Unit Circle

Definition and equation of the unit circle

A circle of radius $1$ centered at the origin $(0,0)$ given by the equation $x^2 + y^2 = 1$.

Representation of coordinates $(x, y)$ on the unit circle in terms of angle $\theta$

$x = \cos(\theta)$ and $y = \sin(\theta)$

Definition of $\tan(\theta)$ using unit circle coordinates $(x, y)$

$\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}$ (where $x \neq 0$)

Radian measure and $(x, y)$ coordinates for $30^\circ$

$\frac{\pi}{6}$ radians with coordinates $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

Radian measure and $(x, y)$ coordinates for $45^\circ$

$\frac{\pi}{4}$ radians with coordinates $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

Radian measure and $(x, y)$ coordinates for $60^\circ$

$\frac{\pi}{3}$ radians with coordinates $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

Coordinates $(x, y)$ for the quadrantal angles $0$, $\frac{\pi}{2}$, $\pi$, and $\frac{3\pi}{2}$

$0$: $(1, 0)$, $\frac{\pi}{2}$: $(0, 1)$, $\pi$: $(-1, 0)$, $\frac{3\pi}{2}$: $(0, -1)$

Rule for determining which trigonometric functions are positive in Quadrants I, II, III, and IV

ASTC ("All Students Take Calculus"): Quadrant I (All positive), Quadrant II ($\sin(\theta)$ positive), Quadrant III ($\tan(\theta)$ positive), Quadrant IV ($\cos(\theta)$ positive).