We now revisit the topic of parametric equations. Consider the following parametric equations of a curve \(C\):
\[x = \cos \theta, y = \sin \theta,\]
where&nbsp;\(\theta \in \mathbb{R}\).
As&nbsp;\(\theta\) ranges over the real numbers, we realize that
\[x^2 + y^2 = \cos^2 \theta + \sin^2 \theta = 1.\]
We recognize that the equation \(x^2 + y^2 = 1\) is that of a circle centred at \(O = (0, 0)\) and of radius \(1\); we call this circle the unit circle. As&nbsp;\(\theta\) increases, say from \(\theta = 0\) to \(\theta = 2\pi\), the point \(Z(\cos \theta, \sin \theta)\) that represents the complex number \(e^{i\theta} := \cos \theta + i \sin \theta\) moves along the unit circle \(C\) in the counter-clockwise direction – performing a rotation (see Figure \(1\)).
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1775874134449-xgbhnbb16k.png" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Figure 1. Unit complex number moves along unit circle." class="aligncenter custom-width" /></figure></div>

We now revisit the topic of parametric equations. Consider the following parametric equations of a curve \(C\):

As \(\theta\) ranges over the real numbers, we realize that

\[x^2 + y^2 = \cos^2 \theta + \sin^2 \theta = 1.\]

We recognize that the equation \(x^2 + y^2 = 1\) is that of a circle centred at \(O = (0, 0)\) and of radius \(1\); we call this circle the **unit circle**. As \(\theta\) increases, say from \(\theta = 0\) to \(\theta = 2\pi\), the point \(Z(\cos \theta, \sin \theta)\) that represents the complex number \(e^{i\theta} := \cos \theta + i \sin \theta\) moves along the unit circle \(C\) in the counter-clockwise direction – performing a rotation (see Figure \(1\)).

The unit complex number and the unit circle

The unit complex number and the unit circle