# Argument of Complex Number/Examples/1+i

## Example of Argument of Complex Number

$\arg \left({1 + i}\right) = \dfrac \pi 4$

## Proof

We have that:

 $\displaystyle \left\lvert{1 + i}\right\rvert$ $=$ $\displaystyle \sqrt 2$ Examples of Complex Modulus: $1 + i$

Hence:

 $\displaystyle \cos \left({\arg \left({1 + i}\right)}\right)$ $=$ $\displaystyle \dfrac 1 {\sqrt 2}$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle \dfrac {\sqrt 2} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \arg \left({1 + i}\right)$ $=$ $\displaystyle \pm \dfrac \pi 4$

 $\displaystyle \sin \left({\arg \left({1 + i}\right)}\right)$ $=$ $\displaystyle \dfrac 1 {\sqrt 2}$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle \dfrac {\sqrt 2} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \arg \left({1 + i}\right)$ $=$ $\displaystyle \dfrac \pi 4 \text { or } \dfrac {3 \pi} 4$

Hence:

$\arg \left({1 + i}\right) = \dfrac \pi 4$

$\blacksquare$