The sides of a triangle inscribed in a given circle subtend angles α , β , γ at the center. The minimum value of the A . M . of cos α + π 2 , cos β + π 2 and cos γ + π 2 is equal to

Step-by-step solution

Let the triangle is A B C , Since the angle made by a chord at the center of a circle is twice the angle made by the chord at the circumference, we get ∠ A = α 2 , ∠ B = β 2   , ∠ C = γ 2 ⇒ α + β + γ = 2 π A . M .= 1 3 [ cos ( α + π 2  )+ cos ( β + π 2  )+ cos ( γ + π 2 )] = −   1 3   [ sinα + sinβ + sinγ ] = − 4 3 sin (   α 2   ) sin (   β 2 ) sin ( &