>>576
それをプログラム化した。 (言語:PARI/GP)

fun1(x) = if(x*x<=1, 1/2*(x*sqrt(1-x*x) + asin(x)), 0.);
fun2(x) = if(x*x<=1, Pi*( x*acos(x)^2 -2*sqrt(1-x*x)*acos(x)-2*x ), 0.);
wine(t) = {
my(a,m,v1,v2);
a = solve(x=-Pi,t-0.0001, cos(x)-cos(t)+sin(t)*(x-t));
m = (a-t)/(cos(a)-cos(t));
v = intnum(y=cos(t), cos(a),
R=acos(y); q=m*(y-cos(t))+t;
2*R^2*(Pi/4 - fun1(q/R)) );
if(a<0, v+= -2*Pi-fun2(cos(a)));
v };

wine_full = -2*Pi-fun2(-1);
t0 = solve( t=Pi/2+0.001, Pi-0.001, wine(t)- 0.5*wine_full );
atan(sin(t0)) *180/Pi
= 9.5126567320359461794301332824985679944

よって θ≒9.5 [deg] 傾ければ良い。