# Argument of Complex Number/Examples/-1-i

## Example of Argument of Complex Number

$\map \arg {-1 - i} = -\dfrac {3 \pi} 4$

## Proof

We have that:

 $\displaystyle \cmod {-1 - i}$ $=$ $\displaystyle \sqrt {\paren {-1}^2 + \paren {-1}^2}$ Definition of Complex Modulus $\displaystyle$ $=$ $\displaystyle \sqrt 2$ simplifying

Hence:

 $\displaystyle \map \cos {\map \arg {-1 - i} }$ $=$ $\displaystyle \dfrac {-1} {\sqrt 2}$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle -\dfrac {\sqrt 2} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \map \arg {-1 - i}$ $=$ $\displaystyle \pm \dfrac {3 \pi} 4$

 $\displaystyle \map \sin {\map \arg {-1 - i} }$ $=$ $\displaystyle \dfrac {-1} {\sqrt 2}$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle \dfrac {\sqrt 2} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \map \arg {-1 - i}$ $=$ $\displaystyle -\dfrac {3 \pi} 4 \text { or } -\dfrac \pi 4$

Hence:

$\map \arg {-1 - i} = -\dfrac {3 \pi} 4$

$\blacksquare$