# Polar Form of Complex Number/Examples/-5 + 5i

## Example of Polar Form of Complex Number

The complex number $-5 + 5 i$ can be expressed as a complex number in polar form as $\polar {5 \sqrt 2, \dfrac {3 \pi} 4}$.

## Proof

 $\displaystyle \cmod {-5 + 5 i}$ $=$ $\displaystyle \sqrt {\paren {-5}^2 + 5^2}$ Definition of Complex Modulus $\displaystyle$ $=$ $\displaystyle \sqrt {2 \times 25}$ $\displaystyle$ $=$ $\displaystyle 5 \sqrt 2$

Then:

 $\displaystyle \map \cos {\map \arg {-5 + 5 i} }$ $=$ $\displaystyle \dfrac {-5} {5 \sqrt 2}$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle -\frac {\sqrt 2} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \map \arg {-5 + 5 i}$ $=$ $\displaystyle \pm \dfrac {3 \pi} 4$ Cosine of $135 \degrees$, Cosine of $225 \degrees$

 $\displaystyle \map \sin {\map \arg {-5 + 5 i} }$ $=$ $\displaystyle \dfrac 5 {5 \sqrt 2}$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle \frac {\sqrt 2} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \map \arg {-5 + 5 i}$ $=$ $\displaystyle \dfrac \pi 4 \text { or } \dfrac {3 \pi} 4$ Sine of $45 \degrees$, Sine of $135 \degrees$

Hence:

$\map \arg {-5 + 5 i} = \dfrac {3 \pi} 4$

and hence the result.

$\blacksquare$