if you haven;t got the hint i wanted to give

let me elaborate

 

(1).draw the graph of sinx from 0 to pi

(2). take 3 pts on it p,q,r as x coordinates (p+q+r=pi)

(3). where will be the centroid of the triangle formed by (p,sinp),(q,sinq),(r,sinq)

it will be on (p+q+r)/3 , (sinp+sinq+sinr)/3 inside the triangle

(4).now, there will be a point just above the centroid on the curve

so that it is (p+q+r)/3 , ( sin(p+q+r)/3 )

it will be this because the x coordinate will be same

 

now since centroid can lie only within the triangle,

y coordinate of the centroid will be always less than the pt in the step (4)

 

so we acn say,  (sinp+sinq+sinr)/3 < sin((p+q+r)/3)

since p+q+r=pi

(sinp+sinq+sinr)/3 < (root3)/2

sinp+sinq+sinr < (3/2)root3

 

correct your statement in the question

 

similarly you can do it for cosx

there will be only one change, i.e. cos(p+q+r)/3 = 1/2

 

this is the simplest and the shortest approach