Question

已知sinC+cosC=1-sinC/2求sinC的值

Answer 1

sinC+cosC=1-sinC/2,移项得 sinC-sinC/2 = 1-cosC

由二倍角公式得 2sinC/2 cosC/2-sinC/2 = 2(sinC/2)^2
因为sinC/2≠0,所以两边消去sinC/2得 2cosC/2-1 = 2sinC/2
整理得 sinC/2-cosC/2=1/2
根据辅助角公式得sin(C/2-π/4)=√2 /4
再由二倍角公式得cos(C-π/2)=1-2sin(C/2-π/4)^2=3/4
∴sinC=cos(C-π/2)=3/4

Answer 2

令x=C/2
则sin2x+cos2x=1-sinx
2sinxcosx+1-2sin²x=1-sinx
sinx(2sinx-2cosx-1)=0
sinx[2√2sin(x-π/4)-1]=0
sinx=0或sin(x-π/4)=√2/4

sinx则sin2x=2sinxcosx=0

sin(x-π/4)=√2/4
则cos(2x-π/2)=1-2sin²(x-π/4)=3/4
sin2x=3/4

所以sinC=sin2x=0或3/4