Buktikan bahwa cosA+cosB+cosC selalu positif pada segitiga ABC.
LHS = cosA+cosB+cosC..
= ( cos A + cos B ) + cos C
= { 2 · cos[ ( A+B) / 2 ] · cos [ ( AB) / 2 ] } + cos C
= { 2 · cos [ (π/2) – (C/2) ] · cos [ (AB) / 2 ] } + cos C
= { 2 · sin( C/2 ) · cos [ (AB) / 2 ] } + { 1 – 2 · sin² ( C/2 ) }
= 1 + 2 sin ( C/2 )· { cos [ (A -B) / 2 ] – sin ( C/2 ) }
= 1 + 2 sin ( C/2 )· { cos [ (AB) / 2 ] – sin [ (π/2) – ( (A+B)/2 ) ] }
= 1 + 2 sin ( C/2 )· { cos [ (AB) / 2 ] – cos [ (A+B)/ 2 ] }
= 1 + 2 sin ( C/2 )· 2 sin ( A/2 )· sin( B/2 ) … … … (2)
= 1 + 4 sin(A/2) sin(B/2) sin(C/2)
Oleh karena itu positif.
10