First slide

Compound angles

Question

If $α, β, γ \in (0, \frac{π}{2})$ then the value of $\frac{\sin (α + β + γ)}{\sin α + \sin β + \sin γ}$ is

Moderate

A

<1
B

>1
C

0
D

None of these

Solution

$\begin{array}{l} \sin (α + β + γ) = \sin αcos βcos γ + \cos αsin βcos γ + \cos αcos βsin γ - \sin αsin βsin γ \\ \Rightarrow \sin (α + β + γ) - \sin α - \sin β - \sin γ \\ = \sin α (\cos βcos γ - 1) + \sin β (\cos αcos γ - 1) + \sin γ (\cos αcos β - 1) - \sin αsin βsin γ \\ \Rightarrow \sin (α + β + γ) - \sin α - \sin β - \sin γ < 0 \\ \Rightarrow \frac{\sin (α + β + γ) < \sin α + \sin β + \sin γ}{\sin (α + β + γ)} < 1 \end{array}$

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