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Express the Indicated Roots of Unity in Standard Form a 1+0i 1+0 \mathrm{i}

Question 190

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Express the indicated roots of unity in standard form a + bi.
-Ninth roots of unity


A) 1+0i 1+0 \mathrm{i}
cos2π9+isin2π90.77+0.64icos4π9+isin4π90.17+0.98icos2π3+isin2π3=0.50+i32cos8π9+isin8π90.94+0.34icos10π9+isin10π90.940.34icos4π3+isin4π3=0.50i32cos14π9+isin14π90.170.98i1i\begin{array}{l}\cos \frac{2 \pi}{9}+i \sin \frac{2 \pi}{9} \approx 0.77+0.64 i \\\cos \frac{4 \pi}{9}+i \sin \frac{4 \pi}{9} \approx 0.17+0.98 i \\\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}=-0.50+i \frac{\sqrt{3}}{2} \\\cos \frac{8 \pi}{9}+i \sin \frac{8 \pi}{9} \approx-0.94+0.34 i \\\cos \frac{10 \pi}{9}+i \sin \frac{10 \pi}{9} \approx-0.94-0.34 i \\\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}=-0.50-i \frac{\sqrt{3}}{2} \\\cos \frac{14 \pi}{9}+i \sin \frac{14 \pi}{9} \approx 0.17-0.98 \mathrm{i} \\1-i\end{array}
B) 1+0i 1+0 \mathrm{i}
cosπ9+isinπ90.94+0.34icos2π9+isin2π90.77+0.64icosπ3+isinπ3=0.50+i32cos4π9+isin4π90.17+0.98icos5π9+isin5π90.17+0.98icos2π3+isin2π3=0.50+i32cos7π9+isin7π90.77+0.64icos8π9+isin8π90.94+0.34i\begin{array}{l}\cos \frac{\pi}{9}+i \sin \frac{\pi}{9} \approx 0.94+0.34 i \\\cos \frac{2 \pi}{9}+i \sin \frac{2 \pi}{9} \approx 0.77+0.64 i \\\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}=0.50+i \frac{\sqrt{3}}{2} \\\cos \frac{4 \pi}{9}+i \sin \frac{4 \pi}{9} \approx 0.17+0.98 i \\\cos \frac{5 \pi}{9}+i \sin \frac{5 \pi}{9} \approx-0.17+0.98 i \\\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}=-0.50+i \frac{\sqrt{3}}{2} \\\cos \frac{7 \pi}{9}+i \sin \frac{7 \pi}{9} \approx-0.77+0.64 i \\\cos \frac{8 \pi}{9}+i \sin \frac{8 \pi}{9} \approx-0.94+0.34 i\end{array}
C) 1+0i 1+0 i
cos2π9+isin2π90.77+0.64i \cos \frac{2 \pi}{9}+i \sin \frac{2 \pi}{9} \approx 0.77+0.64 i
cos4π9+isin4π90.17+0.98i \cos \frac{4 \pi}{9}+i \sin \frac{4 \pi}{9} \approx 0.17+0.98 i
cos2π3+isin2π3=0.50+i32 \cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}=-0.50+i \frac{\sqrt{3}}{2}
cos8π9+isin8π90.94+0.34i \cos \frac{8 \pi}{9}+i \sin \frac{8 \pi}{9} \approx-0.94+0.34 i
cos10π9+isin10π90.940.34i \cos \frac{10 \pi}{9}+i \sin \frac{10 \pi}{9} \approx-0.94-0.34 i
cos4π3+isin4π3=0.50i32 \cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}=-0.50-i \frac{\sqrt{3}}{2}
cos14π9+isin14π90.170.98i \cos \frac{14 \pi}{9}+i \sin \frac{14 \pi}{9} \approx 0.17-0.98 i
cos16π9+isin16π90.770.64i \cos \frac{16 \pi}{9}+i \sin \frac{16 \pi}{9} \approx 0.77-0.64 i

D) 1+0i 1+0 i
cos2π9+isin2π90.77+0.64i \cos \frac{2 \pi}{9}+i \sin \frac{2 \pi}{9} \approx 0.77+0.64 i
cos4π9+isin4π90.17+0.98i \cos \frac{4 \pi}{9}+i \sin \frac{4 \pi}{9} \approx 0.17+0.98 i
cos2π3+isin2π3=0.50+i32 \cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}=-0.50+i \frac{\sqrt{3}}{2}
1+0i -1+0 i
cos10π9+isin10π90.940.34i \cos \frac{10 \pi}{9}+i \sin \frac{10 \pi}{9} \approx-0.94-0.34 i
cos4π3+isin4π3=0.50i32 \cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}=-0.50-i \frac{\sqrt{3}}{2}
cos14π9+isin14π90.170.98i \cos \frac{14 \pi}{9}+i \sin \frac{14 \pi}{9} \approx 0.17-0.98 i
cos16π9+isin16π90.770.64i \cos \frac{16 \pi}{9}+i \sin \frac{16 \pi}{9} \approx 0.77-0.64 i

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