# Inspired by Daniel Liu

Geometry Level 5

$\Large f\left( x \right) ={ 8x }^{ 3 }-{ 12x }^{ 2 }+bx+d$

For variables $$b$$ and $$d$$ independent of $$x$$, we are given the function $$f(x)$$ as described above to have roots $$-1\le\alpha ,\beta ,\gamma \le 1$$ and that

$\alpha +\beta +\gamma =1+4\sin { \frac { \cos ^{ -1 }{ (\alpha ) } }{ 2 } } \times \sin { \frac { \cos ^{ -1 }{ (\beta) } }{ 2 } } \times \sin { \frac { \cos ^{ -1 }{ (\gamma ) } }{ 2 } }$

Evaluate $$b+d$$.

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