Question #117766
Use De Moivre’s theorem to simplify the following (a) (cosπ/5+isinπ/5)10, (b) (cosπ/9+isinπ/9)−3, (c) {cos(−π/6) + isin(−π/6)}−4
1
Expert's answer
2020-05-25T17:42:49-0400

De Moivre's formula is (cos(x)+isin(x))n=(cos(nx)+isin(nx)),nZ(cos(x)+i\,sin(x))^n=(cos(nx)+i\,sin(nx)),\quad n\in{\mathbb{Z}}

By applying it, we obtain:

(a) (cos(π5)+isin(π5))10=cos(2π)+isin(2π)=1(cos(\frac{\pi}{5})+i\,sin(\frac{\pi}{5}))^{10}=cos(2\pi)+i\,sin(2\pi)=1

(b) (cos(π9)+isin(π9))3=cos(π3)+isin(π3)=12i32(cos(\frac{\pi}{9})+i\,sin(\frac{\pi}{9}))^{-3}=cos(-\frac{\pi}{3})+i\,sin(-\frac{\pi}{3})=\frac{1}{2}-i\,\frac{\sqrt{3}}{2}

(c) (cos(π6)+isin(π6))4=cos(2π3)+isin(2π3)=12+i32(cos(-\frac{\pi}{6})+i\,sin(-\frac{\pi}{6}))^{-4}=cos(\frac{2\pi}{3})+i\,sin(\frac{2\pi}{3})=-\frac{1}{2}+i\,\frac{\sqrt{3}}{2}


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