已知角α的顶点与直角坐标系的原点重合,始边在X轴的正半轴上,终边经过点P(-1,2),求sin(2α+2/3π )的值

Question

已知角α的顶点与直角坐标系的原点重合,始边在X轴的正半轴上,终边经过点P(-1,2),求sin(2α+2/3π )的值

Answer 1

终边经过点P(-1,2),则角α在第二象限
sinα=2/√5,cosα=-1/√5
sin(2α+2π/3)=sin(2α)cos(2π/3)+cos(2α)sin(2π/3)
=2sinαcosαcos(2π/3)+[cos²α-sin²α]sin(2π/3)
=2*(2/√5)*(-1/√5)*(-1/2)+[(-1/√5)^2-(2/√5)^2]*√3/2
=2/5-3/5*√3/2
=(4-3√3)/10