Unit Circle

Overview

On the Cartesian coordinate system, the unit circle is the ellipse with center at the origin and radius $1$ .

Suppose closed interval $[a, b]$ is mapped to an arc on the unit circle. Then the point on the unit circle corresponding to $a$ is called the initial point of the arc. The point corresponding to $b$ is called the terminal point of the arc.

Reference Arcs

Let reference arc $\hat{t}$ for an arc $t$ is the smallest non-negative arc between the terminal point of the arc $t$ and the closer of the two $x$ -intercepts of the unit circle.

Certain values are common enough to warrant memorizing:

$\cos (π / 6) = \sqrt{3} / 2$
$\sin (π / 6) = 1 / 2$
$\cos (π / 4) = \sqrt{2} / 2$
$\sin (π / 4) = \sqrt{2} / 2$
$\cos (π / 3) = 1 / 2$
$\sin (π / 3) = \sqrt{3} / 2$

Tangent

The tangent function can also be found on the unit circle. Using the below diagram as a reference,

\tan t = \frac{\sin t}{\cos t} = \frac{y}{x},

the slope of the line through points $⟨ 0, 0 ⟩$ and $⟨ \cos t, \sin t ⟩$ . This line intersects the vertical line through $⟨ 1, 0 ⟩$ at point $⟨ 1, m ⟩$ . Thus the line also has slope $m / 1 = m$ . Hence $\tan t = m$ .