Let x=cosθ, then:
2x2+2x-1=0,
x=(-2±√(4+8))/4=(-1±√3)/2.
cosθ=(-1±√3)/2, θ=cos-1((-1±√3)/2). We can eliminate (-1-√3)/2, because this value is less than -1 and cosine is always between -1 and 1. θ=cos-1((√3-1)/2). 2π-cos-1((√3-1)/2) is also a solution.
(√3-1)/2=cos(π/6)-sin(π/6).
cos(a+b)=cos(a)cos(b)-sin(a)sin(b).
If a=π/4, and b=π/6, cos(π/4+π/6)=cos(5π/12)=
cos(π/4)cos(π/6)-sin(π/4)sin(π/6)=(1/√2)(½√3-½)=cosθ/√2.
Hence cosθ=cos(5π/12)√2, θ=cos-1(cos(5π/12)√2) or 2π-cos-1(cos(5π/12)√2) (the exact answer).
An approximation to this solution is θ=1.1961 or 5.0871 radians (68.53° or 291.47°).