = 0$
	Commented by ajfour last updated on 31/Aug/17

	$t h a n k s i m m e n s e l y .$
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