if-are-the-angles-of-a-triangle-find-sin-2-sin-2-sin-2-sin-sin-sin-

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

Facebook Tweet Pin