设△ABC的内角A B C所对边的长分别为a b c则下列命题正确 ①ab>c^2 ②若a+b>2c,则C

题目

设△ABC的内角A B C所对边的长分别为a b c则下列命题正确 ①ab>c^2 ②若a+b>2c,则C

答案

①?
②由正弦定理,sinA+sinB>2sinC,
∴2sin[(A+B)/2]cos[(A-B)/2]>4sin(C/2)cos(C/2),
sin[(A+B)/2]=cos(C/2)>0,
∴cos[(A-B)/2]>2sin(C/2),
∴sin(C/2)<1/2,0∴C/2<π/6,C<π/3.
③由a^3+b^3=c^3得0∴a^2+b^2>a^2*a/c+b^2*b/c=(a^3+b^3)/c=c^2,
∴C<π/2 .
④当a>c,b>c时(a+b)c=ac+bc此时CC<π/3.
选②、③.