A strange equation II

Algebra Level 5

The three real roots of the polynomial \(P(x)=x^{3}-3x^{2}+1\) can be expressed as:

\[ x_{1}=-b\cos(e°)+a , x_{2}=-c\cos(f°)+a, x_{3}=d\cos(g°)+a,\]

where \(x_{1}<x_{2}<x_{3}\), \(0°<e<90°\), \(0°<f<90°\), \(0°<g<90°\), such that \(a\), \(b\), \(c\), \(d\), \(e\), \(f\) and \(g\) are positive integers; and \(e\), \(f\) and \(g\) are integer degrees. Find \(a+b+c+d+e+f+g\).

You may also try Part III.

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