The Beauty Of Complex Inequalities

$sin \theta_1z^4+sin \theta_2z^3+sin \theta_3z^2+sin \theta_4z+sin \theta_5$ =2

or,2<= $|sin \theta_1z^4|+|sin \theta_2z^3|+|sin \theta_3z^2|+|sin \theta_4z|+|sin \theta_5|$

Since the sines lie in the range [0,1/2] the inequality becomes 2<=(1/2)( $|z|^4+|z|^3+|z|^2+|z|+1$ ) hence 3<= $|z|^4+|z|^3+|z|^2+|z|$ or,3< $|z|+|z|^2+|z|^3+|z|^4+|z|^5+...$ or, $3<\frac{|z|}{|1-z|}$ or |z|>3/4.