Help using Demoivre's Theorem
Use Demoivre's Theorem to find (sqrt(5)-4i)^3. Write the result in Standard form.
r=sqrt(sqrt(5)^2+(-4)^2) = sqrt(5+16) = sqrt(21)
I am stuck after this step. I don't know what to do next or how I find theta.
Use Demoivre's Theorem to find (sqrt(5)-4i)^3. Write the result in Standard form.
r=sqrt(sqrt(5)^2+(-4)^2) = sqrt(5+16) = sqrt(21)
I am stuck after this step. I don't know what to do next or how I find theta.
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(√5 - 4i)=√21(cos -1.06105664796 + isin -1.06105664796)
(√5 - 4i)³=(√21)³(cos -1.06105664796 + isin -1.06105664796)³
(√5 - 4i)³=(√21)³(cos -3.18316994389 + isin -3.18316994389)
(√5 - 4i)³=(√21)³(cos -3.18316994389 + isin -3.18316994389)
(√5 - 4i)³=(√21)³(cos 3.10001536329 + isin 3.10001536329)
(√5 - 4i)³=96.2340895941(-0.999135788971+ i(0.041565312426)
(√5 - 4i)³=-96.1509230325+4i
(√5 - 4i)³=(√21)³(cos -1.06105664796 + isin -1.06105664796)³
(√5 - 4i)³=(√21)³(cos -3.18316994389 + isin -3.18316994389)
(√5 - 4i)³=(√21)³(cos -3.18316994389 + isin -3.18316994389)
(√5 - 4i)³=(√21)³(cos 3.10001536329 + isin 3.10001536329)
(√5 - 4i)³=96.2340895941(-0.999135788971+ i(0.041565312426)
(√5 - 4i)³=-96.1509230325+4i
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θ = tan^-1 (-4/√5) = -1.061 or -60.8°, approximately.
√21 (cos 1.061 - i sin 1.061) or √21 (cos 60.8° - i sin 60.8°)
√21 (cos 1.061 - i sin 1.061) or √21 (cos 60.8° - i sin 60.8°)