Question

已知sin(α+π/3)+sinα=-（4根号3）/5,-π/2＜α＜0,求cosα的值

Answer 1

sin(α+π/3)+sinα=1/2sina+√3/2cosa+sina
=3/2sina+√3/2cosa
=√3(√3/2sina+1/2cosa)
=√3sin(a+π/6)
=-4√3/5
sin(a+π/6)=-4/5 cos(a+π/6)=3/5
cosα=cos[(a+π/6)-π/6]
=cos(a+π/6)cosπ/6+sin(a+π/6)sinπ/6
=3/5*√3/2-4/5*1/2
=(3√3-4)/10

Answer 2

由和差化积得sin(α+π/6)=-4/5.于是：根号3/2sinα+1/2cosα=-4/5
cos(α+π/6)=3/5 根号3/2cosα-1/2sinα=3/5
消去sinα可得：cosα=(-4+3根号3)/10