Prove that the distance of the roots of the equation `|sintheta_1|z^3+|sintheta_2|z^2+|sintheta_3|z+|sintheta_4|=3fromz=0` is greater than `2//3.`

Explanation

We know that `| sin theta_(k)| lt 1`. Given
`|sin theta_(1)|z^(3) + |sin theta_(2)| z^(2) +|sin theta_(3)| z+ |sin theta_(4)| = 3`
`or |3|=||sin theta_(1)|z^(3)+|sin theta_(2)| z^(2) +|sin theta_(3)| z+|sin theta_(4)| = 3`
`lt 1|z^(3)+z^(2) + z + 1`
`lt|z|^(3) +|z|^(2) +|z| +1`
`lt 1+|z| +|z|^(2)+ |z|^(3) +|z|^(4) +....oo`
`or 3 lt (1)/(1-|z|)`
`or 3-3|z| lt 1`
`or 2lt 3|z|`
`or |z| gt (2)/(3)`