# Argument of Complex Number/Examples/-3

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## Example of Argument of Complex Number

$\arg \left({-3}\right) = \pi$

## Proof

We have that:

$-3 = -3 + 0 i$

and so:

 $\displaystyle \left\lvert{-3}\right\rvert$ $=$ $\displaystyle \sqrt{3^2 + 0^2}$ Definition of Complex Modulus $\displaystyle$ $=$ $\displaystyle 3$

Hence:

 $\displaystyle \cos \left({\arg \left({-3}\right)}\right)$ $=$ $\displaystyle \dfrac {-3} 3$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle -1$ $\displaystyle \leadsto \ \$ $\displaystyle \arg \left({-3}\right)$ $=$ $\displaystyle \pi$ Cosine of Multiple of Pi

 $\displaystyle \sin \left({\arg \left({-3}\right)}\right)$ $=$ $\displaystyle \dfrac 0 3$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \arg \left({-3}\right)$ $=$ $\displaystyle 0 \text { or } \pi$ Sine of Multiple of Pi

Hence:

$\arg \left({-3}\right) = \pi$

$\blacksquare$