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:
\(\ds \cmod {-1 - i}\) | \(=\) | \(\ds \sqrt {\paren {-1}^2 + \paren {-1}^2}\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt 2\) | simplifying |
Hence:
\(\ds \map \cos {\map \arg {-1 - i} }\) | \(=\) | \(\ds \dfrac {-1} {\sqrt 2}\) | Definition of Argument of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac {\sqrt 2} 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg {-1 - i}\) | \(=\) | \(\ds \pm \dfrac {3 \pi} 4\) |
\(\ds \map \sin {\map \arg {-1 - i} }\) | \(=\) | \(\ds \dfrac {-1} {\sqrt 2}\) | Definition of Argument of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\sqrt 2} 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg {-1 - i}\) | \(=\) | \(\ds -\dfrac {3 \pi} 4 \text { or } -\dfrac \pi 4\) |
Hence:
- $\map \arg {-1 - i} = -\dfrac {3 \pi} 4$
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Example $\text{(iv)}$