Category:Examples of Polar Form of Complex Number

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This category contains examples of Polar Form of Complex Number.

For any complex number $z = x + i y \ne 0$, let:

\(\displaystyle r\) \(=\) \(\displaystyle \cmod z = \sqrt {x^2 + y^2}\) the modulus of $z$, and
\(\displaystyle \theta\) \(=\) \(\displaystyle \arg z\) the argument of $z$ (the angle which $z$ yields with the real line)

where $x, y \in \R$.

From the definition of $\arg z$:

$(1): \quad \dfrac x r = \cos \theta$
$(2): \quad \dfrac y r = \sin \theta$

which implies that:

$x = r \cos \theta$
$y = r \sin \theta$

which in turn means that any number $z = x + i y \ne 0$ can be written as:

$z = x + i y = r \paren {\cos \theta + i \sin \theta}$

The pair $\polar {r, \theta}$ is called the polar form of the complex number $z \ne 0$.

The number $z = 0 + 0 i$ is defined as $\polar {0, 0}$.