Complex Power/Examples/(2 cis 50)^6
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Example of Complex Power
- $\paren {2 \cis 50 \degrees}^6 = 32 - 32 \sqrt 3 i$
Proof
| \(\ds \paren {2 \cis 50 \degrees}^6\) | \(=\) | \(\ds 2^6 \map \cis {6 \times 50 \degrees}\) | De Moivre's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds 64 \cis 300 \degrees\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 64 \paren {\cos 300 \degrees + i \sin 300 \degrees}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 64 \times \paren {\dfrac 1 2} + 64 i \paren {-\dfrac {\sqrt 3} 2}\) | Cosine of $300 \degrees$, Sine of $300 \degrees$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 32 - 32 \sqrt 3 i\) | 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: $89 \ \text{(b)}$