How to solve this 6 equations (f1,f2,f3,f4,f5,f6) to find out the unknown (A,B,C,D,E,F) given that f1=0,f2=0,​f3=0,f4=0,​f5=0,f6=0.

f1(K)=1-2*cos(A(K)*pi/180)+2*cos(B(K)*pi/180)-2*cos(C(K)*pi/180)+2*cos(D(K)*pi/180)-2*cos(E(K)*pi/180)+2*cos(F(K)*pi/180)-(pi/4)*(.1);
f2(K)=1-2*cos(5*A(K)*pi/180)+2*cos(5*B(K)*pi/180)-2*cos(5*C(K)*pi/180)+2*cos(5*D(K)*pi/180)-2*cos(5*E(K)*pi/180)+2*cos(5*F(K)*pi/180);
f3(K)=1-2*cos(7*A(K)*pi/180)+2*cos(7*B(K)*pi/180)-2*cos(7*C(K)*pi/180)+2*cos(7*D(K)*pi/180)-2*cos(7*E(K)*pi/180)+2*cos(7*F(K)*pi/180);
f4(K)=1-2*cos(11*A(K)*pi/180)+2*cos(11*B(K)*pi/180)-2*cos(11*C(K)*pi/180)+2*cos(11*D(K)*pi/180)-2*cos(11*E(K)*pi/180)+2*cos(11*F(K)*pi/180);
f5(K)=1-2*cos(13*A(K)*pi/180)+2*cos(13*B(K)*pi/180)-2*cos(13*C(K)*pi/180)+2*cos(13*D(K)*pi/180)-2*cos(13*E(K)*pi/180)+2*cos(13*F(K)*pi/180);
f6(K)=1-2*cos(17*A(K)*pi/180)+2*cos(17*B(K)*pi/180)-2*cos(17*C(K)*pi/180)+2*cos(17*D(K)*pi/180)-2*cos(17*E(K)*pi/180)+2*cos(17*F(K)*pi/180);

[Walter Roberson]

There are probably an infinite number of solutions.

Are you still trying to find a solution under the restriction that all of the values are in the range 0 to 90?

[Walter Roberson]

Two of the solutions are:

[A = 30.24118792921818, B = 74.74637378922529, 
 C = 15.63858441243751, D = 43.98352669758213, 
 E = 60.66990948492604, F = 29.14702869113963]

and

[A = 15.64641697375307, B = 89.44314730541587,
 C = 45.55860174021200, D = 14.73701050176809,
 E = 60.65381678359069, F = 44.30297095045943]

Reducing by two variables is not so bad, but the rest are a mess.