Question

已知△ABC的三个内角A,B,C成等差数列,求（sinA）^2+(sinC)^2的取值范围

Answer 1

三角形ABC的三个内角A,B,C成等差数列

则A+C=2B

因为A+B+C=180°

3B=180°

所以B=60°

A+C=120°

(sinA)^2+(sinC)^2

=(sinA+sinC)^2-2sinAsinC

={2sin[(A+C)/2]cos[(A-C)/2]}^2+cos(A+C)-cos(A-C)

=3{cos[(A-C)/2]}^2-1/2-cos(A-C)

=3[1+cos(A-C)]/2-1/2-cos(A-C)

=1+[cos(A-C)]/2

-120°