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Question

If α, β γ, ϵ(0,π2), then sin(α+β+γ)sin α+sin β+sin γ is


A

< 1

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B

> 1

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C

= 1

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D

None of these

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Solution

The correct option is A

< 1


We have sin α+sin β+sin γsin(α+β+γ)
=sin α+sin β+sin γsin α cos β cos γ
cos α sin β cos γcos α cos β sin γ+sin α sin β sin γ
=sin α(1cos β cos γ)+sin β(1cos α cos γ)
+sin γ(1cos α cos β)+sin α sin β sin γ > 0
sin α+sin β+sin γ>sin(α+β+γ)
sin(α+β+γ)sin α+sin β+sin γ < 1.


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