Question

The sides of a triangle inscribed in a given circle subtends angles `alpha,beta,gamma` at the centre.Then, the minimum value of the arithmetic mean of `cos(alpha+pi/2), cos(beta+pi/2), cos(gamma+pi/2)` is

Answer 1

Correct Answer - `-(sqrt(3))/(2)`
Since, sides of a triangle subteands `alpha, beta, gamma` at the centra.

`:." "alpha+beta+gamma=2pi" ...(i)"`
Now, arithmetic mean
`=(cos(pi/2+alpha)+cos(pi/2+beta)+cos(pi/2+gamma))/(3)`
As we know that, `AMgeGM,` i.e.
AM is minimum, when `pi/2alpha=pi/2+beta=pi/2+gamma`
`"or "alpha=beta=gamma=120^(@)`
`:.` Minimum value of arithmetic mean
`=cos(pi/2+alpha)=cos(210^(@))=sqrt3/2`