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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $26 \ \text {(b)}$