Home / The imaginary part of $ e^{i\theta} $ is $ \sin \theta $. We compute $ \sin \theta $ for each angle:

The imaginary part of $ e^{i\theta} $ is $ \sin \theta $. We compute $ \sin \theta $ for each angle:

February 22, 2026

$The imaginary part of $ e^{i\theta} $ is $ \sin \theta $. We compute $ \sin \theta $ for each angle:$

z = e^{-2\pi i / 9},\ e^{-8\pi i / 9},\ e^{-14\pi i / 9} = e^{4\pi i / 9} \quad \text{(since } -14\pi/9 \equiv 4\pi/9 \mod 2\pi).
So the six roots are:
e^{2\pi i / 9},\ e^{8\pi i / 9},\ e^{14\pi i / 9},\ e^{-2\pi i / 9},\ e^{-8\pi i / 9},\ e^{4\pi i / 9}.
$ \sin(2\pi/9) = \sin(40^\circ) \approx 0.6428 $
$ \sin(8\pi/9) = \sin(160^\circ) = \sin(20^\circ) \approx 0.3420 $
$ \sin(14\pi/9) = \sin(280^\circ) = \sin(-80^\circ) = -\sin(80^\circ) $ (negative)