I want to find the smallest positive root of equation: $$x^3-0.75x+b=0$$ when $$b = \frac{1}{64}\left[(\sqrt{5}-1)(\sqrt{6}+\sqrt{2})-(\sqrt{6}-\sqrt{2})\sqrt{10+2\sqrt{5}}\right]$$ I know value of this root is almost $0.01745240643$ but can any one find exact value by roots of integers? not approximate value in the form I wrote? This equation has two other roots that I am not interested.
2026-03-25 21:48:00.1774475280
finding roots of an equation
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The hint: $$\frac{\sqrt5-1}{4}\cdot\frac{\sqrt6+\sqrt2}{4}-\frac{\sqrt{10+2\sqrt5}}{4}\cdot\frac{\sqrt6-\sqrt2}{4}=$$ $$=\sin18^{\circ}\sin75^{\circ}-\cos18^{\circ}\cos75^{\circ}=-\cos93^{\circ}$$ and use $4\cos^3\alpha-3\cos\alpha=\cos3\alpha$.
I got that the smallest positive root it's $\sin1^{\circ}.$