if α,β,γ are the angles of a triangle, find ((sin 2𝛂+sin 2𝛃+sin 2𝛄)/(sin 𝛂 sin 𝛃 sin 𝛄))=?


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Question Number 158391 by mr W last updated on 03/Nov/21

$i f α, β, γ a r e t h e a n g l e s o f a t r i a n g l e,$ $f i n d \frac{\sin 2 α + \sin 2 β + \sin 2 γ}{\sin α \sin β \sin γ} = ?$
Commented by MJS_new last updated on 03/Nov/21

$4$
	Answered by puissant last updated on 03/Nov/21

	$\frac{s i n 2 α + s i n 2 β + s i n 2 γ}{s i n α s i n β s i n γ} = \frac{4 s i n α s i n β s i n γ}{s i n α s i n β s i n γ} = 4 .$
	Answered by mr W last updated on 03/Nov/21

	$2 π - 2 γ = 2 α + 2 β$ $- \sin 2 γ = \sin 2 α \cos 2 β + \sin 2 β \cos 2 α$ $- \sin 2 γ = \sin 2 α (1 - 2 \sin^{2} β) + \sin 2 β (1 - 2 \sin^{2} α)$ $\sin 2 α + \sin 2 β + \sin 2 γ = 2 \sin 2 α \sin^{2} β + 2 \sin 2 β \sin^{2} α$ $\sin 2 α + \sin 2 β + \sin 2 γ = 4 \sin α \sin β (\cos α \sin β + \cos α \sin α)$ $\sin 2 α + \sin 2 β + \sin 2 γ = 4 \sin α \sin β \sin (α + β)$ $\sin 2 α + \sin 2 β + \sin 2 γ = 4 \sin α \sin β \sin γ$ $\Rightarrow \frac{\sin 2 α + \sin 2 β + \sin 2 γ}{\sin α \sin β \sin γ} = 4$
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