Argument of Complex Number/Examples/1+i

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Example of Argument of Complex Number

$\arg \left({1 + i}\right) = \dfrac \pi 4$


Proof

We have that:

\(\displaystyle \left\lvert{1 + i}\right\rvert\) \(=\) \(\displaystyle \sqrt 2\) Examples of Complex Modulus: $1 + i$


Hence:

\(\displaystyle \cos \left({\arg \left({1 + i}\right)}\right)\) \(=\) \(\displaystyle \dfrac 1 {\sqrt 2}\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\sqrt 2} 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \arg \left({1 + i}\right)\) \(=\) \(\displaystyle \pm \dfrac \pi 4\)


\(\displaystyle \sin \left({\arg \left({1 + i}\right)}\right)\) \(=\) \(\displaystyle \dfrac 1 {\sqrt 2}\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\sqrt 2} 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \arg \left({1 + i}\right)\) \(=\) \(\displaystyle \dfrac \pi 4 \text { or } \dfrac {3 \pi} 4\)


Hence:

$\arg \left({1 + i}\right) = \dfrac \pi 4$

$\blacksquare$


Sources