Argument of Complex Number/Examples/-i

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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:

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

$\blacksquare$


Sources