Division of Complex Numbers in Polar Form/Examples/(8 cis 40)^3 (2 cis 60)^-4
Jump to navigation
Jump to search
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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $89 \ \text{(c)}$