<s>[INST] Prove the inequality regarding complex numbers If $\theta_i\in [0,\pi/6],i=1,2,3,4,5$.And $$\sin \theta_1\ z^4 + \sin\theta_2 \ z^3 + \sin\theta_3 \ z^2 + \sin\theta_4 \ z + \sin\theta_5=2$$ Prove that  $|z|\gt \frac{3}{4}$.
 [/INST] $\theta_i\in [0,\pi/6]$ implies that $0 \le \sin \theta_i \le \frac 12$.
So it follows from your equation that 
$$
 2 \le \frac 12 \left( 1 + |z| + |z|^2 + |z|^3 + |z|^4 \right)
$$
or, with $r = |z|$,
$$
f(r) = 1+ r+r^2 +r^3 +r^4 - 4 \ge 0
$$
$f(r)$ is monotonically increasing in $r$ and $f(3/4) < 0$,
therefore a solution must satisfy $r > 3/4$. 
A numerical computation with PARI/GP 
? solve(r=0, 1, 1+r+r^2+r^3+r^4-4)
%1 = 0.88817966758531001825082741056489343552

shows that $r > 0.888$.
</s>