Polar Form of Complex Number/Examples/3 cis -2 pi 3^-1
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Example of Polar Form of Complex Number
The complex number $\polar {3, -\dfrac {2 \pi} 3}$ can be expressed in Cartesian form as:
- $3 \, \map \cis {-\dfrac {2 \pi} 3} = -\dfrac 3 2 - \dfrac {3 \sqrt 3} 2 i$
and depicted in the complex plane as:
Proof
\(\ds 3 \, \map \cis {-\dfrac {2 \pi} 3}\) | \(=\) | \(\ds 3 \paren {\map \cos {\dfrac {4 \pi} 3} + i \, \map \sin {\dfrac {4 \pi} 3} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times \paren {-\dfrac 1 2 + \dfrac {\sqrt 3} 2 i}\) | Cosine of $240 \degrees$ and Sine of $240 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 3 2 - \dfrac {3 \sqrt 3} 2 i\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $84 \ \text {(f)}$
- but beware of the mistake in the solution.