Polar Form of Complex Number/Examples/root 3 over 2 - 3 i over 2

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Example of Polar Form of Complex Number

The complex number $\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2$ can be expressed as a complex number in polar form as $\polar {\sqrt 3, \dfrac {5 \pi} 3}$.


Proof

\(\ds \cmod {\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2}\) \(=\) \(\ds \sqrt {\paren {\dfrac {\sqrt 3} 2}^2 + \paren {\dfrac 3 2}^2}\) Definition of Complex Modulus
\(\ds \) \(=\) \(\ds \sqrt {\dfrac 3 4 + \dfrac 9 4}\)
\(\ds \) \(=\) \(\ds \sqrt {\dfrac {12} 4}\)
\(\ds \) \(=\) \(\ds \sqrt 3\)


Then:

\(\ds \map \cos {\map \arg {\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2} }\) \(=\) \(\ds \dfrac {\sqrt 3 / 2} {\sqrt 3}\) Definition of Argument of Complex Number
\(\ds \) \(=\) \(\ds 1 2\)
\(\ds \leadsto \ \ \) \(\ds \map \arg {\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2}\) \(=\) \(\ds \dfrac \pi 3 \text { or } \dfrac {5 \pi} 3\) Cosine of $60 \degrees$, Cosine of $300 \degrees$


\(\ds \map \sin {\map \arg {\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2} }\) \(=\) \(\ds \dfrac {-3 / 2} {\sqrt 3}\) Definition of Argument of Complex Number
\(\ds \) \(=\) \(\ds -\dfrac {\sqrt 3} 2\)
\(\ds \leadsto \ \ \) \(\ds \map \arg {\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2}\) \(=\) \(\ds \dfrac {4 \pi} 3 \text { or } \dfrac {5 \pi} 3\) Sine of $240 \degrees$, Sine of $300 \degrees$


Hence:

$\map \arg {\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2} = \dfrac {5 \pi} 3$

and hence the result.

$\blacksquare$


Sources

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