Page 1 of 1
O inegalitate trigonometrica intr-un triunghi
Posted: Sun Jun 15, 2008 2:26 pm
by BogdanCNFB
Fie ABC un triunghi ascutitunghic. Aratati ca:
\( \sin A+\sin B>\cos A+\cos B+\cos C \).
Posted: Sun Jun 15, 2008 6:05 pm
by Marius Mainea
Relatia din enunt se scrie \( 2t^2-2t\cos\frac{A-B}{2}+2\sin\frac{C}{2}\cos\frac{A-B}{2}-1<0 \) unde \( t=\cos\frac{C}{2} \). Avem \( \Delta=4(\cos^2\frac{A-B}{2}-4\sin\frac{C}{2}\cos\frac{A-B}{2}+2)>0 \) si se arata ca \( x_1<\cos\frac{C}{2}<x_2 \) de unde rezulta concluzia.