Polar Form of Complex Number/Examples/2 root 2 + 2 root 2 i

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Example of Polar Form of Complex Number

The complex number $2 \sqrt 2 + 2 \sqrt 2 i$ can be expressed as a complex number in polar form as $\polar {4, \dfrac \pi 4}$.


Proof

\(\displaystyle \cmod {2 \sqrt 2 + 2 \sqrt 2 i}\) \(=\) \(\displaystyle \sqrt {\paren {2 \sqrt 2}^2 + \paren { 2 \sqrt 2}^2}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {8 + 8}\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {16}\)
\(\displaystyle \) \(=\) \(\displaystyle 4\)


Then:

\(\displaystyle \map \cos {\map \arg {2 \sqrt 2 + 2 \sqrt 2 i} }\) \(=\) \(\displaystyle \dfrac {2 \sqrt 2} 4\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\sqrt 2} 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \arg {2 \sqrt 2 + 2 \sqrt 2 i}\) \(=\) \(\displaystyle \dfrac \pi 4 \text { or } \dfrac {7 \pi} 4\) Cosine of $45 \degrees$, Cosine of $315 \degrees$


\(\displaystyle \map \sin {\map \arg {2 \sqrt 2 + 2 \sqrt 2 i} }\) \(=\) \(\displaystyle \dfrac {2 \sqrt 2} 4\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\sqrt 2} 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \arg {2 \sqrt 2 + 2 \sqrt 2 i}\) \(=\) \(\displaystyle \dfrac \pi 4 \text { or } \dfrac {3 \pi} 4\) Sine of $45 \degrees$, Sine of $135 \degrees$


Hence:

$\map \arg {2 \sqrt 2 + 2 \sqrt 2 i} = \dfrac \pi 4$

and hence the result.

$\blacksquare$


Sources