dzielenie liczb zespolonych

[luqasz]

\(\displaystyle{ \frac{(i+ \sqrt{3})^{26} }{(-1-i\sqrt{3})^{21} }= \frac{2^{26}(cos26 \frac{\pi}{3}+sin26 \frac{\pi}{3}) }{{2^{21}}(cos21 \frac{4\pi}{3} +sin21 \frac{4\pi}{3}) } =2^{5}(cos-\frac{5\pi}{3} +sin-\frac{5\pi}{3}) }}\)
jesli dobrze wszystko policzyłem to jaki bedzie sinus z \(\displaystyle{ -\frac{5\pi}{3}}\) zeby przedstawic ta liczbe w postaci algebraicznej

[natkoza]

\(\displaystyle{ (\sqrt{3}+i)^{26} =2^{26}(cos(26\cdot \frac{\pi}{3})+isin(26\cdot \frac{\pi}{3}))=2^{26}(cos(\frac{26\pi}{3})+isin(\frac{26\pi}{3}))=2^{26}(cos(8\cdot 2\pi +\frac{2\pi}{3})+isin(8 2\pi +\frac{2\pi}{3}))=2^{26}(cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3})=2^{26}(cos(\pi-\frac{\pi}{3})+isin(\pi-\frac{\pi}{3}))=2^{26}(-cos\frac{\pi}{3}+isin\frac{\pi}{3})=2^{26}(-\frac{1}{2}+\frac{\sqrt{3}}{2})\\
(-1-i\sqrt{3})^{21} =2^{21}(cos(\frac{21\cdot 4\pi}{3})+isin(\frac{21\cdot 4\pi}{3})) = 2^{21}(cos(\frac{84\pi}{3})+isin(\frac{84\pi}{3})) = 2^{21}(cos(28\cdot 2\pi)+isin(28\cdot 2\pi))=2^{21}(cos0+isin0)=2^{21}\\
\frac{(\sqrt{3}+i)^{26}}{(-1-i\sqrt{3})^{21}}=\frac{2^{26}(-\frac{1}{2}+\frac{\sqrt{3}}{2})}{2^{21}}=2^{5}(-\frac{1}{2}+\frac{\sqrt{3}}{2})=-2^{4}+2^4\cdot \sqrt{3}=16+16\sqrt{3}}\)
mam nadzieje, że sie nei pomyliłam

[luqasz]

a Arg z \(\displaystyle{ \frac{(i+ \sqrt{3})^{26} }}\) nie jest \(\displaystyle{ \frac{\pi}{6}}\)