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 $(\alpha +\frac{\pi }{2})cos(\beta +\frac{\pi }{2})andcos(\gamma +\frac{\pi }{2})$ 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
= $\frac{(\frac{\pi }{2}+\alpha )+cos(\frac{\pi }{2}+\beta )+cos(\frac{\pi }{2}+\gamma )}{3}$
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
$=cos(\frac{\pi }{2}+\alpha )=cos(210^{\circ })=-\frac{\sqrt{3}}{2}$