sin18° is a root of which of these quadratic equations?

Let θ = 18°, so 2θ = 90° − 3θ and therefore sin2θ = cos3θ. That is 2sinθ·cosθ = 4co$s^3$θ − 3cosθ = cosθ(1 − 4si$n^2$θ). Cancelling cosθ (which is not zero) gives 2sinθ = 1 − 4si$n^2$θ, i.e. $4x^2+2x-1=0$ with $x$ = sinθ. Its positive root is (√5 − 1)/4 ≈ 0.309 = sin18°. The equation $4x^2-2x-1=0$ has root 0.809 = cos36° instead, and $4x^2+2x+1=0$ has no real root at all.