A system of equations involving sine and cosines can be converted to a polynomial system by:

- use angle addition identitites to replace things like cos(t1+t) with cos(t1)cos(t) - sin(t1)sin(t)

- replace cos(t1) with ct1, sin(t1) with st1, etc.

- for each angle add the equation ct1^2 + st1^2 - 1 = 0

You would have 8 equatins in 8 unknowns. Solve them symboically with Dixon resultant (best) or Groebner bases (often hangs).

It is often better to do one more step.  Use the well-known half angle tangent identity to get rid of all the sines and cosines and have tangents.  In your case, instead of ct1 and st1 you would have just t1 (meaning tan(t1/2)).  Then you would have four messier equations in four unknowns.