Argument of Complex Number/Examples/-1-i

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Example of Argument of Complex Number

$\map \arg {-1 - i} = -\dfrac {3 \pi} 4$


Proof

We have that:

\(\displaystyle \cmod {-1 - i}\) \(=\) \(\displaystyle \sqrt {\paren {-1}^2 + \paren {-1}^2}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle \sqrt 2\) simplifying


Hence:

\(\displaystyle \map \cos {\map \arg {-1 - i} }\) \(=\) \(\displaystyle \dfrac {-1} {\sqrt 2}\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle -\dfrac {\sqrt 2} 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \arg {-1 - i}\) \(=\) \(\displaystyle \pm \dfrac {3 \pi} 4\)


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


Hence:

$\map \arg {-1 - i} = -\dfrac {3 \pi} 4$

$\blacksquare$


Sources