By "nontrivial way", do you mean the cosine or the square root (or both)?
And yes, I could eliminate b and then solve for a, but that's a manual step
I'd rather avoid.
Ok, so I simplified the expressions using Sage (I'm quite surprised to see
that simplify_full() produces better results than Matlab's simplify()
function). Then, replacing the cos(2pi/n) by a new variable x results in a
much clearer system of equations. The complete code to solve for a and b is
included below.
a,b,x = var('a,b,x')
L = 2*b^2*x^2 + sqrt(4*b^2*x^4 - 4*(a + 2*b - 1)*x^2)*b - a - 2*b + 1
M = 8*b^2*x^4 - 8*b^2*x^2 + 2*b^2 + sqrt(64*b^2*x^8 - 128*b^2*x^6 +
16*(6*b^2 - a - 2*b + 1)*x^4 - 16*(2*b^2 - a - 2*b + 1)*x^2 + 4*b^2 - 4*a -
8*b + 4)*b - a - 2*b + 1
Eq1 = L == 1/2
Eq2 = M == 1/4
Sol = solve([Eq1, Eq2], a, b)
Sage starts/tries to solve it, but after half an hour it was still running
so I terminated it. Are there any parameters I can tune for the solve()
function for it to work properly, or can Sage just not solve these types of
equations (yet)? Thanks.
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