Argument of Complex Number/Examples/1+i

From ProofWiki
Jump to navigation Jump to search

Example of Argument of Complex Number

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


Proof

We have that:

\(\ds \size {1 + i}\) \(=\) \(\ds \sqrt 2\) Examples of Complex Modulus: $1 + i$


Hence:

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


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


Hence:

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

$\blacksquare$


Sources