Question

If $\alpha ,\beta \gamma ,ϵ(0,\frac{\pi }{2}),then\frac{sin(\alpha +\beta +\gamma )}{sin\alpha +sin\beta +sin\gamma }is$

• A. < 1
• B. > 1
• C. = 1
• D. None of these

Answer 1

The correct option is A

< 1

We have $sin\alpha +sin\beta +sin\gamma -sin(\alpha +\beta +\gamma )$
$=sin\alpha +sin\beta +sin\gamma -sin\alpha cos\beta cos\gamma$
$-cos\alpha sin\beta cos\gamma -cos\alpha cos\beta sin\gamma +sin\alpha sin\beta sin\gamma$
$=sin\alpha (1-cos\beta cos\gamma )+sin\beta (1-cos\alpha cos\gamma )$
$+sin\gamma (1-cos\alpha cos\beta )+sin\alpha sin\beta sin\gamma$ > 0
$\therefore sin\alpha +sin\beta +sin\gamma >sin(\alpha +\beta +\gamma )$
$\Rightarrow \frac{sin(\alpha +\beta +\gamma )}{sin\alpha +sin\beta +sin\gamma }$ < 1.