Question

The sides of a triangle inscribed in a given circle
subtend angles $ \alpha \beta$ and $\gamma$ at the centre. The minimum
value of the arithmetic mean of
cos $ \bigg( \alpha + \frac{\pi}{2}\bigg) \, cos \, \bigg( \beta + \frac{\pi}{2}\bigg) and \, cos \, \bigg(\gamma + \frac{\pi}{2}\bigg) $ is ...

Answer 1

Since, sides of a triangle subtends $ \alpha \beta $ and $\gamma $ at the centre
$\therefore \alpha + \beta + \gamma = 2 \pi $ ...(i)
Now, arithmetic mean
= $ \frac{ \bigg( \frac{\pi}{2} + \alpha \bigg) + cos \bigg( \frac{\pi}{2} + \beta \bigg) + cos \bigg( \frac{\pi}{2} + \gamma \bigg)}{3}$
As we know that, AM $\ge$ GM, i.e.
AM is minimum, when $ \frac{ \pi}{ 2} + \alpha = \frac{ \pi}{2} + \beta = \frac{ \pi}{2} + \gamma $
or $ \alpha = \beta = \gamma = 120^\circ$
$\therefore $ Minimum value of arithmetic mean
$ = cos \bigg( \frac{\pi}{2} + \alpha \bigg) = cos \, (210^\circ) = - \frac{ \sqrt 3}{2}$