Old post, but since I had the same question, here is a way, with Maple 2021.
First, define a few variables.
They are designed such that a1=b1+b2, a2=b3+b3 and b3=c1+c2. If we can compute the pairwise products as well as a1+a2, we are done, as we can solve easily the equation x+y=S, x*y=P: x and y are the roots of t^2-S*t+P.
First check the signs, they will be useful to identify x and y in the above trinomial roots.
Therefore, the variables come in positive/negative pairs.
Now, we need a few values. Note that in this case it's easier for Maple to simplify the exponential form.
Now define a function to solve the equations x+y=S, x*y=P. You may have to check that for S>0, P<0, the negative solution is the second one.
We are almost done.
Numerical check, the values should be close to zero:
And finally the value of cos(pi/17):