Category:Examples of Arguments of Complex Numbers

From ProofWiki
Jump to navigation Jump to search

This category contains examples of Argument of Complex Number.

Let $z = x + i y$ be a complex number.

An argument of $z$, or $\arg z$, is formally defined as a solution to the pair of equations:

$(1): \quad \dfrac x {\cmod z} = \map \cos {\arg z}$
$(2): \quad \dfrac y {\cmod z} = \map \sin {\arg z}$

where $\cmod z$ is the modulus of $z$.

From Sine and Cosine are Periodic on Reals, it follows that if $\theta$ is an argument of $z$, then so is $\theta + 2 k \pi$ where $k \in \Z$ is any integer.