Argument of Complex Number/Examples/-i

Example of Argument of Complex Number

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

Proof

We have that:

 $\displaystyle \cmod {-i} = 1$ $=$ $\displaystyle$ Example of Complex Modulus: $-i$

Hence:

 $\displaystyle \cos \left({\arg \left({-i}\right)}\right)$ $=$ $\displaystyle \dfrac 0 1$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \arg \left({-i}\right)$ $=$ $\displaystyle \pm \dfrac \pi 2$ Cosine of Half-Integer Multiple of Pi

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

Hence:

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

$\blacksquare$