Division of Complex Numbers in Polar Form/Examples/(8 cis 40)^3 (2 cis 60)^-4

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Example of Use of Division of Complex Numbers in Polar Form

$\dfrac {\paren {8 \cis 40 \degrees}^3} {\paren {2 \cis 60 \degrees}^4} = -16 - 16 \sqrt 3 i$


Proof

\(\ds \dfrac {\paren {8 \cis 40 \degrees}^3} {\paren {2 \cis 60 \degrees}^4}\) \(=\) \(\ds \dfrac {8^3 \paren {\map \cis {3 \times 40 \degrees} } } {2^4 \paren {\map \cis {4 \times 60 \degrees} } }\) De Moivre's Theorem
\(\ds \) \(=\) \(\ds \dfrac {2^9} {2^4} \map \cis {120 \degrees - 240 \degrees}\) Division of Complex Numbers in Polar Form
\(\ds \) \(=\) \(\ds 2^5 \paren {\map \cos {-120 \degrees} + i \, \map \sin {-120 \degrees} }\)
\(\ds \) \(=\) \(\ds 32 \times \paren {-\dfrac 1 2} + 32 i \paren {-\dfrac {\sqrt 3} 2}\) Cosine of $240 \degrees$, Sine of $240 \degrees$
\(\ds \) \(=\) \(\ds -16 - 16 \sqrt 3 i\) simplifying

$\blacksquare$


Sources