Question

If $\alpha ,\beta ϵ[0,\pi ];x=\sin \frac{\alpha +\beta }{2},y=\frac{\sin \alpha +\sin \beta }{2}$ then

• A. $x>y$
• B. $x\ge y$
• C. $x<y$
• D. $x\le y$

Answer 1

The correct option is B $x\ge y$

$x=\sin \left(\frac{\alpha +\beta }{2}\right)$

$y=\frac{\sin \alpha +\sin \beta }{2}=\sin \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)$

$y=x\cos \left(\frac{\alpha -\beta }{2}\right)$

Now as

$\alpha ,\beta \in [0,\pi ]\Rightarrow \frac{\alpha }{2},\frac{\beta }{2}\in [0,\frac{\pi }{2}]\Rightarrow \cos \frac{\alpha }{2}\ge 0,\cos \frac{\beta }{2}\ge 0,\sin \frac{\alpha }{2}\ge 0,\sin \frac{\beta }{2}\ge 0$

$\Rightarrow \cos \frac{\alpha }{2}\cos \frac{\beta }{2}+\sin \frac{\alpha }{2}\sin \frac{\beta }{2}\ge 0\Rightarrow \cos (\frac{\alpha }{2}-\frac{\beta }{2})\ge 0$

but $\cos (\frac{\alpha }{2}-\frac{\beta }{2})$ lies in between $0$ and $1$

$\Rightarrow x\ge y$