Division of Complex Numbers in Polar Form/Examples/(2 cis 15)^7 (4 cis 45)^-3

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Example of Use of Division of Complex Numbers in Polar Form

$\dfrac {\paren {2 \cis 15 \degrees}^7} {\paren {4 \cis 45 \degrees}^3} = \sqrt 3 - i$


Proof

\(\ds \dfrac {\paren {2 \cis 15 \degrees}^7} {\paren {4 \cis 45 \degrees}^3}\) \(=\) \(\ds \dfrac {2^7 \paren {\map \cis {7 \times 15 \degrees} } } {4^3 \paren {\map \cis {3 \times 45 \degrees} } }\) De Moivre's Theorem
\(\ds \) \(=\) \(\ds \dfrac {2^7} {2^6} \map \cis {105 \degrees - 135 \degrees}\) Division of Complex Numbers in Polar Form
\(\ds \) \(=\) \(\ds 2 \paren {\map \cos {-30 \degrees} + i \, \map \sin {-30 \degrees} }\)
\(\ds \) \(=\) \(\ds 2 \times \paren {\dfrac {\sqrt 3} 2} + 2 i \paren {-\dfrac 1 2}\) Cosine of $330 \degrees$, Sine of $330 \degrees$
\(\ds \) \(=\) \(\ds \sqrt 3 - i\) simplifying

$\blacksquare$


Sources