We denote by
the set of all complex numbers whose absolute value is equal to 1.
The image of
in the Cauchy-Argand plane is the unit circle: if
, with real
if, and only if
exists and is an element of
We leave the proof to the reader; use Corollary 1.18.
, we have
And now we will see how complex numbers are tied with trigonometry.
, we have:
. There exists a real number
are real numbers. Then
if, and only if,
the image of
in the complex plane is a point on the unit-circle. For each point on the unit-circle, there
exist a real number
such that the coordinates of this point are
whence the result.
is called an argument
and is denoted
Note that this argument is defined up to an additional
In Fig 5, a value of the argument of the complex number corresponding to a point is
diplayed in green.
The unit circle in Cauchy-Argand plane.
We can now generalize this to any non zero complex number. First note that for any
, we have
, as the following holds:
The trigonometric form of a non zero complex number.
The following proposition is easy to understand. It explains the specific role of the coordinate axes when
representing complex numbers in the Cauchy-Argand plane.
- Please do it yourself.