Colorful Absolutes

plotted graph in complex plane , $z_{n}$ be a point from $z^{5} - 1 = 0$

which $z_{1} = cis\left ( 0 \right )$ , $z_{2} = cis\left ( \frac{2\pi }{5} \right )$ , $z_{3} = cis\left ( \frac{4\pi }{5} \right )$ , $z_{4} = cis\left ( \frac{6\pi }{5} \right )$ , $z_{5} = cis\left ( \frac{8\pi }{5} \right )$

choose $z_{3} , z_{4}$

the absolute value of the difference between these two complex numbers is $\left |z_{3} - z_{4} \right |$

$\left |z_{3} - z_{4} \right | = \left |\left ( cos\left ( \frac{4\pi }{5} \right ) - cos\left ( \frac{6\pi }{5} \right )\right ) + i \left ( sin\left ( \frac{4\pi }{5} \right ) - sin\left ( \frac{6\pi }{5} \right )\right ) \right |$

$\left |z_{3} - z_{4} \right | = \left |0 + 2i sin\frac{4\pi }{5} \right |$

$\left |z_{3} - z_{4} \right | = 2sin\frac{\pi }{5} = 2sin\left ( 36^{\circ} \right ) = \sqrt{\frac{5-\sqrt{5}}{2}} \approx {\color{#D61F06} 1.175}5705045$