The value of 1 + sin 2 π 9 + i cos 2 π 9 1 + sin 2 π 9 − i cos 2 π 9 3 is

${\left(\frac{1+\sin \frac{2\pi }{9}+i\cos \frac{2\pi }{9}}{1+\sin \frac{2\pi }{9}-i\cos \frac{2\pi }{9}}\right)}^3$

$={\left(\frac{1+\cos \frac{5\pi }{18}+i\sin \frac{5\pi }{18}}{1+\cos \frac{5\pi }{18}-i\sin \frac{5\pi }{18}}\right)}^3={\left(\frac{2{\cos }^2\frac{5\pi }{36}+i2\sin \frac{5\pi }{36}·\cos \frac{5\pi }{36}}{2{\cos }^2\frac{5\pi }{36}-i2\sin \frac{5\pi }{36}·\cos \frac{5\pi }{36}}\right)}^3$

$={\left(\frac{\cos \frac{5\pi }{36}+i\sin \frac{5\pi }{36}}{\cos \frac{5\pi }{36}-i\sin \frac{5\pi }{36}}\right)}^3={\left(\cos \frac{5\pi }{36}+i\sin \frac{5\pi }{36}\right)}^6$

$=\cos \left(6\times \frac{5\pi }{36}\right)+i\sin \left(6\times \frac{5\pi }{36}\right)=\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}$

$=-\frac{\sqrt{3}}{2}+i\frac{1}{2}$