size(7.5cm,0);
import geometry;

pair O=(0,0), pI=(1,0), pJ=(0,1);
dot("$O$",O,SW);
draw(unitcircle);

real angle1=63, angle2=130;
pair pM=dir(angle1), pN=dir(angle2);
draw(pJ--O--pI--pM--O--pN--pM);

draw("$\alpha$",arc(pI,O,pM,0.15),blue,Arrow);
draw("$\beta$",arc(pI,O,pN,0.3),blue,Arrow);
pair pcos1=(Cos(angle1),0), psin2=(0,Sin(angle2));
draw(pM--pcos1^^pN--psin2,blue);
perpendicular(pcos1,NE,blue);
perpendicular(psin2,SW,blue);

dot(scale(.7)*"$(\cos\alpha,\sin\alpha)$",pM,dir(O--pM));
dot(scale(.7)*"$(\cos\beta,\sin\beta)$",pN,dir(O--pN));
dot(scale(.7)*"$\cos\alpha$",pcos1,dir(pM--pcos1));
dot(scale(.7)*"$\sin\beta$",psin2,dir(pN--psin2));