If$\ \alpha,\beta,\gamma\ \in \left( 0,\frac{\pi}{2} \right),\$ then the value of $\frac{\sin{(\alpha + \beta + \gamma)}}{\sin{\alpha + \sin{\beta + \sin\gamma}}}$ is

Question

Dr. Abubacker · Accepted Answer

(a)
$\sin{(\alpha + \beta + \gamma) = \sin{\alpha\cos{\beta\cos{\gamma + \cos{\alpha\sin{\beta\cos\gamma}}}}}}\  + \cos{\alpha\cos{\beta\sin{\gamma - \sin{\alpha\sin{\beta\sin\gamma}}}}}$
$\Rightarrow \sin{(\alpha + \beta + \gamma) - \sin{\alpha - \sin{\beta - \sin\gamma}}}$
$= \sin{\alpha\ (\cos{\beta\cos{\gamma - 1) + \sin{\beta\ (\cos{\alpha\cos{\gamma - 1)}}}}}}$ $+ \sin{\gamma(\cos{\alpha\cos{\beta - 1)\sin{\alpha\sin{\beta\sin\gamma}}}}}$
$\Rightarrow \sin{(\alpha + \beta + \gamma) - \sin{\alpha - \sin{\beta - \sin\gamma}} < 0}$
${\Rightarrow sin\ (\alpha + \beta + \gamma) < \sin}{\alpha + \sin{\beta - \sin\gamma}}$
$\Rightarrow \frac{\sin{(\alpha + \beta + \gamma)}}{\sin{\alpha + \sin{\beta + \sin\gamma}}} < 1$

Q. If$\ \alpha,\beta,\gamma\ \in \left( 0,\frac{\pi}{2} \right),\$ then the value of $\frac{\sin{(\alpha + \beta + \gamma)}}{\sin{\alpha + \sin{\beta + \sin\gamma}}}$ is

Explanation

Reviewed by Dr. Abubacker

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