#sin(a_*x_):= if(a<0,-sin(-a*x) ); sin(-x_):= -sin(x); sin(0.5pi+x_):= cos(x); sin(-0.5*pi+x_):= -cos(x); sin(0.5pi-x_):= cos(x); sin(asin(x_)):=x; sin(2pi+x_):=sin(x); sin(-2pi+x_):=sin(x); sin(pi+x_):= -sin(x); sin(-pi+x_):= -sin(x); sin(pi) := 0; sin(2pi):=0; sin(3pi):=0; sin(0.25*pi):=sqrt(2)/2; sin(0.5*pi):=1; sin(1.5*pi):=-1; sin(-2pi):=0; sin(-3pi):=0; sin(-0.5*pi):=1; sin(-1.5*pi):=-1; sin(1,0):=1; sin(2,0):=0;