# Argument of Complex Number/Examples/2i

## Example of Argument of Complex Number

$\arg \left({2 i}\right) = \dfrac \pi 2$

## Proof

We have that:

 $\displaystyle \left\lvert{2 i}\right\rvert$ $=$ $\displaystyle \sqrt{0^2 + 2^2}$ Definition of Complex Modulus $\displaystyle$ $=$ $\displaystyle 2$ simplifying

Hence:

 $\displaystyle \cos \left({\arg \left({2 i}\right)}\right)$ $=$ $\displaystyle \dfrac 0 2$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \arg \left({2 i}\right)$ $=$ $\displaystyle \pm \dfrac \pi 2$ Cosine of Half-Integer Multiple of Pi

 $\displaystyle \sin \left({\arg \left({2 i}\right)}\right)$ $=$ $\displaystyle \dfrac 2 2$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle \arg \left({2 i}\right)$ $=$ $\displaystyle \dfrac \pi 2$ Sine of Half-Integer Multiple of Pi

Hence:

$\arg \left({2 i}\right) = \dfrac \pi 2$

$\blacksquare$