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:


3 cis -2 pi 3^-1.png


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

but beware of the mistake in the solution.