Question

在△ABC中,a,b,c依次是∠A,B,C所对的边,且4sinB.sin^2(π/4+B/2)+cos2B=1+根号3

Answer 1

(1)sin^2(π/4+B/2)=-(1/2)[1-2sin^2(π/4+B/2)]+1/2=-(1/2)cos(π/2+B)+1/2=(1/2)sinB+1/2

4sinB.sin^2(π/4+B/2)+cos2B

=2(sinB)^2+2sinB+cos2B

=2(sinB)^2+2sinB+1-2(sinB)^2

=2sinB+1=√3+1

sinB=√3/2.B=π/3或B=2π/3.

(2)B=π/3,sinC=sinπ/6=1/2.C=π/6或C=5π/6(舍去).

A=π/2,c=asinC=4*(1/2)=2