Argument of Complex Number/Examples/2i

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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:

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

$\blacksquare$


Sources