Division of Complex Numbers in Polar Form/Examples/(3 cis pi by 6) (2 cis -5 pi by 4) (6 cis 5 pi by 3) (4 cis 2 pi by 3)^-2
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Example of Use of Division of Complex Numbers in Polar Form
- $\dfrac {\paren {3 \cis \dfrac \pi 6} \paren {2 \cis \dfrac {-5 \pi} 4} \paren {6 \cis \dfrac {5 \pi} 3} } {\paren {4 \cis \dfrac {2 \pi} 3}^2} = -\dfrac {9 \sqrt 2} 8 - \dfrac {9 \sqrt 2} 8 i$
Proof
| \(\ds \dfrac {\paren {3 \cis \dfrac \pi 6} \paren {2 \cis \dfrac {-5 \pi} 4} \paren {6 \cis \dfrac {5 \pi} 3} } {\paren {4 \cis \dfrac {2 \pi} 3}^2}\) | \(=\) | \(\ds \dfrac {\paren {3 \cis \dfrac \pi 6} \paren {2 \cis \dfrac {-5 \pi} 4} \paren {6 \cis \dfrac {5 \pi} 3} } {16 \cis \dfrac {4 \pi} 3}\) | De Moivre's Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {3 \times 2 \times 6 \, \map \cis {\dfrac \pi 6 - \dfrac {-5 \pi} 4 + \dfrac {5 \pi} 3} } {16 \cis \dfrac {4 \pi} 3}\) | Product of Complex Numbers in Polar Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \times 2 \times 6 / 16} \, \map \cis {\dfrac \pi 6 - \dfrac {-5 \pi} 4 + \dfrac {5 \pi} 3 - \dfrac {4 \pi} 3}\) | Division of Complex Numbers in Polar Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 9 4 \, \map \cis {\dfrac {-3 \pi} 4}\) | simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 9 4 \paren {\dfrac {-\sqrt 2} 2} - \dfrac 9 4 \paren {\dfrac {-\sqrt 2} 2} i\) | Cosine of $225 \degrees$, Sine of $225 \degrees$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\dfrac {9 \sqrt 2} 8 - \dfrac {9 \sqrt 2} 8 i\) | simplifying |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $89 \ \text{(d)}$