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

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.

$\blacksquare$